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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). - Mathematics

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

In triangles and , POA 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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APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 14 Areas of Parallelograms and Triangles
Exercise 14.3 | Q 14 | Page 46
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