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P and Q are the mid-points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram.

1 Answer

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Best answer

According to the question,

P is the midpoint of AB

Q is the midpoint of CD

Now,

AB||CD,

Also,

AP||QC

And, AB = DC

½ AB = ½ DC

AP = QC

Now,

AP||QC and AP = QC

APCQ is a parallelogram.

AQ||PC or SQ||PR

Again,

AB||DC means ½ AB = ½ DC

BP = QD

Now, BP||QD and BP = QD

BPDQ is a parallelogram

So, PD||BQ or PS||QR

Thus, SQ||RP and PS||QR

PQRS is a parallelogram.

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