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P is the mid-point of the side CD of a parallelogram ABCD. A line through C parallel to PA intersects AB at Q and DA produced at R. Prove that DA = AR and CQ = QR.

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Given, ABCD is a parallelogram.

imageBC = AD and BC||AD

Also, DC = AB and DC||AB

Since, P is mid-point of DC.

imageDP = PC = 1/2DC

Now, QC||AP and PC||AQ

imageAPCQ is a parallelogram.

image AQ = PC = 1/2DC = 1/2AB = BQ

Now, in ΔAQR and ΔBQC,

BQ = AQ

∠BQC = ∠AQR

And ∠BCQ = ∠ARQ

image ΔAQR imageBQC

image AR = BC

But BC = DA

image AR = DA

Also, CQ = QR

Hence, Proved.

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