Question

Given a rectangle ABCD and P,Q,R,S midpoint of AB,BC,CD and DA respectively . Length of diagonal of rectangle is 8cm , the quadrilateral PQRS is a)Parallogram with adjacent sides 4 cm b) Rectangle with adjacent side 4cm c)Rhombus side 4cm
d) Square side with 4cm

Answer 1

Let the length and the breadth of the rectangle be 2a and 2b units.

 

From Pythagoras Theorem, we have,

 

 

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.

 

Also we have (linear pair) and .

 

In the parallelogram PQRS all the sides are equal to  and the diagonals are perpendicular and hecne PQRS is a rhombus.

 

Since the diagonal AC = 8 cm (given), we have

 

 

Thus, PQRS is a rhombus with side 4 cm.

 

Hence, (c) is the correct option.