I want to know the best answer of the question from Quadrilaterals chapter of class 9^{th} ncert math. The question from exercise 8.2 of math. Give me the easy way for solving this question of 2 ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.

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# ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle. Q.2

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Given in the question,

ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively.

To Prove,

PQRS is a rectangle.

Construction,

Join AC and BD.

Proof:

In Î”DRS and Î”BPQ,

DS = BQ (Halves of the opposite sides of the rhombus)

âˆ SDR = âˆ QBP (Opposite angles of the rhombus)

DR = BP (Halves of the opposite sides of the rhombus)

, Î”DRS â‰… Î”BPQ [SAS congruency]

RS = PQ [CPCT]â€”â€”â€”â€”â€”â€”â€”- (i)

In Î”QCR and Î”SAP,

RC = PA (Halves of the opposite sides of the rhombus)

âˆ RCQ = âˆ PAS (Opposite angles of the rhombus)

CQ = AS (Halves of the opposite sides of the rhombus)

, Î”QCR â‰… Î”SAP [SAS congruency]

RQ = SP [CPCT]â€”â€”â€”â€”â€”â€”â€”- (ii)

Now,

In Î”CDB,

R and Q are the mid points of CD and BC respectively.

â‡’ QR || BD

also,

P and S are the mid points of AD and AB respectively.

â‡’ PS || BD

â‡’ QR || PS

, PQRS is a parallelogram.

also, âˆ PQR = 90Â°

Now,

In PQRS,

RS = PQ and RQ = SP from (i) and (ii)

âˆ Q = 90Â°

, PQRS is a rectangle.