Tick the correct answer and justify : ABC and BDE are two equilat…

MCQ

Tick the correct answer and justify : $\text{ABC}$ and $\text{BDE}$ are two equilateral triangles such that $D$ is the mid $-$ point of $BC$. Ratio of the areas of triangles $\text{ABC}$ and $\text{BDE}$ is :

A
$2 : 1$
B
$1 : 2$
✓
$4 : 1$
D
$1 : 4$

✓

Answer

Correct option: C.

$4 : 1$

$\triangle\text{ABC}$ and $ \triangle\text{BDE}$ are both equilateral triangles.
$\therefore\ \triangle\text{ABC}\sim\triangle\text{BDE}$
$[$Using $\text{AAA}$ similar condition$]$.
We know that the ratio of the areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides.
$\therefore\ \frac{\text{ar}(\triangle\text{ABC})}{\text{ar}(\triangle\text{BDE})}=\frac{\text{AB}^2}{\text{BD}^2}$
$\Rightarrow\ \frac{\text{ar}(\triangle\text{ABC})}{\text{ar}(\triangle\text{BDE})}=\Big(\frac{\text{BC}}{\text{BD}}\Big)^2$
$[\therefore AB = BC = CA]$
$\Rightarrow\ \frac{\text{ar}(\triangle\text{ABC})}{\text{ar}(\triangle\text{BDE})}= \Big(\frac{2\text{BD}}{\text{BD}}\Big)^2$
$\Rightarrow\ \frac{\text{ar}(\triangle\text{ABC})}{\text{ar}(\triangle\text{BDE})}=\frac{4}{1}$