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The Pythagorean Theorem

Overview

The legend itself.

What is it?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs (a² + b² = c²).

History

Used by rope-stretchers in Egypt to create perfect 90-degree corners for pyramids using a 3-4-5 rope.

Key Idea

Legs are short. Hypotenuse is long. Square them, add them, root them.

Practice This Topic

Concept Guide

Plain English: If you know two sides of a right angle (L-shape), you can calculate the diagonal connecting them. It connects Geometry (shapes) with Algebra (equations).

Real-world example: Navigation. If you fly 300 miles East and 400 miles North, this theorem tells you exactly how far you are from the airport.

How to do it

  1. Identify the Legs (a, b) touching the right angle.
  2. Identify the Hypotenuse (c) across from the right angle.
  3. Formula: a² + b² = c².
  4. Solve for the missing letter (Subtract if finding a leg, Add if finding hypotenuse).
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Common Pitfall

Forgetting the Square Root! If a² + b² = 25, the answer is not 25. It is 5.

Word Problem
"A ladder is 5m long. The base is 3m from the wall. How high does it reach?"
Reasoning: We have leg (3) and hypotenuse (5). 3² + b² = 5². 9 + b² = 25. b² = 16. Height is 4m.

Practice Examples

The 3-4-5
3² + 4² = 5² 9 + 16 = 25
Most common example.