30-60-90 Triangle

30-60-90 Triangle

How to solve 30-60-90 triangle problems: definition, property, examples, and their solutions.

Definition, Property

30-69-90 triangle is a triangle whose measures of angles are 30, 60, and 90 degrees. The ratio of its sides is 1 : square root 3 : 2.

30-60-90 triangle is a triangle
whose measures of interior angles are
30º, 60º, and 90º.

The ratio of its sides is 1 : √3 : 2.

Example 1

Find the value of x. The lengths of the right triangle's legs: 2, x. The measure of the right triangle's non-rignt angle: 30 degrees.

The measure of the right triangle's
non-right angle is 30º.
So this is the 30-60-90 triangle.

Draw a 30-60-90 triangle next to the given triangle.
(with its sides → 1 : √3: 2)

The given triangle is × 2 bigger.

So x = 2⋅√3.

Example 2

Find the value of x. The length of the right triangle's leg: 5. The length of the right triangle's hypotenuse: x. The measure of the right triangle's non-rignt angle: 60 degrees.

The measure of the right triangle's
non-right angle is 30º.
So this is the 30-60-90 triangle.

Draw a 30-60-90 triangle next to the given triangle.
(with its sides → 1 : √3: 2)

It's hard to find the scale factor.
But these two triangles are similar.
So set a proportion between the triangles:
x / 2 = 5 / √3.

Similarity of sides in triangles

Rationalizing the denominator