The right-angled triangle
A right-angled triangle has one angle equal to 90°. The side opposite the right angle is the longest side, called the hypotenuse. The other two sides are the shorter sides (the legs).
Key idea
Pythagoras theorem: in a right-angled triangle, c² = a² + b², where c is the hypotenuse and a, b are the two shorter sides. The square on the hypotenuse equals the sum of the squares on the other two sides.
Finding the hypotenuse
When the two shorter sides are known, square each, add them, then take the square root.
Worked example
A triangle has shorter sides 3 cm and 4 cm. Find the hypotenuse c.
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5 cm.
The set (3, 4, 5) is a well-known Pythagorean triple. Others include (6, 8, 10) and (5, 12, 13).
Finding a shorter side
When the hypotenuse and one short side are known, subtract the squares instead of adding.
Worked example
The hypotenuse is 13 cm and one shorter side is 5 cm. Find the other side a.
a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12 cm.
Remember
- The hypotenuse is always opposite the right angle and is the longest side.
- Finding the hypotenuse: add the squares, then √.
- Finding a shorter side: subtract the squares, then √.
- Keep the units the same throughout.