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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