The Converse

The Pythagorean theorem says: if a triangle has a right angle, then ${a}^{2}+{b}^{2} = {c}^{2}$. Turned around, it is just as true: if the sides of a triangle satisfy ${a}^{2}+{b}^{2} = {c}^{2}$, then the triangle has a right angle, facing the longest side. This is the converse of the theorem.

It means you can test for a right angle with nothing but the three lengths. Builders use it to set out square corners: mark a rope into lengths of 6, 8 and 10, pull it tight into a triangle, and the corner between the 6 and the 8 is square, because 36 + 64 = 100.

Why it holds: a triangle is fixed by its three sides. A right triangle with legs 6 and 8 has hypotenuse 10, so a triangle with sides 6, 8 and 10 is that same right triangle.