## ABCD is a parallelogram in which P and Q are midpoints of opposite sides AB and CD. If AQ intersects DP at S and BQ intersects CP at R, show

Question

ABCD is a parallelogram in which P and Q are midpoints 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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1 month 2021-08-14T15:08:00+00:00 2 Answers 0 views 0

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Step-by-step explanation:

(1)AB║CD and AB=DC

⇒AP║QC and 1/2AB=1/2DC

⇒AP║QC and AP=QC

∴APCQ is a parallelogram

(2)AB║DC and AB=DC

⇒PB║DQ and 1/2AB=1/2DC

⇒PB║DQ and PB=DQ

∴DPBQ is a parallelogram

(3)PC║AQ [∵APCQ is a parallelogram]

∴SO║PR

and DP║QB [∵DPBQ is a parallelogram]

∴PS║QR

∴PSQR is a parallelogram

∴Hence proved

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

Step-by-step explanation:

Given: ABCD is a parallelogram

PQ=CQ=1/2DC

AP=PB=1/2AB

∴AP=CQ=1/2AB

AB||CD

∴AB||CQ  (AQCP)  Therefore, (i) APCQ is a parallelogram.

simillarly,

DP||BQ (DPBQ) Therefore,   (ii) DPBQ is a parallelogram.

PS||QR (PSQR) Therefore, (iii)PSQR is a parallelogram