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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 it called when you try to find the reciprocal of a 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 .
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how would you find the length of the hypotenuse if you did n't have a right triangle and had an acute or obtuse triangle instead ?
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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 do you find a square root of a number ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 .
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ca n't we take the square root of 12^2-6^2 ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and 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 a squared + b squared = c squared , should n't 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 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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does the hypotenuse always have to be the longest side of a triangle when doing 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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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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why does the triangle have to be 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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we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there .
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who is adjacent and opposite in a right triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and 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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can you solve the right triangle if you only know one side and an angle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 .
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what is a principal root ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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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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i was wondering instead of doing the tree method could i just use my calculator typing in the square root of 108 ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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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 do we need to take the positive square root of the number equal 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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so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared .
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how can i know what side is a or b ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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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 .
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if 2/3 of class is 9 years and 1/6 is 10 years what fraction is 12 years ?
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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 the square root of 100 is 10 how come the square root of 108 is 6square root 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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that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 .
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how do you find the hypotenuse when you only know the length of the long leg , which is 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 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 .
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is the hypotenuse just the longest side of a right triangle , or the longest side for all 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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when do we use pythagorean theorem in real life ?
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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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could you find the pythagorean theorem if the triangle is n't a right triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and 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 length of a rectangle ?
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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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can you use the pythagorean theorem if you have only one side that is not 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 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 .
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at the second question is the length of the unknown side sqrt108 ?
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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 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 .
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why are the only the 2 square root answers allowed ?
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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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what happens when you have an isosceles triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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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 phythagoras theorem applicable to triangles not right angled ?
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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 c always the hypotenuse ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical .
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can we just make c = the square root of 108 ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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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 .
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could you use the pythagorean theorem 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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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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why is the last 3 a sq root all on its own ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared .
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does it matter which side is a or b ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math .
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what is 640,100 divided by ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 .
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why does the formula have to be 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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what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 .
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and the other 2 sides are 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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that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 .
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how much distance is saved if he takes main street 5 blocks instead of market street 3 blocks exchange street d blocks ?
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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 the resulted number , such as 108 in does n't have a root or anything that makes it up ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit .
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is the principle root always going to be ... ... ... .. `` equal '' ?
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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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i would like to know , when a number is given how do we find the main 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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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 .
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i do n't understand the second example ... why do you have to do the prime factorization ?
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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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ca n't you do the square root of 108 and simplify to the nearest tenth or hundredth ?
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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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does the hypotenuse always have to be the `` unknown '' ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and 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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why is a 90 degree angle also known as and called a 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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what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 .
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why ca n't you `` cancel out '' the exponents when adding the two sides like , a + b = c instead 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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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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in 7 can the hypotenuse be the butom side if the triangle is fliped ?
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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 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 .
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how do i know where to keep or put a2 and b2 ?
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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 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 .
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i 'm having trouble simplifying radicals , what if your radical is uneven ?
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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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how do you tell the the difference between a and b ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical .
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what is the difference between the principal square root and regular 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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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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what is the purpose of the square in the formula for the hypotenuse , a square + b square = c square ?
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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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i need help with a question of perimeter and area of a triangle , the top of the triangle is 9cm and the hieght is 12cm and i need to find out what is 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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can you also use the pythagorean theorem to solve for angles too ?
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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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how do you find the square roots of numbers other then factorization ?
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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 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 .
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why is there no way to simplify the squares right in the beginning ?
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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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the hypotenuse is 8 how do i get a^2 and b^2 ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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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 solve the square root of 24 using 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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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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do square roots only work in even 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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in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here .
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is the hypotenuse always c ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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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 pythagorean in 2ed sem geometry ?
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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 can you solve for either the leg or hypotenuse of a right triangle if one of the lengths is given in a 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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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 to solve c= square root of 89 ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit .
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why is it that at in in the exercises there is always ... .like a number outside of the square root sign and one inside.i do n't get it ... what does it mean ?
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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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can someone explain to me how sal simplifies square roots , like ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit .
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how do you get the square root , as in 8 ?
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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 pythagorean theorem and pythagoras property mean the same thing ?
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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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is there supposed to be sound ?
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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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should i put a question mark at the end ?
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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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why does the equation have to be a squared + b squared + 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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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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if the base of the ladder is 7 feet away from the bottom of the building , how high up the building can the ladder reach ?
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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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wait , if a right triangle has a 90 degree angle , what does a left triangle have ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology .
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is n't there a way to solve using pythagorean theorem if you only have one side of the triangle ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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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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we already know what a right triangle is , so why does sal say what a right triangle is ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared .
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who actually has come up the idea of pythagorean theorem sir ?
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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 a squared plus b squared is equal to c squared , would n't a plus b equal c ?
