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

1 Answer
sente
Nov 13, 2015

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
enter image source here
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
enter image source here

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

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