Question

∆ABC and ∆BDE are two equilateral triangles such that D is the midpoint of BC. Then, ar(∆BDE) : ar(∆ABC) = ?
(a) 1 : 2
(b) 1 : 4
(c) $\sqrt{3}:2$
(d) 3 : 4

Answer 1

(b) 1:4

∆ABC and ∆BDE are two equilateral triangles​ and D is the midpoint of BC.
Let AB = BC = AC = a
Then BD = BE = ED = $\frac{a}{2}$
∴ $\frac{\mathrm{ar}(∆BDE)}{\mathrm{ar}(∆ABC)}=\frac{{\displaystyle \frac{\sqrt{3}}{4}}A{B}^2}{{\displaystyle \frac{\sqrt{3}}{4}}B{E}^2}=\frac{{\left({\displaystyle \frac{a}{2}}\right)}^2}{a^2}=\frac{1}{4}$
So, required ratio = 1 : 4