Lesson 3.9

Special Right Triangles (45-45-90)

The isosceles right triangle is the first of two “magic” triangles that make exact trig values possible. Half of the unit circle coordinates come from this triangle alone.

Introduction

A 45-45-90 triangle is an isosceles right triangle — two equal legs with a right angle between them. By deriving its exact side ratios, we unlock the trig values for () without a calculator.

Past Knowledge

The Pythagorean Theorem: .

Today's Goal

Derive the side ratio and connect it to the unit circle coordinate .

Future Success

The value appears in every quadrant of the unit circle at .

Key Concepts

Deriving the Ratio

Start with two equal legs of length . By the Pythagorean Theorem:

So the side ratio of a 45-45-90 triangle is:

45-45-90 Side Ratio

Visualizing the 45-45-90 Triangle

Connecting to the Unit Circle

On the unit circle, the hypotenuse IS the radius and equals . To scale a 45-45-90 triangle so the hypotenuse is , divide every side by :

Since the legs correspond to the and coordinates:

Unit circle point at :

Complete Trig Values at 45°

Function	Value

Worked Examples

Basic

Finding the Hypotenuse

Question: A 45-45-90 triangle has legs of length . Find the hypotenuse.

Step 1: Use the ratio. In a 45-45-90 triangle, hypotenuse = leg × .

Final Answer: Hypotenuse

Intermediate

Finding the Leg from the Hypotenuse

Question: A 45-45-90 triangle has hypotenuse . Find the length of each leg.

Step 1: Set up the ratio. Hypotenuse = leg × , so leg = hypotenuse ÷ .

Final Answer: Each leg

Advanced

Unit Circle Application

Question: Using the 45-45-90 triangle, find the exact value of .

Step 1: Recall the values.

Step 2: Add.

Final Answer:

Common Pitfalls

Putting √2 on the Leg Instead of the Hypotenuse

The factor belongs on the hypotenuse, not the legs. The legs are the shorter, equal sides. If you multiply the leg by , you get the hypotenuse. If you divide the hypotenuse by , you get a leg.

Writing √2/2 as √2 or 1/√2

On exams, the standard form is (rationalized denominator). Writing is mathematically correct but typically marked as unsimplified.

Real-Life Applications

Diagonal of a Square

Cutting a square along its diagonal creates two 45-45-90 triangles. Architects use this to calculate the diagonal bracing needed for square window frames and floor tiles. A -inch square tile has a diagonal of inches.