Question

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.​

Answer 1

Answer:

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 = [Math Processing Error]12AC ………….. (i)

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

∴ SR || AC

SR = [Math Processing Error]12AC …………… (ii)

Similarly, in ∆ABD,

SP || BD

SP = [Math Processing Error]12BD ……………….. (iii)

Similarly, in ∆BCD,

QR || BD

QR = [Math Processing Error]12BD ……………… (iv)

From (i), (ii), (iii) and (iv),

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

∴ PQRS is a rhombus

Answer 2

Answer:

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