45-45-90 Triangle

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 45, 45, and 90 degrees. The ratio of its sides is 1 : 1 : sqrt[2]. So it's an isosceles right triangle.

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

Find the value of x.

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

Find the value of x.

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.