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 AB and CD, respectively.

To show PRQS is a parallelogram.

Proof Since, ABCD is a parallelogram.

AB || CD

$\Rightarrow$ AP || QC

Also, AB || DC

$\frac{1}{2}AB=\frac{1}{2}DC$ [dividing both sides by 2 ]

$\Rightarrow$ 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.

$\therefore$ AQ||PC or SQ || PR . . . . . . . . ..(i)

Again, AB||DC or BP||DQ

Also, $AB=DC\Rightarrow \frac{1}{2}AB=\frac{1}{2}DC$ [dividing both sides by 2]

$\Rightarrow$ 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.

$\therefore$ PD||BQ or PS||QR

From Eqs. (i) and (ii), SQ||RP and PS||QR

So, PRQS is a parallelogram. Hence proved.