In triangle ABC, angle C = 90 degrees, angle a = 30 degrees. AC = 39√3 find BC.

To solve the problem, we will use the property of a right-angled triangle. The leg opposite to the 30˚ angle is equal to half of the hypotenuse.

BC = 2 * AB.

Let us express the side BC in terms of x. Then AB = 2 * x.

Let’s use the Pythagorean theorem – the square of the hypotenuse is equal to the sum of the squares of the legs.

(2 * x) ^ 2 = x ^ 2 + (39√3) ^ 2.

4 * x ^ 2 = x ^ 2 + (39) ^ 2 * 3.

4 * x ^ 2 – x ^ 2 = (39) ^ 2 * 3.

3 * x ^ 2 = (39) ^ 2 * 3.

Reduce both parts by 3.

x ^ 2 = (39) ^ 2.

x = 39.

Answer: BC = 39