The perimeter of an equilateral triangle is 108√3, find its height.

Given:

equilateral triangle ABC,

R ABC = 108√3,

ВO – height.

Find the length of the ВO height -?

Decision:

Consider an equilateral triangle ABC. He has all sides are equal, that is, AB = BC = AC. Therefore AB = BC = AC = 108√3: 3 = 36√3.

The AO height is the median and bisector. Then AO = OС = 1/2 * AC = 1/2 * 36√3 = 18√3.

Consider a right-angled triangle ABO.

By the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):

AO ^ 2 + BO ^ 2 = AB ^ 2 (we express the legs BC ^ 2 from this equality);

BO ^ 2 = AB ^ 2 – AO ^ 2;

BO ^ 2 = (36√3) ^ 2 – (18√3) ^ 2;

BO ^ 2 = 3888 – 972;

BO ^ 2 = 2916;

BO = 54.

Answer: 54.