ABCD is a rectangle and P,Q,R …

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

Sol:   In a rectangle ABCD, P is the mid-point of AB, Q is the mid-point of BC, R is the mid-point of CD, S is the mid-point of DA AC is the diagonal.

             From (1) and (2), we get

                                       PQ = SR and PQ || SR

             Similarly, by joining BD, we have

                                       PS = QR and PS || QR

             i.e. Both pairs of opposite sides of quadrilateral PQRS are equal and parallel.

             ∴ PQRS is a parallelogram.

             Now, in ΔPAS and ΔPBQ,

             ∴Their corresponding parts are equal.

                ⇒

PS = PQ

                Also

PS = QR

[Proved]

                and

PQ = SR

[Proved]

PQ = QR = RS = SP

             i.e. PQRS is a parallelogram having all of its sides equal.

             ⇒PQRS is a rhombus.