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.

Question

Philoid · Accepted Answer

Given, ABCD is a parallelogram.

BC = AD and BC||AD

Also, DC = AB and DC||AB

Since, P is mid-point of DC.

DP = PC = DC

Now, QC||AP and PC||AQ

APCQ is a parallelogram.

AQ = PC = DC = AB = BQ

Now, in ΔAQR and ΔBQC,

BQ = AQ

∠BQC = ∠AQR

And ∠BCQ = ∠ARQ

ΔAQR BQC

AR = BC

But BC = DA

AR = DA

Also, CQ = QR

Hence, Proved.