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 (Δ POA) = ar (Δ QOC).

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Solution

Given:

(1) Diagonals AC and BD of a parallelogram ABCD intersect at point O.

(2) A line through O intersects AB at P point.

(3) A line through O intersects DC at Q point.

To find: Area of (ΔPOA) = Area of (ΔQOC)

Proof:

From ΔPOA and ΔQOC we get that

=

OA = OC

=

So, by ASA congruence criterion, we have

So

Area (ΔPOA) = Area (ΔQOC)

Hence it is proved that

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