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Question

ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD . If AQ intersects DP at S and BQ intersects CP at R, show that ;
i) APCQ is a parallelogram
ii) DPBQ is a parallelogram
iii) PSQR is a parallelogram
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Solution

Given:ABCD is a parallelogram
P,Q are the midpoints of AB and CD respectively.
(i)Since ABDC since opposite sides of parallelogram are parallel.
APQC since parts of parallel lines are parallel
Also,AB=CD since opposite sides of parallelogram are equal.
12AB=12CD
AP=QC since P is mid-point of AB and Q is the midpoint of DC
Since APQC and AP=AC
One pair of opposite sides of APCQ are equal and parallel
APCQ is a parallelogram.

(ii)Since ANDC since opposite sides of parallelogram are parallel.
PBDQ since parts of parallel lines are parallel
Also,AB=CD since opposite sides of parallelogram are equal.
12AB=12CD
PB=DQ since P is mid-point of AB and Q is the midpoint of DC
Since PBDQ and PB=DQ
One pair of opposite sides of DPBQ are equal and parallel
DPBQ is a parallelogram.

(iii)So,DPQB since opposite sides of parallelogram are parallel.
SPQR since parts of parallel lines are parallel .......(1)
Also,in part(i) we proved APCQ is a parallelogram
So,AQPC since opposite sides of parallelogram are parallel
SQPR since parts of parallel lines are parallel ........(2)
From (1) and (2) we have SPQR and SQPR
Now, in PSQR both pairs of opposite sides of PSQR are paralle.
PSQR is a parallelogram.

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