In a triangle ABC, E is the mid-point of median AD. Show that ar (BED) =1/4 ar(ABC)

Given: Δ ABC, with AD as median i.e., BD = CD and E is the mid-point of AD, i.e., AE = DE

To prove: ar (BED) = 1/4 ar (ABC).

Proof: AD is a median of Δ ABC and median divides a triangle into two triangles of equal area

∴ ar(ABD) = ar(ACD)

⇒ ar (ABD) = 1/2 ar (ABC)       …(1)

In Δ ABD,

BE is median (As E is mid-point of AD)

median divides a triangle into two triangles of equal area

∴ ar(BED) = ar(BEA)

⇒ ar (BED) = 1/2 ar (ABD)

⇒ ar (BED) = 1/2 × 1/2 ar (ABC) (from (1): ar (ABD) = 1/2 ar (ABC))

⇒ ar (BED) = 1/4 ar (ABC)

Hence proved.