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CCSS.Math:

let's now talk about what is easily one of the most famous theorems in all of mathematics and that's the Pythagorean theorem by [ __ ] Pythagorean theorem theorem and it deals with right triangles so a right triangle is a triangle is a triangle that has a 90 degree angle in it so the way I drew it right here this is our 90 degree angle if you've never seen a 90 degree angle before the way to think about it is if this side goes straight left to right this side goes straight up and down these sides are perpendicular or the angle between them is 90 degrees or it is a right angle and the Pythagorean theorem tells us that if we're dealing with a right triangle let me write that down if we're dealing with a right right triangle not a wrong triangle if we're dealing with a right triangle which is a triangle that has a right angle or a 90 degree angle in it then the relationship between their sides is this so this side is a this side is B and this side is C the fact and remember the C that we're dealing with right here is the side opposite is the side opposite the 90 degree angle it's important to keep track of which side is which the Pythagorean theorem tells us that if and only if this is a right triangle then a squared plus B squared is going to be equal to C squared and we can use this information we know two of these we can then use this theorem this formula to solve for the third I'll give you one more piece of terminology here this long side the side that is the longest side of our right triangle the side that is opposite of our right angle this right here it's C in this example this is called a hypotenuse hypotenuse very fancy word for a very simple idea the longest side of a right triangle the side that is opposite the 90-degree angle is called the hypotenuse now that we know the Pythagorean theorem let's actually solve let's actually use it because it's you know it's one thing to know something but it's a lot more fun to use it so let's say I have the following let's say I have the following right triangle let me draw a little bit neater than that the following right triangle it's a right triangle this side over here has length 9 this side over here has length 7 and my question is what is this side over here maybe we can call that we'll call that C well C in this case once again is the hypotenuse it is the longest side so we know that the sum of the squares of the other side is going to be equal to C squared so by the Pythagorean theorem 9 squared 9 squared plus 7 squared is going to be equal to C squared 9 squared is 81 plus 7 squared is 49 80 plus 40 is 120 then we're going to have the 1 plus the 9 that's another 10 so this is going to be equal to 130 so this is going let me write it this way so this is the left-hand side is going to be equal to 130 and that is equal to C squared so what's C going to be equal to let me rewrite it over here C squared is equal to 130 or we could say that C is equal to the square root of 130 and notice I'm only taking the principal root here because C has to be positive we're dealing with a distance so we can't take the negative square root so we'll only take the principal square root right here and if we want to simplify this a little bit we know how to simplify our radicals 130 is 2 times 2 times 65 which is 5 times 13 well these are all prime numbers so that's about as simple as I can get C is equal to the square root of 100 and thirty let's do another one of these let's do another one of these maybe I want to keep this Pythagorean theorem right there just so we always remember what we're referring to so let's say I have a triangle that looks like this let's see let's say it looks like that it looks like that and this is the right angle up here that's my right angle let's say that this side I'm going to call it a this side I'm going to call twenty or it's going to have length twenty one and this side right here is going to be of length 35 so your instinct to solve for a might say a21 squared plus 35 squared is going to be equal to a squared but notice notice in this situation 35 is the hypotenuse 35 is our C it's the side it's the longest side of our right triangle so what the Pythagorean theorem tells us is that a squared a squared plus the other non longest side the other non hypotenuse squared so a squared plus 21 squared is going to be equal to 35 squared you always have to remember the C squared right here the C that we're talking about is always going to be the longest side of your right triangle the side that is opposite side that is opposite of our right angle this is a side that's opposite of the right angle so a squared plus 21 squared is equal to 35 squared and what do we have here so 21 squared I'm tempted to use a calculator but I won't so 21 times 21 one times 21 is 21 2 times 21 is 42 it is 441 35 squared once again I'm tempted to use a calculator but I won't 35 times 35 5 times 5 is 25 carry the 2 5 times 3 is 15 plus 2 is 17 put a 0 here get rid of that thing 3 times 5 is 15 3 times 3 is 9 plus 1 is 10 so it is 11 let me do it in order 5 + 5 0 is 5 7 plus 5 is 12 1 plus 1 is 2 bring down the 1 12 25 so this tells us that a squared plus 440 441 is going to be equal to 35 squared which is 1225 now we could subtract 441 from both sides of this equation 441 the left-hand side just becomes a squared the right-hand side what do we get we get 5 minus 1 is 4 we want to let me write this a little bit neater here let me write this a little bit neater so minus 441 so the left-hand side once again they cancel out a squared is equal to and then on the right hand side what we have to do that's larger than that but 2 is not larger than 4 so we're going to have to borrow so that becomes a 12 or regroup depending on how you want to view it that becomes a 1 1 is not greater than 4 so we're gonna have to borrow again get rid of that and then this becomes an 11 5 minus 1 is 4 12 minus 4 is 8 11 minus 4 is 7 so a squared is equal to 784 and we could write then that a a is equal to the square root of 784 and once again I'm very tempted to use a calculator but let's well let's let's not let's not use it so this is 2 times what 392 392 392 and then this and then write 300 390 times 2 is 78 yeah and then this is 2 times what this is 2 times one hundred and ninety six 196 that's right right 190 times two is yeah that's on two times 186 196 is two times this is I want to make sure I don't make a careless mistake 196 is two times 98 98 let's keep going down here 98 is two times two times 98 is 49 and of course we know what that is so notice we have 2 times 2 times 2 times 2 so this is 2 to the fourth power so it's 16 times 49 so a is equal to the square root of 16 times 49 I pick those numbers because they're both perfect squares so this is equal to equal to the square root of 16 is 4 times the square root of 49 is 7 it's equal to 28 so this side right here is going to be equal to 28 by the Pythagorean theorem let's do one more of these let's do one more you can never get enough can never get enough practice so let's say I have another triangle I'll draw this one big there you go that's my triangle that is the right angle this side is 24 this side is 12 we'll call this side right here B now once again always identify the hypotenuse that's the longest side the side opposite the 90 degree angle you might say hey I don't know that's the longest side I don't know what B is yet how do I know this is long ago and they're in that situation you just say well it's the side opposite the 90 degree angle so if that's the hypotenuse that right there is the hypotenuse then this squared plus that squared is going to be equal to 24 squared so the Pythagorean theorem B squared plus 12 squared is equal to 24 squared or we could subtract 12 squared from both sides we say B squared is equal to 24 squared minus 12 squared which we know is 144 and that B is equal to the square root of 24 squared minus 12 squared now I'm tempted to use the calculator and I'll give in to them too into the temptation so let's do the last one was so painful I'm still recovering so 24 squared minus 12 squared is equal to 24 point seven eight so this actually turns into let me let me do it without it well I'll do it half way 24 squared minus twelve squared is equal to 432 so B is equal to the square root of 432 and let's factor this again we saw what the answer is but we can write it in kind of a simplified radical form so this is two times two 16 to 16 I believe is they let me see I believe that's a perfect square so let me take the square root of 216 nope not a perfect square so 216 let's just keep going to 16 is 2 times 108 108 is we could say 4 times 4 times what 25 plus another 2 4 times 27 which is 9 times 3 so what do we have here we have 2 times 2 times 4 so this right here is a 16 16 times 9 times 3 is that right I'm using a different calculator 16 times 9 times 3 is equal to 432 so this is going to be equal to V is equal to the square root of 16 times 9 times 3 which is equal to the square root of 16 which is 4 times the square root of 9 which is 3 times the square root of 3 which is equal to 12 roots of 3 12 so B is 12 times the square root of 3 hopefully you found that useful