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  • Write true or false and justify your answer. ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Then ar (BDE) =  1/4 ar ( ABC).

Write true or false and justify your answer.
ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Then ar (BDE) = \frac{1}{4} ar ( ABC).

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best_answer

Answer: [True]

Solution.

Given \triangle ABC and \triangle BDE are two equilateral triangles.
D is the mid-point of BC, so BD = CD = \frac{1}{2} BC

 

Let each sides of  \triangle ABC be x.
So each side of \triangle BDE is \frac{x}{2}.
Area of an equilateral triangle is given as \frac{\sqrt{3}}{4}(side)^{2}
Now,  \frac{ar(\triangle BDE)}{ar(\triangle ABC)}=\frac{\frac{\sqrt{3}}{4}\left ( \frac{x}{2} \right )^{2}}{\frac{\sqrt{3}}{4}x^{2}}=\frac{1x^{2}}{4x^{2}}=\frac{1}{4}

Hence, ar(\triangle BDE)=\frac{1}{4}ar(\triangle ABC)

Therefore the given statement is true. 

 

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