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In the below figure, ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD. If AQ intersects DP at S abd BQ intersects CP at R, show that:

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## Answers

(i) In quadrilateral APCQ,

AP || QC (Since AB || CD) (1)AP = ½AB, CQ = ½CD (Given)

Also, AB = CD (Why?)

So, AP = QC (2)

Therefore, APCQ is a parallelogram [From (1) and (2) and Theorem 8.8]

(ii) Similarly, quadrilateral DPBQ is a parallelogram, because

DQ || PB and DQ = PB.

(iii) In quadrilateral PSQR,

SP || QR (SP is a part of DP and QR is a part of QB)

Similarly, SQ || PR

So, PSQR is a parallelogram.