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Sal introduces the famous and super important Pythagorean theorem!  Created by Sal Khan and CK-12 Foundation.
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Let's now talk about what is easily one of the most famous theorems in all of mathematics. And that's the Pythagorean theorem. And it deals with right triangles. So a right 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 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. And remember, the c that we're dealing with right here 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. If we know two of these, we can then use this theorem, this formula to solve for the third. And 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. A 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 hypotentuse. Now that we know the Pythagorean theorem, let's actually use it. Because 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 right triangle. Let me draw it a little bit neater than that. 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, it 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 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 let me write it this way. 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 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 130. 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. And this is the right angle, up here. Let's say that this side, I'm going to call it a. The side, it's going to have length 21. And this side right here is going to be of length 35. So your instinct to solve for a, might say, hey, 21 squared plus 35 squared is going to be equal to a squared. But notice, in this situation, 35 is a hypotenuse. 35 is our c. It's the longest side of our right triangle. So what the Pythagorean theorem tells us is that 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 of our right angle. This is the 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: 1 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 plus 0 is 5, 7 plus 5 is 12, 1 plus 1 is 2, bring down the 1. 1225. So this tells us that a squared plus 441 is going to be equal to 35 squared, which is 1225. Now, we could subtract 441 from both sides of this equation. 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. 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 do 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 regrouped, depending on how you want to view it. That becomes a 1. 1 is not greater than 4, so we're going to 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 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 not. Let's not use it. So this is 2 times, what? 392. And then this-- 390 times 2 is 78, yeah. And then this is 2 times, what? This is 2 times 196. That's right. 190 times 2 is-- yeah, that's 2 times 196. 196 is 2 times-- I want to make sure I don't make a careless mistake. 196 is 2 times 98. Let's keep going down here. 98 is 2 times 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 picked those numbers because they're both perfect squares. So this is 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. 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 longest? And there, in that situation, you say, well, it's the side opposite the 90 degree angle. So if that's 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 a calculator, and I'll give into the temptation. So let's do it. The last one was so painful, I'm still recovering. So 24 squared minus 12 squared is equal to 24.78. So this actually turns into-- let me do it without a-- well, I'll do it halfway. 24 squared minus 12 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 maybe we can write it in kind of a simplified radical form. So this is 2 times 216. 216, I believe, is a-- 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. 216 is 2 times 108. 108 is, we could say, 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-- b 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. So b is 12 times the square root of 3. Hopefully you found that useful.