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SQA National 5 Mathematics

Pythagoras

Finding missing sides in right-angled triangles.

Formula referenceMixed revision

Formula reference

Check the National 5 rules and formulae linked to this topic.

Which method?

Match exam clues to a suitable method.

Before you start

  • Identify the right angle.
  • Label the hypotenuse as the longest side, opposite the right angle.
  • Decide whether you are finding the hypotenuse or a shorter side.
National 5 Mathematics lesson

Explanation

Pythagoras' theorem works only in right-angled triangles. It connects the two shorter sides and the hypotenuse.

If you are finding the hypotenuse, add the squares of the two shorter sides. If you are finding a shorter side, subtract the known shorter-side square from the hypotenuse square.

Always square the lengths first, then take the square root at the end.

Visual support

Right-angled triangle

6 cm8 cmc

Key formulae and rules

  • a² + b² = c²
  • c is the hypotenuse
  • shorter side² = c² − other shorter side²

Watch out

Using Pythagoras on a triangle that is not right-angled.

Check

Compare your answer with the size you expected from the question.

Exam tip

State which side is the hypotenuse before substituting. This prevents adding when you should subtract.

Calculator tip

Use brackets for fractions, powers and square roots, then round only at the final line.

Worked examples

Worked example 1

Find the hypotenuse

A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.

  1. c² = 6² + 8²
  2. c² = 36 + 64 = 100
  3. c = √100 = 10

Answer: The hypotenuse is 10 cm.

Worked example 2

Find a shorter side

A right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm. Find the other shorter side.

  1. x² + 5² = 13²
  2. x² + 25 = 169
  3. x² = 144
  4. x = 12

So: The missing side is 12 cm.

Worked example 3

Use Pythagoras in context

A ladder reaches 4.8 m up a wall and its foot is 1.4 m from the wall. Find the ladder length.

  1. The ladder is the hypotenuse.
  2. l² = 4.8² + 1.4²
  3. l² = 23.04 + 1.96 = 25
  4. l = 5

Answer: The ladder is 5 m long.

Watch out

  • Using Pythagoras on a triangle that is not right-angled.
  • Adding when the missing side is not the hypotenuse.
  • Forgetting to take the square root at the end.