Pythagoras' theorem
Pythagoras' theorem
- For a right-angled triangle, the longest side (the hypotenuse, opposite the right angle) is $c$:
$$a^2 + b^2 = c^2$$
- Use it to find a missing side. The classic example is the $3$-$4$-$5$ triangle: $3^2 + 4^2 = 25 = 5^2$.
Practice
A right-angled triangle has short sides 3 cm and 4 cm. Find the hypotenuse (cm).
√(3² + 4²) = √25 = 5 cm.
Practice
The hypotenuse of a right-angled triangle is:
The hypotenuse is the longest side and lies opposite the 90° angle.
Worked example
- Hypotenuse $13\ \text{cm}$, one short side $5\ \text{cm}$. Find the other short side.
- $b^2 = 13^2 - 5^2 = 169 - 25 = 144$, so $b = \sqrt{144} = 12\ \text{cm}$.
Practice
The hypotenuse is 13 cm and one short side is 5 cm. Find the other short side (cm).
√(13² − 5²) = √(169 − 25) = √144 = 12 cm.
You've got it
Key idea
- $a^2 + b^2 = c^2$, with $c$ the hypotenuse (opposite the right angle)
- to find a short side, subtract: $b = \sqrt{c^2 - a^2}$
- $3$-$4$-$5$ and $5$-$12$-$13$ are worth remembering