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($△BOE$)=?

Answer 1

ANSWER: 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 BD , 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 ) = $\frac{1}{2}$⨯ar(∆ABC)
...(i)
ar(∆ABE ) = $\frac{1}{2}$⨯​ar(∆ABD)
...(ii)
ar(∆BOE) = ​$\frac{1}{2}$⨯​ar(∆ABE)
...(iii)
From (i), (ii) and (iii), we have:
ar(∆BOE ) = ​$\frac{1}{2}$ ⨯ ar(∆ ABE)
ar(∆BOE ) = $\frac{1}{2}$ ⨯​ $\frac{1}{2}$ ⨯​ $\frac{1}{2}$​ ⨯​ ar(∆ABC)
∴​​ ar(∆BOE )​ = $\frac{1}{8}$⨯ ar(∆ABC)