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

A. x²

B. $$\frac{1}{2}{x^2}$$

C. $$\frac{{\sqrt 3 }}{2}{x^2}$$

D. $$\frac{{\sqrt 3 }}{3}{x^2}$$

Answer: Option D

Solution(By Examveda Team)

Let the side of the triangle be a
Then,
$$\eqalign{ & {a^2} = {\left( {\frac{a}{2}} \right)^2} + {x^2} \cr & \Leftrightarrow \frac{{3{a^2}}}{4} = {x^2} \cr & \Leftrightarrow {a^2} = \frac{{4{x^2}}}{3} \cr} $$
∴ Area :
$$\eqalign{ & = \frac{{\sqrt 3 }}{4}{a^2} \cr & = \frac{{\sqrt 3 }}{4} \times \frac{{4{x^2}}}{3} \cr & = \frac{{{x^2}}}{{\sqrt 3 }} \cr & = \frac{{\sqrt 3 {x^2}}}{3} \cr} $$