GCSE Revision

Pythagoras' theorem

M6 formulae

Pythagoras' theorem

In a right-angled triangle, the is the longest side, opposite the right angle. links the three sides:

a² + b² = c², where c is the hypotenuse (on the formulae sheet)

abc

To find the hypotenuse:

  1. Square the two shorter sides.
  2. Add them.
  3. Square root the answer.

Example: the two shorter sides are 6 cm and 8 cm. Find the hypotenuse.

  1. 6² + 8² = 36 + 64 = 100
  2. √100 = 10 cm

Example: the two shorter sides are 5 cm and 7 cm. Find the hypotenuse, to 1 decimal place.

  1. 5² + 7² = 25 + 49 = 74
  2. √74 = 8.60… = 8.6 cm

Key terms

Finding a shorter side

To find one of the shorter sides, subtract instead of adding.

  1. Square the hypotenuse and the side you know.
  2. Subtract the smaller from the bigger.
  3. Square root the answer.

Example: the hypotenuse is 13 cm and one side is 5 cm. Find the other side.

  1. 13² − 5² = 169 − 25 = 144
  2. √144 = 12 cm

Example: a 4 m ladder leans against a wall. Its foot is 1.5 m from the wall. How high up the wall does it reach, to 2 decimal places?

  1. The ladder is the hypotenuse: 4² − 1.5² = 16 − 2.25 = 13.75
  2. √13.75 = 3.708… = 3.71 m

Common mistake

Check your answer is shorter than the hypotenuse. If it's longer, you added when you should have subtracted.

Is a triangle right-angled?

Pythagoras only works for right-angled triangles, so you can use it to test one.

  1. Square the longest side.
  2. Square the other two sides and add them.
  3. If the answers are equal, the triangle is right-angled.

Example: is a triangle with sides 9 cm, 12 cm and 15 cm right-angled?

  1. 15² = 225
  2. 9² + 12² = 81 + 144 = 225
  3. Yes, because they are equal

Example: is a triangle with sides 6 cm, 7 cm and 9 cm right-angled?

  1. 9² = 81
  2. 6² + 7² = 36 + 49 = 85
  3. No, because 85 isn't 81

Key terms

Distance between two points

The line between two points is the hypotenuse of a right-angled triangle. Its other sides are the differences in x and in y.

  1. Find the difference in the x-coordinates.
  2. Find the difference in the y-coordinates.
  3. Use Pythagoras.

Example: find the distance between (−3, 1) and (2, 13).

  1. Difference in x: 2 − (−3) = 5
  2. Difference in y: 13 − 1 = 12
  3. √(5² + 12²) = √169 = 13 units

The order you subtract in doesn't matter, because squaring makes the difference positive.

Take the quiz