If x is the length of a median of an equilateral triangle, then its area is :

Answer: Option D

Let the side of the triangle be a
Then,
$$\begin{array}{rl}& a^2={\left(\frac{a}{2}\right)}^2+x^2\\ & \iff \frac{3a^2}{4}=x^2\\ & \iff a^2=\frac{4x^2}{3}\end{array}$$
∴ Area :
$$\begin{array}{rl}& =\frac{\sqrt{3}}{4}{a}^2\\ & =\frac{\sqrt{3}}{4}\times \frac{4x^2}{3}\\ & =\frac{x^2}{\sqrt{3}}\\ & =\frac{\sqrt{3}{x}^2}{3}\end{array}$$