The Pythagorean Theorem
📐 Big Idea
The Pythagorean Theorem is one of the most fundamental relationships in all of mathematics. It connects the three sides of a right triangle in a beautifully simple equation that has been known for over 2,500 years.
The Theorem
In any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides (called legs).
$$ a^2 + b^2 = c^2 $$
- $a$ and $b$ are the lengths of the two legs (the sides that form the right angle)
- $c$ is the length of the hypotenuse (the side opposite the right angle — always the longest side)
Key Vocabulary
| Right Triangle | A triangle with exactly one $90°$ angle |
| Hypotenuse | The longest side, opposite the right angle |
| Legs | The two shorter sides that form the right angle |
| Pythagorean Triple | A set of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$ |
| Converse | If $a^2 + b^2 = c^2$, then the triangle is a right triangle |
🔍 Interactive: Explore the Right Triangle
Interactive visualization — available in the full lesson