Question

$△ABC$ and $△BDE$ are two equilateral triangles such that $D$ is the midpoint of $BC$. Ratio of $A\left(△ABC\right)$ and $A\left(△BDE\right)$ is:

• A. $2:1$
• B. $1:2$
• C. $4:1$
• D. $1:4$

Answer 1

The correct option is C $4:1$

Given: $△ABC$ and $△BDE$ are equilateral triangles.
D is midpoint of BC.
Since, $△ABC$ and $△BDE$ are equilateral triangles.

All the angles are ${60}^{\circ }$ and hence they are similar triangles.
Ratio of areas of similar triangles is equal to ratio of squares of their sides:

Now, $\frac{A\left(△BDE\right)}{A\left(△ABC\right)}=\frac{B{C}^2}{B{D}^2}$

$\frac{A\left(△ABC\right)}{A\left(△BDE\right)}=\frac{(2BD{)}^2}{B{D}^2}$ ....Since $BC=2BD$

$\frac{A\left(△ABC\right)}{A\left(△BDE\right)}=4:1$