Question

ABCD is a rectangle. P, Q, R and S are the mid-points of sides AB, BC, CD and AD respectively. What kind of figure is PQRS? If AB = 32 cm and BC = 24 cm, find the perimeter of PQRS.

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 = `1/2` AC (Mid-point theorem)   ...(1)

Similarly in ΔADC,

SR || AC and SR = `1/2` 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 ABCD, Q and R are the mid-points of side BC and CD respectively.

∴ QR || BD and QR = `1/2` 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.