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
Answers
Answered by
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.
Similar questions