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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 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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how do you do pythagorean theorem when the sides arent whole numbers so that the sides are radicals ?
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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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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 .
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why could n't we just find the square root and that would be the sum of 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 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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why do a factor tree when you can divide ?
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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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so why do you have to divide by 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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how you solve the numbers are the side 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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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 .
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how would you find the length of one of the sides if the hypotenuse is 11 and the other leg is 5 times as long as the other ?
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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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secondly if the longest leg was 5 cm longer than the other and the hypotenuse is 11 ?
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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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can i just take the positive root of both sides to solve for 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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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 does there have to be a right triangle to do 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 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 with 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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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 u find the 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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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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so , you should be able to do a + b = c , 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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where would i find a `` pythagorean theorem '' of cylinders ?
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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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for the equation of finding a non-hypotenuse length , instead of solving it the way in the video , could n't it also be solved using c squared minus a or b squared equals 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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why is pythagorean theorem 2 before pythagorean theorem 1 ?
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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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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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can we use the pythagorean theorem for triangles that 's not 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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what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 .
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for the problem why would you not just divide 108 by 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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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 formula a+b=c or a2+b2=c2 ?
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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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find the unknown length of this right triangle : leg=15 ft. , leg=36 ft. 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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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 i stopped and i was wondering why each equation has to be like squared such as his example 4 squared + 3 squared = c squared , how come ?
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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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should n't he write the theorem in lowercase letters since capital letters means something else ?
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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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you go right what it opens into . that is the hypotenuse . that is the longest side .
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is the other two sides of the triangle besides the hypotenuse equal to 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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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 .
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why did sal square 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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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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just to be clear , the exact uses of pythagorean theorem is to : a ) find the length of 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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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 .
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b ) find the length of any shorter side ?
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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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c ) find out if a triangle is a right triangle or not ?
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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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also , 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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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 .
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how does sal come up with sqr 36 and then 6 sqr 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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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 is the square root of tau , multiplied by pi , plus tau 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 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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= unknown area 3 squared plus 10 squared equals ?
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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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squared ( i do n't know how to do the square root sign so just picture it where it should be ) 29 equals ?
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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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if you 're going to square root the numbers back again at the end , then why do we have to square the numbers at first ?
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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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did anyone else get taught this as `` pythagorus '' bot `` pythagorean '' ?
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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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in the 2nd triangle example , why would you leave out the third 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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what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 .
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in the 2nd triangle example , would n't you add 6 ^2 + 12 ^2 to find out 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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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 .
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what would happen with the negative root numbers ?
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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 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 .
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does c always equal the length of 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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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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if the child holds the string 1.5m from the ground how high is the kite ?
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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 n't it just 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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what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 .
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how did pythagoras prove that this equation , of a^2+b^2=c^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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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 .
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is pythagoras used in trigonometry ?
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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 for triangles or for other things to ?
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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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the pythagorean theorem works , but who developed the theorem and is it even possible ?
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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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i 'm going into algebra 1 next year and i will be in the 8th grade ( i get 2 skip 8th-grade math ) do any of y'all have any tips on what to practice or go over along the course of summer ?
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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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why did sal put the last 3 into it 's own square root why not together ?
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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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how does one pull out ?
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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 the hypotenuse the longest side of the right triangle , or the side 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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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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or is the longest side 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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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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or does the hypotenuse needs to be both the longest side , and the side 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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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 shouldnt the intro for the pyagrean theorem be on the top ?
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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 do you teach people pythagorean theorem 1. ?
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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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if the legs of a right triangle are x , ( 2x - 1 ) and its hypotenuse is ( 2x + 1 ) , what is the value of x ?
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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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do you use scale factors 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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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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square roots are the opposite of powers ?
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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 long is the walkway along the diagonal ?
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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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could n't you take the square root of a^2 , b^2 , and c^2 to simplify leaving 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 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 .
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how do you find the area with the given length of `` a '' and the given area.. ?
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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 hypotenuse apply to only right triangles , or all types 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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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 the pythagorean theorem be use on triangles that are not 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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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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why does he do square root of 2x2x3x3 and put the last 3 in a seperate 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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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 do we find out the rest 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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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 .
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what scratch pad does sal use ?
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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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where does the squared 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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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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is the hypotenuse always the longest measurement ?
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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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would the hypotenuse then become b or a ?
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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 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 .
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what is the measure of each 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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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 .
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what is the measure of each 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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we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there .
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does the hypotenuse have to be 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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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 13 ?
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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 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 .
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how come 0 sal put the last factor 3 as its own separate self and not together with the other factors ?
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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 i find the height of where the ladder is leaning on the wall ?
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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 are the pythagorean triples , and how do i know if a set of numbers is a pythagorean triple or not ?
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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 ca n't you use the pythagorean theorem on triangles that are not 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 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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if you have a squared + b squared = c squared , could n't you just un-square a , b , and 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 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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why does the pythagorean theoruem only work with 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 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 same formula for the volume of 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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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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wait so for the last problem could you not do 3 square root 12 and would that not be correct ?
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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 a triangle have to be a right triangle to be used 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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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 .
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in : 2x2x3x3x3 0 , why does sal square the last 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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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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edit oh sorry , i meant why does sal take the square root of the last 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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you go right what it opens into . that is the hypotenuse . that is the longest side .
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is hypotenuse always longer than any other 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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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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how would you know how it is 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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why can we not form a right angled triangle with sides 1 cm,1 cm , 1 cm ?
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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 it possible to use the pythagorean theorem for other 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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the other thing that 's a little more confusing is why did he break the square root of 2 x 2 x 3 x 3 x 3 into 2 x 2 x 3 x 3 and then separated the last 3 into a square root of 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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is equal to 12 squared . and now we can solve for b . and notice the difference here .
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for example ; if you knew side a was 7 inches , is there any way you could solve for c without knowing 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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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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is the hypotenuse always equals to c 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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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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how would you put a square on the right triangle on a graph ?
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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 many feet above the ground does the ladder touch the wall ?
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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 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 .
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0 , why did you just take out one of the threes and make it its own 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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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 .
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if i had an equilateral triangle , how would i find 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 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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is the hypotenuse only found in 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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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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what is the c 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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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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so the hypotenuse is always c 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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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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could anyone pls tell me how to 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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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 .
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so should n't lowercase letters be used ?
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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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are n't the smaller sides called legs ?
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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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if the slant height of the sail is 20 square root of 3 ft and its height is 10 square root of 3 ft , how many feet is its base length ?
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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 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 .
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what do you do if they are asking you to find the pythagorean theorem of a triangle that does n't have a right angle ?
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