Question

ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the area of triangles ABC and BDE is

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

Answer 1

The correct option is C

$4:1$

As $\text{∆ABC}$ and $\text{∆EBD}$ are equilateral triangles, it means all the angles are equal in both the triangles i.e. ${60}^0$.

Therefore, by AA similarity criteria, we can say that,

$\text{∆ABC∼∆EBD}$

Now, the triangles are similar, it means,

$\frac{\mathrm{Area}(∆\mathrm{ABC})}{\mathrm{Area}(∆\mathrm{EBD})}=\frac{{\mathrm{BC}}^2}{{\mathrm{BD}}^2}$

Let the length of each side of $\text{∆ABC}$ is $2\mathrm{x}$.

D is the midpoint of BC, it means

$\mathrm{BD}=\mathrm{DC}=\frac{1}{2}\mathrm{BC}$

$\mathrm{BD}=\frac{1}{2}\times 2\mathrm{x}=\mathrm{x}$

Therefore,

$\begin{array}{rcl}\frac{\mathrm{Area}(∆\mathrm{ABC})}{\mathrm{Area}(∆\mathrm{EBD})}& =& \frac{{\mathrm{BC}}^2}{{\mathrm{BD}}^2}\\ & =& \frac{{\left(2\mathrm{x}\right)}^2}{{\mathrm{x}}^2}\\ & =& \frac{4{\mathrm{x}}^2}{{\mathrm{x}}^2}\\ & =& \frac{4}{1}\\ & =& 4:1\end{array}$

Hence, option C is the correct option.