ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.
Solution:
It is given that ABCD is a rectangle. P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. We need to show that PQRS is a rhombus.
First step is to Join AC and BD.
In
, S is the mid-point of AD and R is the mid-point of CD.
Therefore, by mid-point theorem, we have
and
(1)
Similarly in
, by mid-point theorem, we have
and
(2)
From (1) and (2), we can say that
(3)
In
, Q is the mid-point of BC and R is the mid-point of CD.
Therefore, by mid-point theorem, we have
and
(4)
Similarly, in
by mid-point theorem, we have
and
(5)
From (4) and (5), we can say that
(6)
We also have
(Diagonals of rectangle are equal) (7)
From (3), (6) and (7), we can say that
which shows that PQRS is a rhombus.
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