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Pythagorean theorem example

Sal uses the Pythagorean theorem to find the height of a right triangle with a base of 9 and a hypotenuse of 14. Created by Sal Khan and Monterey Institute for Technology and Education.

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Video transcript

Say we have a right triangle. Let me draw my right triangle just like that. This is a right triangle. This is the 90 degree angle right here. And we're told that this side's length right here is 14. This side's length right over here is 9. And we're told that this side is a. And we need to find the length of a. So as I mentioned already, this is a right triangle. And we know that if we have a right triangle, if we know two of the sides, we can always figure out a third side using the Pythagorean theorem. And what the Pythagorean theorem tells us is that the sum of the squares of the shorter sides is going to be equal to the square of the longer side, or the square of the hypotenuse. And if you're not sure about that, you're probably thinking, hey Sal, how do I know that a is shorter than this side over here? How do I know it's not 15 or 16? And the way to tell is that the longest side in a right triangle, and this only applies to a right triangle, is the side opposite the 90 degree angle. And in this case, 14 is opposite the 90 degrees. This 90 degree angle kind of opens into this longest side. The side that we call the hypotenuse. So now that we know that that's the longest side, let me color code it. So this is the longest side. This is one of the shorter sides. And this is the other of the shorter sides. The Pythagorean theorem tells us that the sum of the squares of the shorter sides, so a squared plus 9 squared is going to be equal to 14 squared. And it's really important that you realize that it's not 9 squared plus 14 squared is going to be equal to a squared. a squared is one of the shorter sides. The sum of the squares of these two sides are going to be equal to 14 squared, the hypotenuse squared. And from here, we just have to solve for a. So we get a squared plus 81 is equal to 14 squared. In case we don't know what that is, let's just multiply it out. 14 times 14. 4 times 4 is 16. 4 times 1 is 4 plus 1 is 5. Take a 0 there. 1 times 4 is 4. 1 times 1 is 1. 6 plus 0 is 6. 5 plus 4 is 9, bring down the 1. It's 196. So a squared plus 81 is equal to 14 squared, which is 196. Then we could subtract 81 from both sides of this equation. On the left-hand side, we're going to be left with just the a squared. These two guys cancel out, the whole point of subtracting 81. So we're left with a squared is equal to 196 minus 81. What is that? If you just subtract 1, it's 195. If you subtract 80, it would be 115 if I'm doing that right. And then to solve for a, we just take the square root of both sides, the principal square root, the positive square root of both sides of this equation. So let's do that. Because we're dealing with distances, you can't have a negative square root, or a negative distance here. And we get a is equal to the square root of 115. Let's see if we can break down 115 any further. So let's see. It's clearly divisible by 5. If you factor it out, it's 5, and then 5 goes in the 115 23 times. So both of these are prime numbers. So we're done. So you actually can't factor this anymore. So a is just going to be equal to the square root of 115. Now if you want to get a sense of roughly how large the square root of 115 is, if you think about it, the square root of 100 is equal to 10. And the square root of 121 is equal to 11. So this value right here is going to be someplace in between 10 and 11, which makes sense if you think about it visually.