CBSE Class 9 Answered
4) Given a rectangle ABCD and P, Q, R, S are the mid-points of AB, BC, CD and DA respectively. Length of diagonal of a rectangle is 8cm. Then the quadrilateral PQRS is a
(a) parallelogram with adjacent sides 4cm and 6cm
(b) rectangle with adjacent sides 4cm and 6cm
(c) rhombus with side 4cm
(d) square with side 4cm
Asked by Madhuchhanda Chakraborty | 25 Dec, 2013, 03:11: PM
Expert Answer
Consider the following figure.
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.
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 
.
(linear pair) and .In the parallelogram PQRS all the sides are equal to 
and the diagonals are perpendicular and hecne PQRS is a rhombus.
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.
Answered by Vimala Ramamurthy | 26 Dec, 2013, 09:12: AM
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