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Question 18
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 CW =QR.


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Solution

Given in a parallelogram ABCD, P is the mid-point of DC.

To prove DA=AR and CQ=QR

Proof ABCD is a parallelogram.


BC=AD and BC||AD

Also,DC=AB and DC||AB

Since, P is the mid-point of DC.

DP=PC=12DC

Now,QC ∥AP and PC ∥ AQ

So, APCQ is a parallelogram,

AQ=PC=12DC

=12AB=BQ [DC=AB] . . . . . .(i)

Now, in ΔAQR and ΔBQC,AQ=BQ[from Eq. (i)]

AQR =∠BQC[vertically opposite angles]

andARQ=BCQ [Alternate opposite angles]

ΔAQR=ΔBQC [by AAS congruence rule]

AR = BC [by CPCT rule]

ButBC = DA

AR = DA

Also,CQ = QR [by CPCT rule]

Hence proved.


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