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# Intro to the Pythagorean theorem

CCSS.Math:

## Video transcript

in this video we're going to get introduced to the Pythagorean theorem Pythagorean Pythagorean theorem which is fun on its own but you'll see as you learn more and more mathematics it's one of those cornerstone theorems of of really all of math it's useful in geometry it's kind of the backbone of trigonometry you're also going to use it to calculate distances between points so it's a good thing to to really make sure we know well so let's let them enough talk on my and let me tell you what the Pythagorean theorem is so if we have a triangle and the triangle has to be a right triangle so it has to be a right triangle which means that one of the three angles in the triangle have to be 90 degrees and you specify that it's 90 degrees by drawing that little box right there so that right there is let me do this in a different color that right there is a 90 degree 90 degree angle or we could call it a right a right angle and a triangle that has a right angle and it is called a right triangle so this is called a right right triangle now with the Pythagorean theorem if we know two sides of a right triangle we can always figure out the third side and before I show you how to do that let me give you one more piece of terminology the longest side of a pythagorean of a right triangle is the side opposite the 90 degree angle or opposite the right angle so in this case it is this side right here this is the longest side and the way to tell figure out where that right triangle is and you kind of it opens into that longest side that longest side is called the hypotenuse hypotenuse and it's good to know because we'll keep referring to it and I just just so we always are good at identifying the hypotenuse let me draw a couple of more right triangles so let's say I have a triangle that looks like that a triangle that looks let me draw it a little bit nicer so let's say I have a triangle that looks like that and I were to tell you that this angle right here is 90 degrees in this situation this is the hypotenuse because it is opposite it is opposite the 90 degree angle it is the longest side let me do one more just so that we're good at recognizing the hypotenuse so let's say that that is my triangle and this is the 90 degree angle right there and I think you know how to do this already you go right what it opens into that is the hypotenuse that is the longest side the longest side so that is the hypotenuse so once you have identified the hypotenuse so let's say that that has length C and now we're going to learn what the Pythagorean theorem tells us so let's say that C is equal to the length of the hypotenuse so let's call this C that side is C let's call this side right over here let's call this side right over here a and let's call this side over here B so the Pythagorean theorem tells us that a squared so one of the shorter sides the length of one of the shorter sides squared plus the length of the other shorter side squared is going to be equal to the length of the hypotenuse squared now let's do that with an actual problem and you'll see that it's actually not so bad so let's say that I have a triangle that looks like this let me draw it let's say that this is my triangle looks something like this and let's say that they tell us that this is the right angle that this length right here let me do this in different colors this length right here is 3 and that this length right here is 4 and they want us to figure out they want us to figure out that length right there now the first thing you want to do before you even apply the Pythagorean theorem is to make sure you have your hypotenuse straight you make sure you know what you're solving for and in this circumstance we're solving for the height and we know that because this side over here it is the side opposite opposite the right angle if we look at the Pythagorean theorem this is C this is the longest side so now we're ready to apply we're ready to apply the Pythagorean theorem it tells us that four squared one of the shorter sides plus three squared plus three squared the square of another of the shorter sides is going to be equal to this longer side squared the hypotenuse squared is going to be equal to C squared and then you just solve for C so 4 squared is the same thing as 4 times 4 that is 16 and 3 squared is the same thing as 3 times 3 so that is 9 and that is going to be equal to is equal to C squared now what is 16 plus 9 it's 25 so 25 is equal to C squared and we could take the positive square root of both sides see I guess just look at it mathematically could be negative 5 as well but we're dealing with distances so we only care about the positive roots so you take the principal root of both sides and you get 5 is equal to C or the length of the longest side is equal to is equal to 5 now you could use the Pythagorean theorem if we give you two of the sides to figure out the third side no matter what the third side is so let's do another one right over here let's say let's say that our triangle looks like let's say our triangle looks something like let's say our triangle looks like this and that is our right angle let's say that this side over here has length 12 and let's say that this side over here has length 6 and we want to figure out we want to figure out this length right over there now like I said the first thing you want to do is identify the hypotenuse and that's going to be the side opposite the right angle we have the right angle here you go opposite the right angle the longest side the hypotenuse is right there so if we think about the if we think about the Pythagorean theorem that a squared plus B squared is equal to C squared 12 you could view as C this is the hypotenuse the hypotenuse the C squared is the hypotenuse squared so you could say 12 is equal to C and then we could set these sides it doesn't matter whether you call one of them a or one of them B so let's just call this side right here let's say a is equal to 6 and then we say B this colored B is equal to question mark and now we can apply the Pythagorean theorem a squared which is 6 squared 6 squared plus the unknown B squared plus B squared is equal to the hypotenuse squared is equal to C squared is equal to 12 squared and now we can solve for B and notice the difference here now we're not solving for the hypotenuse we're solving for one of the shorter sides in the last example we solved for the hypotenuse we solved for C so that's why it's always important to recognize that the a squared plus B squared plus C squared C is the length of the hypotenuse let's just solve for B here so we get 6 squared is 36 plus b squared plus b squared is equal to 12 squared that's 12 times 12 is 144 now we can subtract 36 from both sides of this equation subtract 36 those cancel out on the left hand side we're left with just a b squared is equal to now 144 minus 36 is what that is 144 minus 30 is 114 and then you might subtract 6 it's 108 so this is going to be 108 so that's what B squared is and now we want to take the principal root or the positive root of both sides and you get B is equal to the square root the principal root of 108 now let's see if we can simplify this a little bit the square root of 108 what we could do is we could take the prime factorization of 108 and see how we can simplify this radical so 108 is the same thing as let's see it's the same thing as 2 times 54 which is the same thing as 2 times 27 which is the same thing as 3 times 9 so we have the square root square root of 108 is the same thing as the square root of 2 times 2 times actually I'm not done 9 can be factorized into 3 times 3 so it's 2 times 2 times 3 times 3 times 3 and so we have a couple of perfect squares in here let me rewrite it a little bit neater and this is all an exercise in simplifying radicals that you will bump into a lot while doing the Pythagorean theorem so it doesn't hurt to do it right here so this is the same thing as so this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there and this is the same thing and you know you would you wouldn't have to do all of this in your head well all of this on paper you could do it in your head what is this 2 times 2 is 4 4 times 9 this is 36 this is the square root of 36 times the square root of 3 the principal root of 36 is 6 so this simplifies to 6 square roots of 3 so the length of B you could write a square root of 108 or you could say it's equal to 6 times the square root of 3 this is 12 this is 6 and the square root of 3 well this is going to be a 1 point something something so it's going to be a little bit larger than 6