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Question

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


A
2:1
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B
1:2
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C
4:1
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D
1:4
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Solution

The correct option is C $$4:1$$
$$\triangle ABC \sim \triangle BDE$$                            (both are equilateral triangles)

According to the given condition, $$BC=\dfrac{BD}{2}$$

$$\Rightarrow \triangle ABC : \triangle BDE = AB^2 : BD^2$$


                                 $$= AB^2 :  (\dfrac{1}{2} BC)^{2} $$
                                          
                                 $$ = AB^2 : \dfrac{1}{4} BC^2 $$

                                 $$= 4 : 1$$           $$(\because AB = BC)$$


Mathematics

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