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

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

To Prove: PQRS is a rhombus. 

Construction : 

Diagonals AC and BD are drawn. 

Proof: 

In ∆ABC, P and Q are the mid-points of AD and BC. 

∴ PQ || AC (Mid-point theorem) 

PQ = \(\frac{1}{2}\)AC ………….. (i) 

Similarly, in ∆ADC, S and R are the mid-points of AD and CD. 

∴ SR || AC 

SR = \(\frac{1}{2}\)AC …………… (ii) 

Similarly, in ∆ABD, 

SP || BD 

SP = \(\frac{1}{2}\)BD ……………….. (iii) 

Similarly, in ∆BCD, 

QR || BD 

QR = \(\frac{1}{2}\)BD ……………… (iv) 

From (i), (ii), (iii) and (iv), 

PQ = QR = SR = PS and Opposite sides are parallel. 

∴ PQRS is a rhombus.

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