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Question 5
The quadrilateral formed by joining the mid – points of the side of quadrilateral PQRS, taken in order, is a rhombus, if
(A) PQRS is a rhombus
(B) PQRS is a parallelogram
(C) diagonals of PQRS are perpendicular
(D) diagonals of PQRS are equal.


Solution

Given, the quadrilateral ABCD is a rhombus.

So, sides AB, BC, CD and AD are equal.

Now, in ΔPQS, we have

D and C are the mid – points of PQ and PS.
So,                    DC=12QS     [by mid –d point theorem] . . . . . (i)
Similarly, in ΔPSR            BC=12PR     [by mid – point theorem] . . . .  . .(ii)
As                                       BC = DC         [since, ABCD is a rhombus]
12QS=12PR    [from Eqs. (i) and (ii) ]
QS=PR
Hence, diagonals of PQRS are equal.


Mathematics

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