Why does the Pythagorean theorem work on only right triangles?

Answer 1

There is a Pythagorean theorem that actually works for every triangle. Its just that for right triangles it can be simplified to A2+B2=C2 due to the properties of cosines. The law of cosines states that for a triangle with sides A, B, and C, and angles a, b, and c (with side C being opposite angle c), C2 = A2 + B2 - (2 x A x B x cos c). This formula will work for any triangle. Now imagine that we are talking about a right triangle, with side C the hypotenuse (just like in the classic Pythagorean theorem) and angle c the right angle. The cosine of a 90 degree angle is 0, which means that the part in bold would completely drop out of the equation, leaving us with A2+B2=C2 . The cosine of any other angle possible on a triangle would result in some other number, making A2+B2=C2 not work.