Using the Converse of the Pythagorean Theorem

Show that a triangle whose sides have lengths \(5\), \(12\), and \(13\) is a right triangle. Identify the hypotenuse.

Solution We square the lengths of the sides. \[ 5^{2}=25\qquad 12^{2}=144\qquad 13^{2}=169 \]

Notice that the sum of the first two squares (25 and 144) equals the third square (169). So, the triangle is a right triangle. The longest side, 13, is the hypotenuse.

See Figure 9.