The Converse

The Pythagorean theorem also works in reverse. If the three sides of a triangle satisfy a^2 + b^2 = c^2, where c is the longest side, then the angle between a and b is a right angle. The rule is called the converse of the Pythagorean theorem. It lets you test whether a triangle is right-angled from its three side lengths alone. You do not need to measure the angle.

Sides 6, 8 and 10: is the corner between the 6 and the 8 a right angle?

Builders use the converse to lay out a square corner. Take a rope with knots at 3, 4 and 5 units and pull it tight into a triangle. The angle between the 3 and the 4 is a right angle, because 9 + 16 = 25. Now take the triangle in the figure, with sides 6, 8 and 10. The two shorter sides give 36 + 64 = 100, and the longest side gives 10^2 = 100. The two results match, so the corner between the 6 and the 8 is a right angle. If the longest side were 11 instead, its square would be 11^2 = 121. That does not match 100, so the corner would not be a right angle.