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in Parallelograms by (26.9k points)
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We get a rhombus by joining the mid-points of the sides of a :

A. Parallelogram 

B. Rhombus 

C. Rectangle 

D. Triangle

1 Answer

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Best answer

Option : (B)

Let ABCD is a rectangle such as AB = CD and BC = DA 

P, Q, R and S are the mid points of the sides AB, BC, CD and DA respectively 

Construction : 

Join AC and BD

In ΔABC,

P and Q are the mid-points of AB and BC respectively 

Therefore,

PQ ‖ AC and PQ = \(\frac{1}{2}\)AC (Mid-point theorem) ...(i)

Similarly,

In ΔADC,

SR ‖ AC and SR = \(\frac{1}{2}\)AC (Mid-point theorem) ..(ii)

Clearly, from (i) and (ii)

PQ ‖ SR and PQ = SR

Since, 

In quadrilateral PQRS one pair of opposite sides is equal and parallel to each other, it is a parallelogram.

Therefore, 

PS ‖ QR and PS = QR (Opposite sides of a parallelogram) ...(iii)

In ΔBCD,

Q and R are the mid-points of side BC and CD respectively 

Therefore,

QR ‖ BD and QR = \(\frac{1}{2}\)BD(Mid-point theorem) ...(iv)

However, 

The diagonals of a rectangle are equal 

Therefore, 

AC = BD ...(v) 

Now, 

By using equation (i), (ii), (iii), (iv), and (v), we obtain 

PQ = QR = SR = PS 

Therefore, 

PQRS is a rhombus.

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