Pythagoras’ Theorem

Pythagoras’ Theorem applies to right-angled triangles and states that:

“The square of the hypotenuse is equal to the sum of the squares of the other two sides.”

Formula: \( a^2 + b^2 = c^2 \)

• \( c \) = hypotenuse (longest side, opposite the 90° angle)
• \( a, b \) = other two sides (legs)

Important Properties and Notes:

• Only applies to right-angled triangles.
• The hypotenuse is always the longest side and opposite the right angle.
• The theorem can be used to:
  • Find the length of a missing side.
  • Check if a triangle is right-angled by verifying \( a^2 + b^2 = c^2 \).
• The converse of Pythagoras’ theorem: If \( a^2 + b^2 = c^2 \), then the triangle is right-angled.
• Extended to 3D: \( d = \sqrt{x^2 + y^2 + z^2} \) for the distance in 3D space.
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Steps to Use Pythagoras’ Theorem:

1. Check if the triangle is right-angled (or if given).
2. Identify the hypotenuse (longest side).
3. Apply the formula: \( a^2 + b^2 = c^2 \).
4. Solve for the unknown side (take square root if needed).

Common Uses in Real Life:

• Finding the shortest distance between two points (straight line).
• Construction and architecture for ensuring right angles.
• Navigation and mapping (finding distances using coordinates).