The height of an equilateral triangle is 3cm, find the perimeter of the triangle?

Given:
equilateral triangle ABC,
ВН – height,
BH = 3 centimeters.
Find the perimeter of the triangle ABC, that is, P ABC -?
Decision:
1. Consider an equilateral triangle ABC. The ВН height is the median. Then НС = AН.
2. Consider a right-angled triangle ВНС.
Let BC = x centimeters, then HC = x / 2 centimeters. Then, by the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):
HC ^ 2 + BH ^ 2 = BC ^ 2:
(x / 2) ^ 2 + 3 ^ 2 = x ^ 2;
x ^ 2/4 + 9 = x ^ 2;
x ^ 2 + 36 = 4x ^ 2;
36 = 3x ^ 2;
x ^ 2 = 36: 3;
x ^ 2 = 12;
x = 2√3 centimeters.
3. P ABC = 2√3 * 3 = 6√3 centimeters.
Answer: 6√3 centimeters.