The height of an equilateral triangle is 13√3. Find the side of this triangle.

It is known from the condition that the height of an equilateral triangle is 13√3. In order to find the side of this triangle, we will use the Pythagorean theorem.

So, we know that in an equilateral triangle, the height is the median.

Let us introduce the notation: h is the height, a is the side of the triangle.

We got two identical right-angled triangles formed by side, height and half side (a / 2) on which the height rests and apply the Pythagorean theorem to it.

a ^ 2 = h ^ 2 + (a / 2) ^ 2;

a ^ 2 = h ^ 2 + a ^ 2/4;

3a ^ 2/4 = h ^ 2;

a ^ 2 = 4h ^ 2/3;

a ^ 2 = 4 (13√3) ^ 2/3 = (4 * 13 ^ 2 * 3) / 3 = (2 * 13) ^ 2;

a = 2 * 13 = 26.

Answer: side of triangle 26.