Question

If x is the length of a median of an equilateral triangle, then its area in terms of x = ?

• A. $x^2$
• B. $\frac{x^2\sqrt{3}}{2}$
• C. $\frac{x^2\sqrt{3}}{3}$
• D. $\frac{x^2}{2}$

Answer 1

The correct option is C $\frac{x^2\sqrt{3}}{3}$

Consider an equilateral triangle $ABC$ having sides $a$ and a median $AD$ of length $x$ unit'.
In an equilateral triangle, the median is always the perpendicular bisector of the triangle.
So, $BD=a/2$
In triangle $ABD$, by pythagoras theorem, we have
$A{B}^2=A{D}^2+B{D}^2⟹a^2=x^2+(\frac{a}{2}{)}^2⟹a^2=x^2+\frac{a^2}{4}⟹\frac{{3a}^2}{4}=x^{2}or,a^2=\frac{{4x}^2}{3}$
Now, area of equilateral triangle $=\frac{\sqrt{3}{a}^2}{4}=\frac{\sqrt{3}}{4}\times \frac{{4x}^2}{3}=\frac{{\sqrt{3}x}^2}{3}$