Pythagoras' Theorem

Understand Pythagoras’ Theorem

  • Pythagoras’ Theorem is a fundamental rule in Geometry that applies to any right-angled triangle.
  • The theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
  • This is often represented as a² + b² = c² where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

Applying Pythagoras’ Theorem

  • Pythagoras’ theorem can be used to find a missing side in a right-angled triangle. You always need to know the lengths of two of the sides.
  • If you know the lengths of the two shorter sides (a and b), square each of these lengths, add the results together, and then find the square root of this sum to find the length of the hypotenuse (c).
  • If you know the length of the hypotenuse (c) and one of the shorter sides (a), square these lengths, subtract the square of the known shorter side from the square of the hypotenuse, and then find the square root of this difference to find the length of the unknown side (b).

Pythagorean Triples

  • A Pythagorean triple is a set of three whole-number side lengths that satisfies Pythagoras’ theorem.
  • Common examples of Pythagorean triples include (3,4,5), (5,12,13), and (8,15,17). These can be scaled up – for example (6,8,10) is a Pythagorean triple because it’s a multiple of (3,4,5).
  • Pythagorean triples can save you time on calculations if you recognise them.

Real-world Applications of Pythagoras’ Theorem

  • Pythagoras’ theorem isn’t just theoretical – it has practical real-world applications.
  • For example, it can be used in construction and navigation, in computer graphics and game development, in architecture, and in field such as astronomy and physics.
  • Any time a right-angled triangle can be formed and you know two of the sides, Pythagoras’ theorem can be used to find the third side.

Remember to practice a range of problem types to ensure you’re fully prepared to use Pythagoras’ theorem in any situation. Also, always show your working out in exams to make sure you get all possible method marks.