45-45-90 Triangle

How to find the sides of the given 45-45-90 triangle: definition, 2 examples, and their solutions.

Definition

Definition

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

Example

Solution

The given interior angles of the triangle are
90º and 45º.

So this triangle is a 45-45-90 triangle.

So the top angle is 45º.

A 45-45-90 triangle is an isosceles right triangle.

So the blue sides are congruent.

So x is equal to the other side: 7.

So x = 7.

Example 2

Example

Solution

The given triangle is an isosceles right triangle.

So the given triangle is a 45-45-90 triangle.

So draw a 45-45-90 triangle
whose sides are 1, 1, and √2.

These two triangles are similar.

Then their sides are proportional.

So x/√2 = 11/1.

Similar Triangles

11/1 = 11

x/√2 = 11

Multiply √2 to both sides.

Then x = 11√2.

So x = 11√2.