National 5 Maths worked question

Find a missing rectangle side using Pythagoras

Practise using Pythagoras’ theorem to find a missing side, then check your method with a full worked solution.

Standard3 marksNon-calculator

Question

A rectangle has a diagonal of 17 cm and one side of 8 cm. Find the length of the other side.

Show the worked solution

Worked solution

  1. The diagonal and two sides form a right-angled triangle.

    A rectangle has right angles, and its diagonal is the hypotenuse of this triangle.

  2. x2 + 82 = 172

    Use Pythagoras with x as the unknown shorter side and 17 cm as the hypotenuse.

  3. x2 = 172 − 82 = 289 − 64 = 225

    Subtract the known shorter-side square from the hypotenuse square.

  4. x = √225 = 15 cm

    Take the positive square root because a side length must be positive.

Final answer: 15 cm

Why this method works

The rectangle's diagonal is opposite a right angle, so it is the hypotenuse. When a shorter side is unknown, subtract the square of the known side from the square of the hypotenuse before taking the square root.

Check your work

Substitute the result back into Pythagoras: 82 + 152 = 64 + 225 = 289 = 172. The 8–15–17 lengths are consistent.

Common mistakes

  • Adding 82 and 172 even though 17 cm is the hypotenuse
  • Stopping at x2 = 225 instead of finding x
  • Using the negative square root for a physical length.