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. $$5^{2}=25\qquad 12^{2}=144\qquad 13^{2}=169$$

Figure 9

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.