Deriving the Pythagorean Theorem ✅

When we're young, when we're in school, we were taught the fabulous theorem. We were taught that given a right triangle with legs A and B and the hypotenuse C, that A squared plus B squared equals C squared, the hypotenuse squared. But you ever ask yourself why this is true or even if this is actually true? Well, let me show you one of the ways why this is true and also how to derive the Pythagorean theorem. So to begin, I'm going to draw a square and inside the square, I'm going to take another square and tilt it a little bit like that. And let's go ahead and label one of the sides as A right there. And then we'll go ahead and label the longer side next to it B. And because it's a square within a square, if this is A, then this side up here will be A as well. And this will be B also. And this continues. A, B, A, and B. And then we'll let the length of the smaller square inside be C. So it's going to be C all around. Now I'm going to be using this figure to show that this right triangle or this right triangle here that A squared plus B squared equals C squared is true. And you can even see how it is true. So here's the game plan. What we're going to do is take the area of the small square inside and add the four right triangles inside and set it equal the area of the large square around it. And the side of the large square is going to be A plus B as you see it here. So we'll take the area of the large square, set it equal the area of the small square plus area of the small or the right triangles. However, since we have four of them, we'll simply go ahead and multiply four to the area of that right triangle. Area of the large square is going to be A plus B times A plus B. Area of the small square is C squared. And then the area of a triangle is one half times the legs or one half times A. A plus B times A plus B is A squared plus 2AB plus B squared. We have C squared plus four times half is going to be 2AB. And then we'll go ahead and subtract the 2AB from the left and the right side, leaving us with A squared plus B squared equals C squared. So as you can see, indeed when you take A squared, one of the legs, plus B squared, the other leg squared, it does equal the hypotenuse squared. So there you go. Now you could be happy knowing that the Pythagorean Theorem is actually true. And you also know how to derive the Pythagorean Theorem. You need to.