In the figure, $ABC$ and $BDC$ are two equilateral triangles such that $D$ is the mid-point of $BC$. $AE$ intersects $BC$ in $F$. Prove that $ \operatorname{ar}(\triangle \mathrm{BDE})=\frac{1}{4} \operatorname{ar}(\Delta \mathrm{ABC}) $. "
Given:
$ABC$ and $BDC$ are two equilateral triangles such that $D$ is the mid-point of $BC$. $AE$ intersects $BC$ in $F$.
To do:
We have to prove that \( \operatorname{ar}(\triangle \mathrm{BDE})=\frac{1}{4} \operatorname{ar}(\Delta \mathrm{ABC}) \).
Solution:
Let $\mathrm{AB}=\mathrm{BC}=\mathrm{CA}=x$
This implies,
$\mathrm{BD}=\frac{x}{2}$ ($\mathrm{D}$ is the mid point of $\mathrm{BC}$)
Area of equilateral triangle $\mathrm{ABC}=\frac{\sqrt{3}}{4}(\text { side })^{2}$
$=\frac{\sqrt{3}}{4} x^{2} \mathrm{~cm}^{2}$
Area of equilateral triangle $\mathrm{BED}=\frac{\sqrt{3}}{4}(\frac{x}{2})^{2}$
$=\frac{\sqrt{3}}{4} \times \frac{x^{2}}{4}$
$=\frac{1}{4}(\frac{\sqrt{3}}{4} x^{2})$
$=\frac{1}{4}(\text { area of } \triangle \mathrm{ABC})$
Hence proved.
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