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Right-triangle lengths: Pythagoras and altitude

2026-09-30Plane GeometryMs Ho

Choose Pythagoras, an altitude relation, or trigonometry according to the given data.

Key formulas

a² = b² + c²
h² = mn
sin θ = opposite / hypotenuse

Choosing the tool

Use Pythagoras when two sides of a right triangle are known. Use h² = mn when an altitude to the hypotenuse splits it into segments m and n. Use sine, cosine or tangent when an acute angle is given.

Worked example A

Legs 3 cm and 4 cm. Hypotenuse c satisfies c² = 9 + 16 = 25, so c = 5 cm. This 3-4-5 triangle is worth memorising because it appears often in exam diagrams.

Worked example B

An altitude to the hypotenuse creates segments 4 cm and 9 cm. Then h² = 4 × 9 = 36, so h = 6 cm. Check: the hypotenuse is 13 cm, and 6 is clearly shorter, as required.

Worked example

Worked example

Problem. In right triangle ABC with ∠C = 90°, AC = 6 and BC = 8. Find AB and the altitude from C to AB.

Solution. AB = √(6² + 8²) = 10. Altitude to hypotenuse: h = (AC × BC)/AB = 48/10 = 4.8. So AB = 10 and the altitude is 4.8.

Trigonometry note

In a right triangle with opposite 5 and hypotenuse 13, sin θ = 5/13.

Review quiz

Right triangle with legs 5 and 12. Hypotenuse is:

Practice questions

  1. Find the hypotenuse of a right triangle with legs 5 cm and 12 cm.
  2. If the hypotenuse parts are 3 and 12, find the altitude to the hypotenuse.

Study notes

How do I spot the hypotenuse?

It is the side opposite the right angle and is always the longest side.

What if my answer exceeds the hypotenuse?

Recheck the formula: a leg or altitude cannot be longer than the hypotenuse.

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