Pythagoras' Theorem

Pythagoras’ Theorem

  • Pythagoras’ Theorem is named after the ancient Greek mathematician, Pythagoras.
  • The theorem applies to right-angled triangles.
  • It states: “In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.”
    • This is usually written 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 the Theorem

  • You can apply the theorem to find the length of one side of a right-angled triangle if you know the lengths of the other two sides.
  • To find the length of the hypotenuse (c):
    • Square the lengths of the other two sides (a and b).
    • Add these two values together.
    • Take the square root of the result.
  • To find the length of one of the other sides (a or b):
    • Square the length of the hypotenuse (c).
    • Subtract the square of the other side (b or a).
    • Take the square root of the result.

Problem Solving with Pythagoras’ Theorem

  • Pythagoras’ Theorem is used in various problem-solving and real-life situations.
  • For instance:
    • Determining the shortest path (as the crow flies).
    • Calculating the diagonal distance on a screen or canvas.
    • Measuring the height of a tree using a shadow.
  • Always sketch and label the triangle implied in the problem.
  • Remember to use appropriate units in your answer.

Checking for Right Angles

  • If a triangle has sides of length a, b and c, and a² + b² = c², then the triangle is right-angled.
  • Conversely, if a² + b² ≠ c² then the triangle is not right-angled.
  • This check can be useful in geometrical proofs and problem-solving.

Practice and Proficiency

  • The key to mastery of Pythagoras’ theorem is practice.
  • Work through as many example problems as possible.
  • Regularly test your understanding and improve your speed at using the theorem.
  • Making mistakes is an important part of the learning process, so don’t be discouraged if you get things wrong occasionally.
  • Keep checking your rough work and make sure your final answers are reasonable or expected based on the given problem.