Question

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

Answer 1

Let us join AC and BD.  

In ΔABC,  

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

∴ PQ || AC and PQ = AC (Mid-point theorem) ... (1)  

Similarly in ΔADC,  

SR || AC and SR = AC (Mid-point theorem) ... (2)  

Clearly, 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.  

∴ PS || QR and PS = QR (Opposite sides of parallelogram)... (3)  

In ΔBCD, Q and R are the mid-points of side BC and CD respectively.  

∴ QR || BD and QR =BD (Mid-point theorem) ... (4)  

However, the diagonals of a rectangle are equal.  

∴ AC = BD …(5)  

By using equation (1), (2), (3), (4), and (5), we obtain  

PQ = QR = SR = PS  

Therefore, PQRS is a rhombus

Answer 2

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