Special Right Triangles: 45-45-90
A 45-45-90 triangle is an isosceles right triangle. Its sides are always in the ratio .
Introduction
Cut a square diagonally — you get two 45-45-90 triangles. This is the simplest “special” right triangle because both legs are equal. If you know one leg, you know everything.
Past Knowledge
Pythagorean Theorem (8.1.1). Isosceles triangles (5.1). Square roots.
Today's Goal
Find all sides of a 45-45-90 triangle given any one side.
Future Success
30-60-90 (8.2.2), trigonometry (8.2.3+), unit circle.
Key Concepts
The 45-45-90 Ratio
- Given a leg : hypotenuse =
- Given hypotenuse : each leg =
Theorem & Proof
Derivation: 45-45-90 Side Ratio
Given: Isosceles right triangle with legs and
Derive: hypotenuse in terms of
| # | Statement | Reason |
|---|---|---|
| 1 | Pythagorean Theorem with both legs = | |
| 2 | Combine like terms | |
| 3 | Take the positive square root of both sides |
∎ The hypotenuse is always times the leg.
Worked Examples
Leg → Hypotenuse
Each leg = 7. Find the hypotenuse.
Hypotenuse → Leg
Hypotenuse = 10. Find each leg.
Square Diagonal
A square has side length 6. Find the diagonal.
A diagonal cuts a square into two 45-45-90 triangles. Legs = 6.
Common Pitfalls
Multiplying by √2 When You Should Divide
Given a leg → multiply by to get the hypotenuse. Given the hypotenuse → divide by to get a leg. Don't mix them up.
Not Rationalizing
should be written as . Always rationalize the denominator on tests.
Real-Life Applications
TV Screen Sizes
A square computer monitor with 20" sides has a diagonal of inches — that's the “screen size” that would be advertised.
Baseball — Throwing Distance
A baseball diamond is a 90 ft square. The throw from home to second base crosses the diagonal: feet — the longest throw an infielder makes.
Practice Quiz
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