Question

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.

Answer 1

Given: In a parallelogram ABCD, P and Q are the mid-points of AS and CD, respectively.

To show: PRQS is a parallelogram.

Proof: Since, ABCD is a parallelogram.

AB || CD

⇒ AP || QC

Also, AB = DC

`1/2`AB = `1/2`DC   ...[Dividing both sides by 2]

⇒ AP = QC  ...[Since, P and Q are the mid-points of AB and DC]

Now, AP || QC and AP = QC

Thus, APCQ is a parallelogram.

∴ AQ || PC or SQ || PR  ...(i)

Again, AB || DC or BP || DQ

Also, AB = DC

⇒ `1/2`AB = `1/2`DC   ...[Dividing both sides by 2]

⇒ BP = QD  ...[Since, P and Q are the mid-points of AB and DC]

Now, BP || QD and BP = QD

So, BPDQ is a parallelogram.

∴ PD || BQ or PS || QR   ...(ii)

From equations (i) and (ii),

SQ || RP and PS || QR 

So, PRQS is a parallelogram.

Hence proved.