Write True or False and justify your answer:
$ABC$ and $BDE$ are two equilateral triangles such that $D$ is the mid-point of $BC.$ Then $\operatorname{ar}(\triangle BDE) = \frac{1}{4} \operatorname{ar}(\triangle ABC).$

(A) Let the side length of the equilateral triangle $ABC$ be $a$.
Since $D$ is the mid-point of $BC$,the side length of the equilateral triangle $BDE$ is $\frac{a}{2}$.
The area of an equilateral triangle with side $s$ is given by $\frac{\sqrt{3}}{4} s^2$.
$\operatorname{ar}(\triangle ABC) = \frac{\sqrt{3}}{4} a^2$.
$\operatorname{ar}(\triangle BDE) = \frac{\sqrt{3}}{4} \left(\frac{a}{2}\right)^2 = \frac{\sqrt{3}}{4} \cdot \frac{a^2}{4} = \frac{1}{4} \left(\frac{\sqrt{3}}{4} a^2\right)$.
Therefore,$\operatorname{ar}(\triangle BDE) = \frac{1}{4} \operatorname{ar}(\triangle ABC)$.
Hence,the given statement is True.