20.1 Pythagoras' theorem — Learn

Prerequisites: square numbers, square roots, and identifying a right angle.

Objectives: use Pythagoras' theorem to find a missing side in a right-angled triangle, including the hypotenuse.

A hypotenuse is the longest side, opposite the right angle. If the shorter sides are $a$ and $b$, and the hypotenuse is $c$, then $a^2+b^2=c^2$.

Example 1. A right-angled triangle has shorter sides $6$ cm and $8$ cm. Find its hypotenuse.

The positive root is used because a length cannot be negative.

Example 2. The hypotenuse is $13$ cm and one shorter side is $5$ cm. Find the other side $x$.

Watch out. Do not subtract when finding the hypotenuse. The hypotenuse is opposite the right angle, so its square equals the sum of the other two squares. Check that it is the longest side.

Try it: change a side

Predict: if one shorter side stays at $6$ units and the other grows, will the hypotenuse grow or shrink? Test: adjust the side length in the figure. Explain: why does the hypotenuse change in that direction?

Progress check

The chapter problem asks for a direct distance across a rectangular garden measuring $9$ m by $12$ m. Which side is the hypotenuse, and what distance do you predict?

You should now be able to

• identify the hypotenuse from the right angle;
• use $a^2+b^2=c^2$ to find a missing side;
• give a length with suitable units and check it is plausible.