In an equilateral triangle ABC, the height is CH = 39√3, find the side AB.

Since in any equilateral triangle the height is at the same time the median, then in this triangle ABC the point H divides the segment AB in half and the length of the segment AH is half the length of the segment AB.

Let us denote the length of the segment AB by x.

Since the triangle ABC is equilateral, then | AB | = | AC | = x and | AН | = x / 2.

Consider the ACН triangle.

Since the CH segment is the height, the ACH angle is 90 °.

Using the Pythagorean theorem, we can form the following equation:

(x / 2) ^ 2 + (39√3) ^ 2 = x ^ 2,

solving which, we get:

(x / 2) ^ 2 + 4563 = x ^ 2,

x ^ 2 – (x / 2) ^ 2 = 4563;

3x ^ 2/4 = 4563;

x ^ 2 = 4563 * 4/3;

x ^ 2 = 1521 * 4;

x = √ (1521 * 4) = 39 * 2 = 78.

Answer: | AB | = 78.