In ∆ABC, D is the midpoint of AB and P Point is any point on BC. If CQ ‖ PD meets AB in Q, then prove that ar(∆BPQ) = ar(∆ABC).

Question

Philoid · Accepted Answer

Given: D is the midpoint of AB and P Point is any point on BC, CQ‖ PD

In Quadrilateral DPQC

Area (Δ DPQ) = Area (Δ DPC)

Add Area (Δ BDP) on both sides

We get,

Area (Δ DPQ) + Area (Δ BDP) = Area (Δ DPC) + Area (Δ BDP)

Area (Δ BPQ) = Area (Δ BCD) –1

D is the midpoint BC, and CD is the median

∴ Area (Δ BCD) = Area (Δ ACD) = 1/2 × Area (Δ ABC) –2

Sub –2 in –1

Area (Δ BPQ) = 1/2 × Area (Δ ABC) (∵Area (Δ BCD) = 1/2 × Area (Δ ABC))