The height of an equilateral triangle is 13√3. Find a side of it.

Given:

isosceles triangle ABC,

ВН – height,

BH = 13√3.

Find the length of the side of an equilateral triangle ABC, that is, AB -?

Decision:

1. Consider an equilateral triangle ABC. The HL height is also the median. Then HC = AH = AB / 2;

2. Consider a right-angled triangle ABН.

Then, by the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):

AH ^ 2 + BH ^ 2 = AB ^ 2;

Let AB = x, then BH = x / 2;

169 * 3 + x ^ 2/4 = x ^ 2;

3 * x ^ 2 = 169 * 3 * 4;

x ^ 2 = 169 * 4;

x = 13 * 2;

x = 26.

Answer: 26.