45-45-90 Triangle

45-45-90 Triangle

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

Definition, Properties

45-45-90 triangle is a triangle whose measures of angles are 45, 45, and 90 degrees. The ratio of its sides is 1 : 1 : square root 2.

45-45-90 triangle is a triangle
whose measures of interior angles are
45º, 45º, and 90º.

It's the only isosceles right triangle.

The ratio of its sides is 1 : 1 : √2.

Example 1

Find the value of x. The lengths of the right triangle's legs: 7, x. The measure of the right triangle's non-rignt angle: 45 degrees.

The measure of the right triangle's
non-right angle is 45º.
So this is the 45-45-90 triangle.

Then the ratio of its sides is 1 : 1 : √2.

So x = 7.

Example 2

Find the value of x. The lengths of the right triangle's legs: 11, 11. The length of the right triangle's hypotenuse: x.

The given triangle is an isosceles right triangle.

So this is the 45-45-90 triangle.

Draw a 45-45-90 triangle next to the given triangle.
(with its sides → 1 : 1 : √2)

The given triangle is × 11 bigger.

So x = 11⋅√2.