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in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it to calculate distances between points . so it 's a good thing to really make sure we know well . so enough talk on my end . let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . so that right there is -- let me do this in a different color -- a 90 degree angle . or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . this is the longest side . and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . let me draw it a little bit nicer . so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . it is the longest side . let me do one more , just so that we 're good at recognizing the hypotenuse . so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . that is the hypotenuse . that is the longest side . so once you have identified the hypotenuse -- and let 's say that that has length c. and now we 're going to learn what the pythagorean theorem tells us . so let 's say that c is equal to the length of the hypotenuse . so let 's call this c -- that side is c. let 's call this side right over here a . and let 's call this side over here b . so the pythagorean theorem tells us that a squared -- so the length of one of the shorter sides squared -- plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared . now let 's do that with an actual problem , and you 'll see that it 's actually not so bad . so let 's say that i have a triangle that looks like this . let me draw it . let 's say this is my triangle . it looks something like this . and let 's say that they tell us that this is the right angle . that this length right here -- let me do this in different colors -- this length right here is 3 , and that this length right here is 4 . and they want us to figure out that length right there . now the first thing you want to do , before you even apply the pythagorean theorem , is to make sure you have your hypotenuse straight . you make sure you know what you 're solving for . and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . so now we 're ready to apply the pythagorean theorem . it tells us that 4 squared -- one of the shorter sides -- plus 3 squared -- the square of another of the shorter sides -- is going to be equal to this longer side squared -- the hypotenuse squared -- is going to be equal to c squared . and then you just solve for c. so 4 squared is the same thing as 4 times 4 . that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . and that is going to be equal to c squared . now what is 16 plus 9 ? it 's 25 . so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . but we 're dealing with distances , so we only care about the positive roots . so you take the principal root of both sides and you get 5 is equal to c. or , the length of the longest side is equal to 5 . now , you can use the pythagorean theorem , if we give you two of the sides , to figure out the third side no matter what the third side is . so let 's do another one right over here . let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . and we want to figure out this length right over there . now , like i said , the first thing you want to do is identify the hypotenuse . and that 's going to be the side opposite the right angle . we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides , it does n't matter whether you call one of them a or one of them b . so let 's just call this side right here . let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . is equal to 12 squared . and now we can solve for b . and notice the difference here . now we 're not solving for the hypotenuse . we 're solving for one of the shorter sides . in the last example we solved for the hypotenuse . we solved for c. so that 's why it 's always important to recognize that a squared plus b squared plus c squared , c is the length of the hypotenuse . so let 's just solve for b here . so we get 6 squared is 36 , plus b squared , is equal to 12 squared -- this 12 times 12 -- is 144 . now we can subtract 36 from both sides of this equation . those cancel out . on the left-hand side we 're left with just a b squared is equal to -- now 144 minus 36 is what ? 144 minus 30 is 114 . and then you subtract 6 , is 108 . so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . so 108 is the same thing as 2 times 54 , which is the same thing as 2 times 27 , which is the same thing as 3 times 9 . so we have the square root of 108 is the same thing as the square root of 2 times 2 times -- well actually , i 'm not done . 9 can be factorized into 3 times 3 . so it 's 2 times 2 times 3 times 3 times 3 . and so , we have a couple of perfect squares in here . let me rewrite it a little bit neater . and this is all an exercise in simplifying radicals that you will bump into a lot while doing the pythagorean theorem , so it does n't hurt to do it right here . so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there . and this is the same thing . and , you know , you would n't have to do all of this on paper . you could do it in your head . what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . so this simplifies to 6 square roots of 3 . so the length of b , you could write it as the square root of 108 , or you could say it 's equal to 6 times the square root of 3 . this is 12 , this is 6 . and the square root of 3 , well this is going to be a 1 point something something . so it 's going to be a little bit larger than 6 .
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or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side .
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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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does the pythagorean theorem work for acute and obtuse triangles as well ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 .
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so how do i know which fraction i need to swap the numerator and the denominator ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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what is the logic behind a number being divided by a fraction being equal to the number being multiplied by the fraction 's reciprocal ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 .
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do you always use the 2nd fraction , or do you use the one which is a improper fraction ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators .
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how does flipping the divisor and then multiplying the dividend by the flipped divisor give you the correct answer ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators .
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can you explain why multiplying by the reciprocal is the same than dividing fractions ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward .
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why did sal find the reciprocal of only one fraction and not the other ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 .
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why do you multiply if you are trying to divide the number ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators .
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why do you have to change the denominator in adding and subtracting fraction and why not in multiplying and dividing fractions ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward .
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can you multiply the reciprocal of the first number ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 .
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how does cross multiplication work ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators .
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what is a reciprocal and why are we flipping the fractions around and multiplying ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward .
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how do i know which fraction is the reciprocal ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators .
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does the `` keep change flip '' work out with negative fractions ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward .
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why do we have to inverse 7/3 ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward .
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how many 7/3 in 2/5 ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 .
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how do you know when you have to flip a number ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 .
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you have to multiply the fraction ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward .
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is it to be considered as 4/2 divided by 3 or 4 divided by 2/3 ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward .
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what is 3/8 divided by 4/5 ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators .
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for all of the dividing of the fractions would the process be the same or would be different ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators .
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how do you divide fractions by using fraction circles to find the quotient ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators .
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how to add and subtract whole number with fractions ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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is 16 , 8 , and 0.14 equal in any way ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 .
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i wonder why does sal use different colors ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 .
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what is a squared + b squared= ?
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 . and then multiplying two fractions is pretty straightforward . this is just going to be equal to the product of the numerators . so 2 times 3 over the product of the denominators , over 5 times 7 -- i 'm trying to keep the colors consistent -- which of course is going to be equal to 2 times 3 is equal to 6 . and 5 times 7 -- i 'll do this in a new color , let 's see , i have n't used this shade of blue yet -- 5 times 7 of course is equal to 35 . so this is equal to 6/35 .
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so let 's calculate what 2/5 divided by 7/3 is , and i encourage you to pause this video and try to calculate this on your own . well we just have to remind ourselves that this is going to be the exact same thing as 2 over 5 times the reciprocal of 7/3 , which is 3 over 7 .
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what would happen if the numarator was bigger than the denomnator ?
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