Proving the Pythagorean Theorem There are many ways to prove the Pythagorean Theorem. Today I would like to share one of my favorite methods. I like this method because it’s simple, elegant, and easy to follow. Let’s begin with a few basic definitions, followed by a statement of the theorem: The picture above shows a right triangle. The vertical and horizontal segments (labeled a and b, respectively) are called the legs of the right triangle, and the side opposite the right angle (labeled c) is called the hypotenuse of the right triangle The Pythagorean Theorem: In a right triangle with legs of lengths a and b, and a hypotenuse of length c, c2 = a2 + b2. Notes: There are many ways to prove the Pythagorean Theorem. Here, we will provide a simple geometric argument. For the proof we will want to recall the following: The area of a square with side length s is A = s2. The area of a triangle with base b and height h is A = 1/2 bh. Notice that in our right triangle drawn above, the base is labeled b, and the height is labeled a. So, the area of that right triangle is A = 1/2 ba = 1/2 ab. Now, let’s prove the theorem! Proof of the Pythagorean Theorem: We draw 2 squares, each of side length a +b, by rearranging 4 copies of the given triangle in 2 different ways: We can get the area of each of these squares by adding the areas of all the figures that comprise each square. The square on the left consists of 4 copies of the given right triangle, a square of side length a and a square of side length b. It follows that the area of this square is 4 ⋅ 1/2 ab + a2 + b2 = 2ab + a2 + b2. The square on the right consists of 4 copies of the given right triangle, and a square of side length c. It follows that the area of this square is 4 ⋅ 1/2 ab + c2 = 2ab + c2. Since the areas of both squares of side length a + b are equal (both areas are equal to (a + b)2 ), we have 2ab + a2 + b2 = 2ab + c2. Cancelling 2ab from each side of this equation yields a2 + b2 = c2. And there you have it…a relatively simple proof of the Pythagorean Theorem. If you enjoy thinking about why things like this are true or you want to begin learning some more advanced mathematics, you may want to check out either Pure Mathematics for Beginners or Set Theory for Beginners. These books contain no prerequisites and are perfect for anyone just starting out in theoretical math. Comments comments