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

Given: D is the midpoint of BC; E is the midpoint of BD and O is the midpoint of AE.

Here,

D is the midpoint of BC and AD is the median of ΔABC

Area (Δ ABD) = Area (Δ ADC) (∵ median divides the triangle into two triangles of equal areas)

∴ Area (Δ ABD) = Area (Δ ADC) = Area (∆ABC)

Now, consider Δ ABD

Here, AE is the median

Area (Δ ABE) = Area (Δ BED)

∴ Area (Δ ABE) = Area (Δ BED) = Area (∆ABD)

Area (Δ ABE) = Area (∆ABD)

Area (Δ ABE) = × (∵Area (Δ ABD) = Area (∆ABC) ) –1

Area (Δ ABE) = Area (∆ABC)

Consider Δ ABE

Here, BO is the median

Area (Δ BOE) = Area (Δ BOA)

∴ Area (Δ BOE) = Area (Δ BOA) = Area (∆ABE)

Area (Δ BOE) = × (∵Area (Δ ABE) = Area (∆ABC) )

Area (Δ BOE) = Area (∆ABC)

∴ Area (Δ BOE) = Area (∆ABC)