Question

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:
(i) APCQ is a parallelogram
(ii) DPBQ is a parallelogram
(iii) PSQR is a parallelogram

Answer 1

(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.

Answer 2

AB = CD (opp. Sides of a llgm )

1/2× on both sides

Therefore AP = CQ

AP || CQ (parts of parallel lines )

So , APCQ is a llgm

So ,AQ || PC

Similarly ,PBQD is a llgm

DP || QB

Consider PSQR

QS || PR (parts of parallel lines )

PS||QR (parts of parallel lines )

Therefore PSQR is a llgm