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Pythagoras' Theorem Overview

Sep 3, 2025

Overview

This lecture explains how to use Pythagoras' theorem to find the missing side length in right-angled triangles, with step-by-step worked examples.

When to Use Pythagoras' Theorem

  • Only use Pythagoras’ theorem with right-angled triangles (triangle with a 90° angle).
  • You must know the lengths of two sides and be asked to find the third side.
  • The theorem does not apply if there is no right angle.

The Pythagoras' Theorem Formula

  • The formula is: a² + b² = c².
  • 'c' is always the hypotenuse (longest side, opposite the right angle).
  • 'a' and 'b' are the other two sides; order does not matter.

Solving for the Hypotenuse

  • Label sides with a, b, and c, with c as the hypotenuse.
  • Substitute known values into the formula (e.g., 4² + 3² = c²).
  • Simplify and solve (e.g., 16 + 9 = 25; √25 = 5).
  • The missing length is the positive square root (e.g., c = 5).

Solving for Other Sides

  • If finding a missing non-hypotenuse side, rearrange: c² - b² = a² (or similar).
  • Always use the square root at the end to solve for the side length.

Worked Example Summaries

  • For sides 1.7 and 3.2, use 1.7² + 3.2² = x²; x = 3.62 (rounded to three significant figures).
  • For sides 5.6 and 10.5, 5.6² + 10.5² = c²; c = 11.9.
  • For sides 8 and 11, 8² + 11² = c²; c = 13.6.

Interpreting Triangle Notation

  • Exam questions may use letters (like X, Y, Z) for corners and side labels.
  • Identify which side is being asked for, then label sides a, b, and c for theorem use.
  • Ignore original corner labels once sides are relabeled for calculation.

Key Terms & Definitions

  • Right-Angled Triangle — a triangle with one 90° angle.
  • Hypotenuse — longest side opposite the right angle, labeled 'c'.
  • Pythagoras’ Theorem — formula a² + b² = c² relating side lengths in a right-angled triangle.
  • Significant Figures — number of important digits to round the answer for precision.

Action Items / Next Steps

  • Practice identifying right-angled triangles and labeling sides correctly.
  • Memorize the formula a² + b² = c² for exams.
  • Complete homework or practice problems involving Pythagoras’ theorem as assigned.

Full transcript