P and Q are the midpoints of sides AB and DC of parallelogram ABCD, let’s prove that the area of quadrilateral shaped area of parallelogram shaped region ABCD.

Given.

P and Q are the midpoints of sides AB and DC of parallelogram ABCD

Formula used

Area of parallelogram = Base × Perpendicular

⇒ Property of parallel lines

Perpendicular distance between 2 parallel is always same.

As AB = CD and AB || CD ∵ property of parallelogram

Their mid points will also be equal and parallel

∴ AP=CQ and AP || CQ

∴ APCQ is a parallelogram

As we know that if both parallelogram lies on same 2 parallel lines

Because opposite lines are parallel in parallelogram

AB || CD

Perpendicular height of both will be same

⇒ By Property of parallel lines

As both parallelogram are on same base line

But CD = 2 × CQ ∵ Q is mid-point of CD

CQ =

Area of parallelogram ABCD = CD × AE

Area of parallelogram APCQ = Base × Height

= CQ × AE

= × AE

=×CD×AE

× [Area of Parallelogram ABCD]

Hence proved;