The Pythagorean Theorem
In any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: .
Introduction
The Pythagorean Theorem is arguably the most famous theorem in all of mathematics. Known for over 4,000 years with hundreds of distinct proofs, it connects the three sides of every right triangle in one elegant equation.
Past Knowledge
Right triangles (5.2). Similarity & geometric mean (7.3.2). Area of squares.
Today's Goal
State, prove, and apply the Pythagorean Theorem to find missing sides.
Future Success
Pythagorean Triples (8.1.2), Converse (8.1.3), Trigonometry (8.2), Distance Formula.
Key Concepts
The Pythagorean Theorem
In a right triangle with legs and and hypotenuse :
The hypotenuse is always the longest side, opposite the right angle.
Finding a Missing Side
- Missing hypotenuse:
- Missing leg:
Theorem & Proof
Two-Column Proof: The Pythagorean Theorem (via Similar Triangles)
Given: Right with right angle at , altitude to hypotenuse
Prove: where
Strategy: Use the Altitude-on-Hypotenuse Theorem (7.3.2) — the leg rule — then add the results.
| # | Statement | Reason |
|---|---|---|
| 1 | Let and , so | Segment Addition on |
| 2 | and | AA Similarity (altitude to hypotenuse creates similar triangles — Lesson 7.3.2) |
| 3 | , so | Leg rule from |
| 4 | , so | Leg rule from |
| 5 | Add steps 3 and 4; factor out | |
| 6 | Substitute from step 1 |
∎ This elegant proof by similar triangles dates back to Euclid. Each leg squared equals the hypotenuse times the adjacent segment, and those segments sum to the whole hypotenuse.
Worked Examples
Finding the Hypotenuse
A right triangle has legs 6 and 8. Find the hypotenuse.
Finding a Missing Leg
Hypotenuse = 13, one leg = 5. Find the other leg.
Irrational Hypotenuse
Legs are 5 and 9. Find the hypotenuse in simplest radical form.
106 has no perfect square factors (106 = 2 × 53), so this is already simplified.
Common Pitfalls
Using a² + b² = c² When It's Not a Right Triangle
The Pythagorean Theorem only works for right triangles. For other triangles, you need the Law of Cosines (8.3.4).
Putting the Hypotenuse on the Wrong Side
must be the hypotenuse (longest side, opposite 90°). Writing will give wrong answers if is actually the hypotenuse.
Real-Life Applications
Construction — Squaring a Corner
Builders use a 3-4-5 rope to check if corners are perfectly square. If a 3-foot and 4-foot measurement along two walls gives exactly 5 feet diagonally, the corner is 90°.
Navigation — Shortest Distance
GPS systems calculate “as the crow flies” distances using the Pythagorean Theorem on coordinate grids, computing the diagonal from east-west and north-south components.
Practice Quiz
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