The Pythagorean theorem and how exams actually use it
The formula is easy; identifying the hypotenuse inside a word problem or a 3D solid is where marks disappear.
What this covers
- a² + b² = c² and its converse
- Pythagorean triples
- Distance formula in 2D and 3D
- Diagonals of rectangles and boxes
- Applied word problems
Worked example
Problem
A ladder 13 m long leans against a wall with its base 5 m out. How high does it reach?
- 1Identify the hypotenuseThe ladder is the longest side, so c = 13.
- 2Set up5² + h² = 13² → 25 + h² = 169.
- 3Solveh² = 144, so h = 12 (length is positive).
Answer: 12 metres
Mistakes that cost marks
- Assigning the hypotenuse to the wrong side.
- Adding instead of subtracting when solving for a leg.
- Forgetting units in the final answer.
Questions students ask
Does it work for non-right triangles?
No — use the cosine rule instead, which reduces to Pythagoras when the angle is 90°.
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