Question

$ABC$ and $BDE$ are two equilateral triangles such that $D$ is the midpoint of $BC$.Ratio of the areas of triangles $ABC$ and $BDE$ is

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

Answer 1

Answer: (C)

$4:1$

Given:$△ABC$ and $△BDE$ are equilateral.

$BD=\frac{1}{2}BC$ as $D$ is the midpoint of $BC$

Since $△ABC$ and $△BDE$ are equilateral.

Their sides would be in the same ratio

$\frac{AB}{BE}=\frac{AC}{ED}=\frac{BC}{BD}$

Hence by SSS similarity,$△ABC\sim △BDE$

And, we know that the ratio of area of triangle is equal to the ratio of the square of corresponding sides.

So,$\frac{areaof△ABC}{areaof△BDE}=\frac{{BC}^2}{{BD}^2}$

$=\frac{{BC}^2}{{\left(\frac{BC}{2}\right)}^2}$ since $BD=\frac{BC}{2}$

$=\frac{4{BC}^2}{{BC}^2}$

$=\frac{4}{1}$

Hence $\frac{areaof△ABC}{areaof△BDE}=\frac{4}{1}$

$=4:1$