Question

If x is the length of a median of an equilateral triangle then what is its area?

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

Answer 1

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

In an equilateral triangle the median and the altitude coincide Let a be the length of each side of the equilateral triangle
In rt $\angle d\Delta ADB$
$A{B}^2=A{D}^2+B{D}^2$
$\Rightarrow$$a^2=x^2+{\left(a/2\right)}^2\Rightarrow a^2-\frac{a^2}{4}=x^2$
$\Rightarrow$$\frac{3a^2}{4}=x^2\Rightarrow a^2=\frac{4x^2}{3}$
Area of equilateral $\Delta =\frac{\sqrt{3}}{4}{a}^2$

$=\frac{\sqrt{3}}{4}\times \frac{4}{3}\times x^2=\frac{x^2}{\sqrt{3}}$