Question

In $△ABC$,it is given that D is the midpoint of BC;E is the midpoint of BD and O is the midpoint of AE.Then,ar($△BOE$)=?

(a) $\frac{1}{3}ar\left(△ABC\right)$

(b) $\frac{1}{4}ar\left(△ABC\right)$

(c) $\frac{1}{6}ar\left(△ABC\right)$

(d) $\frac{1}{8}ar\left(△ABC\right)$

Answer 1

ANSWER:
(d) 1/8 ar (∆ ABC)
Given: D is the midpoint of BC, E is the midpoint of BD and O is the mid point of AE.
Since D is the midpoint of BC, AD is the median of ∆ ABC.
E is the midpoint of BC , so AE is the median of ∆ ABD. O is the midpoint of AE , so BO is median of ∆ABE.
We know that a median of a triangle divides it into two triangles of equal areas.
i.e., ar(∆ ABD ) = 1/2 ⨯ ​ ar(∆ ABC)
...(i)
ar(∆ ABE ) =1/2​ ⨯​ ar(∆ ABD)
...(ii)
ar(∆ BOE) = ​ 1/2⨯​ ar(∆ ABE)
...(iii)
From (i), (ii) and (iii), we have:
ar(∆ BOE ) = ​ 1/2ar(∆ ABE)
ar(∆ BOE ) = 1/2 ⨯​ 1/2​ ⨯​ 1/2​ ⨯​ ar(∆ ABC)
∴​​ ar(∆ BOE )​ = 1/8 ar(∆ ABC)