The Pythagorean Theorem is one of the most interesting theorems for two reasons: First, it’s very elementary; even high school students know it by heart. Second, it has hundreds of proofs. The proof below uses triangle similarity.
In a right triangle with side lengths and and hypotenuse , the following equation always holds:
In the figure above, triangle is similar to triangle . In similar triangles, the corresponding sides are proportional. It follows that
which is equivalent to
Cross multiplying, we have . Adding to both sides of the equation, we have , which is what we want to prove.