The hypotenuse and Pythagoras’ theorem

In a right-angled triangle, the side opposite the right angle is the hypotenuse and is the longest side. If its length is \(c\) and the shorter sides are \(a,b\), then \(c^2=a^2+b^2\). The squares built on the two shorter sides have total area equal to the square on the hypotenuse.

Example

A right triangle has shorter sides 5 cm and 12 cm. \(c=\sqrt{5^2+12^2}=13\text{ cm}\). If instead the hypotenuse is 10 cm and one shorter side is 6 cm, the other is \(\sqrt{10^2-6^2}=8\text{ cm}\).

Add squares to find the hypotenuse; subtract to find a shorter side. Take the positive square root because a length is positive. Use matching units before squaring.