# 45-45-90 Triangle

How to solve 45-45-90 triangle problems: definition, property, examples, and their solutions.

## Definition, Property

A 45-45-90 triangle is a triangle

whose interior angles are 45º, 45º, and 90º.

The ratio of its sides is

1 : 1 : √2.

Its legs are congruent.

So a 45-45-90 triangle is an isosceles right triangle.

Isosceles triangle

## Example 1

The interior angles are 90º and 45º.

So this is a 45-45-90 triangle.

So the upper angle is 45º.

The ratio of its sides is 1 : 1 : √2.

So the legs are congruent.

The legs are *x* and 7.

So *x* = 7.

## Example 2

The given triangle is an isosceles right triangle.

Isosceles triangle

So this is a 45-45-90 triangle.

Draw a 45-45-90 triangle,

whose sides are 1, 1, and √2,

next to the given right triangle.

Then these two triangles are similar.

Since these two triangles are similar,

their sides are proportional.

So *x*/√2 = 11/1.

Similar triangles

Multiply √2 on both sides.

Then *x* = 11√2.