Deriving the Pythagorean Theorem

https://alexandertutoring.com/ This is Sean with Alexander Mathematics and Physics Tutoring and today I want to talk about the Pythagorean theorem, something that everyone learns but very few understand where it comes from. In most classes the students are just given the Pythagorean theorem as gospel with no explanation as to where it comes from. Does it come from outer space? No. It’s very important to us at Alexander Mathematics and Physics Tutoring that our students understand where formulas come from, it's not enough to just memorize the formula and go through the motions. I want you to see how Pythagoras figured this out thousands of years ago. The Pythagorean theorem applies to right triangles, right means that there is a 90 degree angle as we can see here. Let's go ahead and label the legs of the triangle. We call the two legs A and B, it doesn’t matter which is which. However it is very important that we label the hypotenuse as C. There's a lot of ways to prove this formula but I’m going to show you one way that is very intuitive. I'm going to draw four equal sized triangles and arrange them such that you get a perfect square in the middle. Then we are going to write down the area of the object in two different ways. One option is to multiply the two sides of the entire object to get the area of a square. So we have (A+B) times (A+B). The other way to get the area is to add up the areas of the individual shapes. We have 4 triangles and a square. The area of a triangle is one half the base times the height and we have 4 of them. Therefore the area of all four triangles is 2AB. We also have a square in the middle of area C squared. Setting the two areas equal we have (A+B) times (A+B) equals 2AB plus C squared. We can distribute the left hand side to get A squared plus B squared plus two AB. The left hand side equals C squared plus 2AB. The 2AB’s cancel and we are left with the Pythagorean theorem!