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

Solution:

Let the side of the triangle be a
Then,

\begin{aligned} a^{2} = {(\frac{a}{2})}^{2} + x^{2} \\ \Leftrightarrow \frac{3 a^{2}}{4} = x^{2} \\ \Leftrightarrow a^{2} = \frac{4 x^{2}}{3} \end{aligned}

∴ Area :

\begin{aligned} = \frac{\sqrt{3}}{4} a^{2} \\ = \frac{\sqrt{3}}{4} \times \frac{4 x^{2}}{3} \\ = \frac{x^{2}}{\sqrt{3}} \\ = \frac{\sqrt{3} x^{2}}{3} \end{aligned}

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