Question

$△$ABC and $△$BDE are two equilateral triangles and BD = DC. Find the ratio between areas of $△$ABC and $△$BDE.

Answer 1

In the given figure, it is given that $△ABC$ and $△BDF$ are equilateral triangles that is $△ABC\sim △BED$ and $BD=DC$.

We know that the area of similar triangles are proportional to squares of their corresponding altitude, therefore,

$\frac{Ar(△ABC)}{Ar(△BDE)}=\frac{B{C}^2}{B{D}^2}$

$=\frac{(2BD{)}^2}{B{D}^2}=\frac{4B{D}^2}{B{D}^2}(\because BC=BD+DC,BD=DC)$

$=4$

Hence the ratio between areas of $△ABC$ and $△BDE$ is $4:1$.