In the give figure, X and Y are the midpoints of AC and AB respectively, QP ‖ BC and CYQ and BXP are straight lines. Prove that ar(∆ABP) = ar(∆ACQ).
Given: X and Y are the midpoints of AC and AB respectively, QP ‖ BC and CYQ and BXP are straight lines.
Construction: Join QB and PC
In Quadrilateral BCQP
Area (Δ QBC) = Area (Δ BCP) (Triangles on same base BC and between same parallel lines are equal in area) –1 and,
Area (||gm ACBQ) = Area (||gm ABCP) (parallelograms on same base BC and between same parallel lines are equal in area) –2
Subtract –1 from –2
Area (||gm ACBQ) – Area (Δ QBC) = Area (||gm ABCP) – Area (Δ BCP)
Area (∆ACQ) = Area (∆ABP)
∴ Area(∆ABP) = Area(∆ACQ)