Question
If $\triangle\text{ABC}$ and $\triangle\text{BDE}$ are equilateral triangles, where D is the midpoint of BC, find the ratio of areas of $\triangle\text{ABC}$ and $\triangle\text{BDE}.$
✓
Answer
We have,$\triangle\text{ABC}$ and $\triangle\text{BDE}$ are equilateral triangles then both triangles are equiangular
$\therefore\triangle\text{ABC}\sim\triangle\text{BDE}$ [By AAA similarity]
By area of similar triangle theorem
$\frac{\text{ar}(\triangle\text{ABC)}}{\text{ar}(\triangle\text{BDE})}=\frac{\text{BC}^2}{\text{BD}^2}$
$=\frac{(2\text{BD})^2}{\text{BD}^2}$ [D is the mid-point of BC]
$=\frac{4(\text{BD})^2}{\text{BD}^2}$
$=\frac{4}{1}$
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