Question

ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD (see the below figure). 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.

Answer 1

Answer:

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Step-by-step explanation:

(I) APCQ is a parallelogram

Answer 2

If P is the mid point of AB and Q is the mid point of CD then

AP = PB = AB/2

CQ = QD = CD/2

But ABCD is a parallelogram

∴ AB = CD

∴ AP = PB = CQ = QD

Also

∵ AB ║ CD

∴ AP ║ CQ ║ QD

And PB ║ CQ ║ QD

In quadrilateral APCQ

∵ AP ║ CQ

And, AP = CQ

Therefore, APCQ is a parallelogram   (By theorem: If in a quadrilateral the opposite sides are equal and parallel then the quadrilateral is a parallelogram)

In quadrilateral DPBQ

∵ PB ║ QD

And, PB = QD

Therefore, DPBQ is a parallelogram

It can be easliy proved that APQD and PBCQ are parallelograms

AQ, DP and PC, QB are diagonals of the parallelograms APQD and PBCQ

∵ AS = SQ and DS = SP

And PR = RC and BR = RQ

∴ APCQ is a parallelogram

∴ AQ = PC

Therefore, (1/2)AQ = (1/2)PC

or, SQ = PR

And AQ ║ PC

Therefore, (1/2)AQ ║ (1/2)PC

or, SQ ║ PR

? SQ = PR

And SQ ║ PR

Therefore, quadrilateral PSQR is a parallelogram     (Hence Proved)

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