Find the hypotenuse of a right-angled triangle if one of the legs is 5 ^ 3 cm, and the angle adjacent to this leg is 30 degrees.

Given: right-angled triangle ABC;
angle C = 90;
leg AC = 5√3 centimeters = 125 centimeters;
angle A = 30 degrees.
Find: hypotenuse AB -?
Solution:
Consider a right-angled triangle ABC. In a right-angled triangle opposite an angle of 30 degrees lies a leg, which is half the hypotenuse. Then
BC = 1/2 * AB;
AB = 2 * BC
By the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):
AC ^ 2 + BC ^ 2 = AB ^ 2;
AC ^ 2 + BC ^ 2 = 4BC ^ 2;
75 + BC ^ 2 = 4BC ^ 2;
3 * BC ^ 2 = 75;
BC ^ 2 = 25;
BC = √25;
BC = 5;
AB = 2 * 5 = 10 centimeters.
Answer: AB = 10 centimeters.