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 | 25th 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.
 
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.

Answered by Vimala Ramamurthy | 26th Dec, 2013, 09:12: AM

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