Question

What is a 30-60-90 triangle? Please give an example.

Answer 1

A 30-60-90 triangle is a right triangle with angles `30^@`, `60^@`, and `90^@` and which has the useful property of having easily calculable side lengths without use of trigonometric functions.

Explanation:

A 30-60-90 triangle is a special right triangle, so named for the measure of its angles. Its side lengths may be derived in the following manner.

Begin with an equilateral triangle of side length `x` and bisect it into two equal right triangles. As the base is bisected into two equal line segments, and each angle of an equilateral triangle is `60^@`, we end up with the following

Because the sum of the angles of a triangle is `180^@` we know that `a = 180^@ - 90^@ - 60^@ = 30^@`

Furthermore, by the Pythagorean theorem, we know that
`(x/2)^2 + h^2 = x^2`
`=>h^2 = 3/4x^2`
`=>h = sqrt(3)/2x`

Therefore a 30-60-90 triangle with hypotenuse `x` will look like

For example, if `x = 2`, the side lengths of the triangle will be `1`, `2`, and `sqrt(3)`