Pythagoras’ Theorem

Statement

In a right-angled triangle with shorter sides \(a\), \(b\), and hypotenuse \(c\):

\[ a^2 + b^2 = c^2 \]

Why it’s true

• Squares built on the sides of a right triangle have areas related this way.
• The area of the square on the hypotenuse equals the sum of the areas on the other two sides.
• It is a geometric property unique to right-angled triangles.

Recipe (how to use it)

1. Square both shorter sides \(a\) and \(b\).
2. Add them together to find \(c^2\).
3. Square root the result to get \(c\).
4. If solving for a shorter side: rearrange as \(a^2 = c^2 - b^2\).

Spotting it

This formula applies only in right-angled triangles.

Common pairings

• Trigonometry (sine, cosine, tangent).
• Distance formula in coordinate geometry.
• 3D Pythagoras problems (space diagonals).

Mini examples

1. Given: \(a=3, b=4\).
   Answer: \(c=\sqrt{3^2+4^2}=\sqrt{25}=5\).
2. Given: \(c=13, b=5\).
   Answer: \(a=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12\).

Pitfalls

• Forgetting to square root at the end.
• Using the formula in triangles that are not right-angled.

Exam strategy

• Check for a right angle before applying the formula.
• Always square root your final result if solving for a length.

Summary

Pythagoras’ theorem links the sides of a right triangle: \(a^2+b^2=c^2\). It is essential in geometry and distance calculations.