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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.

**Solution:**

Given, ABCD is a __parallelogram__

P and Q are the midpoints of the opposite sides AB and CD of the parallelogram

AQ intersects DP at S

BQ intersects CP at R

We have to show that PQRS is a parallelogram

Since P is the midpoint of AB

AP = PB

AB = AP + PB

AB = AP + AP

AB = 2AP

AP = 1/2 AB ----------- (1)

Since Q is the midpoint of CD

QC = QD

CD = QC + QD

CD = QC + QC

CD = 2QC

QC = 1/2 CD ------------ (2)

We know that the opposite sides of a parallelogram are parallel and __congruent__

AB || CD

Also, AB = CD

Dividing by 2 on both sides,

1/2 AB = 1/2 CD

From (1) and (2),

AP = QC

Also, AP || QC

Therefore, APCQ is a parallelogram

So, AQ || PC or SQ || PR

Similarly , AB || DC or BP || DQ

So, AB = DC

Dividing by 2 on both sides,

1/2 AB = 1/2 DC

Since P is the midpoint of AB

AP = PB

AB = AP + PB

AB = BP + BP

AB = 2BP

BP = 1/2 AB ----------- (3)

Since Q is the midpoint of CD

QC = QD

CD = QC + QD

CD = QD + QD

CD = 2QD

QD = 1/2 CD ------------ (4)

From (3) and (4),

BP = QD

Therefore, BPDQ is a parallelogram

We know that the opposite sides of a parallelogram are parallel and congruent

PD || BQ

Similarly, PS || QR

So, SQ || RP

PS || QR

Therefore, PQRS is a parallelogram.

**✦ Try This: **ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD. If AQ intersects DP at S and BQ intersects CP at R, show that APCQ is a parallelogram

**☛ Also Check:** NCERT Solutions for Class 9 Maths Chapter 8

**NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 7**

## 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.

**Summary:**

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. It is shown that PRQS is a parallelogram

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