How can I express the area (A) of an equilateral triangle as a function of its side (X)?

Question

Although I have the solution to this problem, I'm completely unsure of how to reach that solution. The illustration displays an equilateral triangle (although split into two triangles) with a base of x, a height of h, and the angles being 30°, 60°, and 90°. Since the triangle is essentially split into two triangles, the bases are labeled as 1/2x.

The solution

The first step of the solution shows...

(h)² + (1/2x)² = x²

and I honestly have no idea how this point was reached. The problem was worked through until a solution for h was found, and it was then substituted into the equation for the area of a triangle, a = (1/2)(x)(h).

My first attempt at this problem yielded an incorrect A(x) = (1/2)(x)(h)

The final correct solution was A(x) = √3/4 x²

My apologies if I have not included enough information. I understand the illustration would have been very beneficial.

Answer 1

The first step (h)² + (1/2x)² = x² comes from using the Pythogorean Theorem (a²+b²=c²)on one of the smaller triangles. The a and b are the legs and c is the hypotenuse. Are you able to manipulate the equation to solve for h and then substitute in to find the are from there?