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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this .
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can a exercise where the answer is rounded to one or two decimals be created ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle .
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why does the triangle have to be a right triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 .
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how do you simplify sqrt 36 sqrt 3 into sqrt 6 sqrt 3 ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b .
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why is c the hypotenuse ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here .
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how do you find c if it asks you ti give the radical sign ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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can you give me a few examples of real world scenarios proving the use of the pythagorean theorem ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here .
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is hypotenuse the opposite side of the right angle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side .
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or the longest side of the triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side .
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why ca n't i use the trigonometry rule to figure out the hypotenuse of the triangle called ( soh cah toa ) is n't that easier ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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can you use the pythagorean theorem on triangles other than right ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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why does pythagorean theorem 1 follow theorem 2 ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it .
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why is the longest side called the hypotenuse ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical .
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how would you find the square root of those ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 .
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with 4^2+3^2=c^2 why could we not simple square root both sides and then simplify from there ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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can you apply the pythagorean theorem to the 3rd dimension ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 .
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like a 3 dimensional pythagorean theorem ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there .
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how would we solve a problem if we were only given one side of the triangle and on of the angles ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology .
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why the theorem always has to be a right triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there .
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how would you find the opposite and the adjacent of a right triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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who made the pythagorean theorem ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 .
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what exactly is a principal root ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle .
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why does the triangle have to be a right triangle in order for the pythagorean theorem to work ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math .
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what is the law of cosines ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 .
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why do you power the 2 numbers you know and the unknown ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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what is the meaning of the term theorem in pythagorean theorem ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared .
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how would we solve the problem if there is already c squared there like nine and we have to find a or b squared ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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how important is the pythagorean theorem ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 .
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why , did sal subtract 6 from 114 ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology .
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does the pythagorean theorem only work if it 's a right triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology .
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what if the right triangle has 2 equal sides and a hypotenuse , how do you find the perfect ratio ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 .
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how can the length of a side of a triangle be the square root of a number ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology .
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what are the other two sides of the triangle called ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle .
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if i am given a triangle with a=3 , b=4 , c=5 , then is it necessarily a right triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit .
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if you square root the whole equation , should n't it be a+b=c ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here .
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should a simpler version be a + b = c ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here .
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does the hypotenuse have to be on the opposite side of the right angle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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does the pythagorean theorm only apply to right triangles ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem .
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and i also do n't understand the logic behind using an equation to find the length of b ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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and is the pythagorean theorem only used to find lengths ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit .
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and why does sal use the principle root of 108 to give to b instead of just giving b 108 ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle .
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when using the pythagorean theorem , must the triangle be a right angle , or can it be just a triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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does the pythagorean theorem also work for rectangles ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that .
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does the pythagorean theorm only work for right triangles ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there .
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can you use the pythagorean theorem on a different type of triangle , for example an isosceles triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 .
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what is a principal root ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here .
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do you always have to use a , b , c ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared .
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so , if a squared + b squared = c squared , does that mean that a squared + c squared = b squared ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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what is the logic behind the pythagorean theorm ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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what is unique about a pythagorean triplet ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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where did the pythagorean theorem come from ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical .
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why do you have to use the square root ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology .
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why do you think the pythagorean theorem only works on a right triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here .
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is there a proper place to put a/b/c on the triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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would the pythagorean theorem work with an equilateral or obtuse triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math .
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how do you rationalize a number ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical .
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what i mean is like figuring out that square root to 50 would be something like 5-2 ( five to the square root of two ) how do you do that ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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does n't a capital letter mean something different in the pythagorean theorem ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle .
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is there a case that a right triangle is n't a `` 3-4-5 '' triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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or does the pythagorean theorem make that false ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here .
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how can you use pythagorean theorem to find isosceles side lengths ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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does the pythagorean theorem only work for right triangles ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there .
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why does the hypotenuse have to be the opposite of the right angle i know how to do the work but ca n't you just tell what it will be ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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in what ways can you apply pythagorean theorem to the world ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle .
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is the pythagorean theorem only used for figuring out the side of an triangle only with a right triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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maybe i 'm missing something here ... for the second practice problem , sal is simplifying the radical ... why does he stick the last little three under a radical sign of its own ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here .
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how come the hypotenuse is on the opposite side of the right angle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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does the pythagorean theorem work on triangles with negative sides ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here .
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is the hypotenuse always `` c '' ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared .
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i thought side b = 54 why is it not 54 ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here .
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do you always put the longest side of a right angle first when solving ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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why does the pythagorean theorem only work on right triangles ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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how can we know that in which case should we use pythagorean theorem ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared .
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does it matter which side is a or b ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 .
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if a^2 *b^2= c^2 , would b^2= c^2 -a ^2 and a^2= c^2-b^2 ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 .
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should n't sal have put a times sign ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well .
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why would you want to square root both sides ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b .
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can the side with the question mark be c ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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how can you use the pythagorean theorem to derive the equation of a circle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle .
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is a right angle always equal to 90 degrees ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here .
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why ca n't we use a+b=c ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical .
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what does the square stand for ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here .
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is the hypotenuse always represented by the variable c ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side .
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can you only find the hypotenuse in a right triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared .
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is there a particular reason why the numbers are squared ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared .
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in this theorem 's case , can the hypotenuse ever be any other letter than c , such as a or b ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology .
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do you always need to know at least two sides of the right triangle to put the pythagorean theorem into effect ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem .
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does it matter what leg is a or b ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical .
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why ca n't i find a real square using the theorem ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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is the pythagorean theorem only used for right triangles ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation .
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if ab= 12 and bc = 13 , what is the area of the parallelogram ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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how come the pythagorean theorem only works for right triangles ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 .
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what are some perfect angle lengths , like a 3 , 4 , 5 triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here .
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is the hypotenuse always going to be c ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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was the pythagorean theorem named after pythagoras ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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what is statement of pythagoras theorem ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it .
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and if the hypotenuse is stated and you 're trying to figure out what side `` b '' would be , would you treat it like an algebraic expression ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that .
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does pythgorean theorem only work for right triangles ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there .
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how is it possible that the hypotenuse is always opposite of the right angle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side .
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also he says has to be right triangle , is it possible to do it with any other kinds of triangles ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 .
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3 , can the 3 ( a ) and the 4 ( b ) be switched around ?
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