NCERT Solutions for Class 9 Maths Chapter 8

Important NCERT Questions

Quadrilaterals

NCERT Books for Session 2022-2023

CBSE Board and UP Board Others state Board

EXERCISE 8.2

Page No:150

Questions No:2

# ABCD is a rhombus and P, Q, R and S are ©wthe mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.

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In ΔABC,

P is mid-point to AB [∵ Given]

Q is mid-point of BC [∵ Given]

Hence, PQ ∥ AC and PQ = (1/2)AC … (1) [∵ Mid-point Theorem]

Similarly, in ΔACD,

S is mid-point of AD [∵ Given]

R is mid-point of CD [∵ Given]

Hence, SR ∥ AC and SR = (1/2) AC …(2) [∵ Mid-Point Theorem]

From (1) and (2), we have

PQ ∥ SR …(3) [∵ PQ ∥ AC and SR ∥ AC]

and PQ = SR …(4) [∵ SR = (1/2)AC and PQ = (1/2)AC]

Hence, PQRS is a parallelogram.

Similarly, in ΔBCD,

Q is mid-point of BC [∵ Given]

R is mid-point of CD [∵ Given]

Hence, QR ∥ BD [∵ Mid-point Theorem]

⇒ QN ∥ LM …(5)

and LQ ∥ MN …(6) [∵ PQ ∥ AC]

From (5) and (6), we have

LMNQ is a parallelogram.

Hence, ∠LMN = ∠LQN [∵ Opposite angles of a parallelogram]

But, ∠LMN = 90° [∵ Digonals of a rhombus are perpendicular to each other]

Hence, ∠LQN = 90°

A parallelogram whose one angle is a rectangle. Hence, PQRS is a rectangle.