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Question

The quadrilateral formed by joining the midpoints of the sides of a quadrilateral ABCD, taken in order, is a rhombus, if

(a) ABCD is a parallelogram

(b) ABCD is a rhombus

(c) diagonals of ABCD are equal

(d) diagonals of ABCD are perpendicular to each other


Solution

Given:
The quadrilateral ABCD is a rhombus.
So, the sides AB, BC, CD and AD are equal.
Now, in ∆PQS, we have
DC=1/2QS
(Using mid-point theorem)-----(1)
Similarly, in ∆PSR,
BC=1/2PR----(2)
As, BC = DC
⇒1/2QS = 1/2PR
[From (1) and (2)]
So, QS = PR
Thus, the diagonals of PQRS are equal.
Hence, the correct option is (c).


Mathematics
Secondary School Mathematics IX
Standard IX

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