Question

D is the midpoint of side BC of ∆ABC and E is the midpoint of BD. If O is the midpoint of AE, prove that $\mathrm{ar}(∆BOE)=\frac{1}{8}\mathrm{ar}(∆ABC)$.

Answer 1

Given: D is the midpoint of BC; E is the midpoint of BD; O is the mid point of AE.
To prove: ar(∆BOE) = $\frac{1}{8}$ ⨯ ar(∆ABC)
Proof:
D is the midpoint of BC, so AD is the median of ∆ABC.
E is the midpoint of BD, so AE is the median of ∆ABD.
O is the mid point of AE, so BO is median of ∆ABE.
We know that a median of a triangle divides it into two triangles of equal areas. So, we have:
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)​ = $\left(\frac{1}{8}\right)$ ⨯ ar(∆ABC)