The height of an equilateral triangle is 7. Find its area.

By condition, triangle ABC is equilateral, its height BH is also the bisector and median of the triangle, therefore it divides the base of the AC into two equal segments. AH = CH.

Let the segment AH = X cm, then the sides of the triangle AB = BC = AC = 2 * X cm.

From the right-angled triangle ABH, by the Pythagorean theorem, we define the leg AH.

AH ^ 2 = AB ^ 2 – BH ^ 2.

(2 * X) ^ 2 = X ^ 2 – 7 ^ 2.

4 * X ^ 2 = X ^ 2 – 49.

3 * X ^ 2 = 49.

X = 7 / √3.

AH = 7 / √3 cm, then AB = BC = AC = 2 * 7 / √3 = 14 / √3 cm.

Determine the area of the triangle.

S = AC * BH / 2 = (14 / √3) * 7/2 = 49 / √3 cm2.

Answer: The area of the triangle is 49 / √3 cm2.