1.

In figure. ABCD is a parallelogram. P and Q are the mid-points of opposite sides AB and DC of a parallelogram ABCD. Prove that PRQS is a parallelogram.

Answer»

Given: ABCD is a parallelogram. P and Q are respectively the mid-points of opposite sides AB and DC of a parallelogram ABCD. 

AQ and DP are joined intersecting in S and CP and BQ are joined intersecting in R.

To prove: Quadrilateral PQRS is a parallelogram.

Proof: ∵ ABCD is a parallelogram.

∴ AB = DC and AB || DC

⇒ \(\frac { 1 }{ 2 }\)AB = \(\frac { 1 }{ 2 }\)DC

⇒ AP = QC and AP || QC

⇒ APCQ is a parallelogram.

Similarly. PBQD is also a parallelogram. 

Since in parallelogram APCQ.

AQ || PC

(opposite sides of a parallelogram)

∴ SQ || PR

Now in parallelogram PBQD

PD || BQ

∴ SP || QR

Now in quadrilateral PRQS, SQ || PR and SP || QR.

Therefore, PRQS is a parallelogram.

Hence proved.


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