Find the side of an equilateral triangle if its height is 4 cm.

From the condition, we know that in an equilateral triangle, the height is 4 cm.

Let’s write the condition:

We are given Δ ABC, in which AB = BC = AC (equilateral triangle).

CH = 4 cm (triangle height).

We need to find the AC.

We know that in an equilateral triangle, the height is also the median. I.e:

AH = AB / 2.

Let’s introduce a variable, let АН = x, then АС = 2x.

We apply the Pythagorean theorem and solve the resulting incomplete quadratic equation:

(2x) ^ 2 – x ^ 2 = 4 ^ 2;

4x ^ 2 – x ^ 2 = 16;

3x ^ 2 = 16;

x ^ 2 = 16/3;

x = 4 / √3;

x = 4√3 / 3.

Looking for the side of the triangle:

AC = 2x = 2 * 4√3 / 3 = 8√3 / 3 cm.