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+1 vote
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in Mathematics by (35.7k points)

ABCD is a parallelogram whose diagonals AC and BD intersect at O. A line through O intersects AB at P and DC at Q. Prove that ar (∆ POA) = ar (∆ QOC).

1 Answer

+1 vote
by (62.0k points)
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Best answer

In triangles POA and QOC, we have 

∠AOP = ∠COQ [vertically opposite angles] 

OA = OC [Diagonals of a parallelogram bisect each other] 

∠PAC = ∠QCA [ AB || DC; alternative angles] 

So, by ASA congruence criterion, we have 

ΔPOA  ≅ QOC 

Area ( ΔPOA) = area (ΔQOC)

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