Pythagoras' Theorem
Pythagoras' theorem
Pythagoras' theorem links the three sides of a right-angled triangle. If the two shorter sides are a and b, and the longest side (the hypotenuse) is c, then:
a² + b² = c²The hypotenuse is always opposite the right angle and is the longest side.
Finding the hypotenuse
When you know the two shorter sides and want the longest side, use it directly.
Example: a = 3, b = 4, find c.
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5Finding a shorter side
When you know the hypotenuse and one short side, rearrange by subtracting:
a² = c² − b²Example: c = 13, b = 5, find a.
a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12Deciding which version to use
- Finding the hypotenuse (longest side)? → add (a² + b²).
- Finding a shorter side? → subtract (c² − b²).
Using it in context
Pythagoras is used whenever a right angle is involved — ladders against walls, diagonals of rectangles, distances between two points on a grid.
Distance between two points (x₁, y₁) and (x₂, y₂): treat the horizontal and vertical gaps as the short sides:
distance = √[(x₂ − x₁)² + (y₂ − y₁)²]Worked example
A ladder 10 m long leans against a wall with its base 6 m from the wall. How high up the wall does it reach?
- The ladder is the hypotenuse (10). Find a short side:
height² = 10² − 6² = 100 − 36 = 64
height = √64 = 8 m✓
Common mistakes
- Adding when you should subtract (when finding a shorter side).
- Forgetting to square root at the end (you found c², not c).
- Using it on a triangle that isn't right-angled.
Exam tips
- Always identify the hypotenuse first (opposite the right angle, longest side).
- Add for the hypotenuse; subtract for a shorter side.
- Keep answers exact (leave as a surd like √52) if the question asks for it.
Key facts to remember
- a² + b² = c² for right-angled triangles; c is the hypotenuse (opposite the right angle).
- Hypotenuse: add the squares. Shorter side: subtract (c² − b²). Always square root to finish.
- Works for distances, ladders, diagonals and points on a grid.