In a right-angled triangle, the leg is 18 and the included angle is 30 degrees. Find the angle and sides.

The sum of the angles of a triangle is 180 °, so the unknown angle of a given right triangle is

180 ° – 90 ° – 30 ° = 60 °.

By the property of a right-angled triangle, if one of the angles is 30 °, then the leg opposite to this angle is half the hypotenuse. If the leg against an angle of 30 ° is x, then the hypotenuse is 2x. By the Pythagorean theorem

(2x) ^ 2 = x ^ 2 + 18 ^ 2,

where

4 * x ^ 2 – x ^ 2 = 18 ^ 2;

3 * x ^ 2 = 324;

x ^ 2 = 108;

x = 6 * sqrt (3) – length of the unknown leg.

Then the length of the hypotenuse is 2 * x = 2 * 6 * sqrt (3) = 12 * sqrt (3).