The side of an equilateral triangle is 12√3. Find the height of the triangle.

Given:
equilateral triangle ABC,
AB = 12√3,
ВO – height.
Find the length of the height, that is, ВO -?
Decision:
1. Consider an equilateral triangle ABC. By the definition of an equilateral triangle, his sides are equal to each other, that is, AB = BC = AC = 12√3.
2. In an equilateral triangle, the height is also the median, then AO = 1/2 * AC = 1/2 * 12√3 = 6√3.
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 BO ^ 2 from this equality);
BO ^ 2 = AB ^ 2 – AO ^ 2;
BO ^ 2 = (12√3) ^ 2 – (6√3) ^ 2;
ВO ^ 2 = 432 – 108;
BO ^ 2 = 324;
BO = 18.
Answer: 18.