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Question
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rhombus, if ______.
Options
PQRS is a rhombus
PQRS is a parallelogram
diagonals of PQRS are perpendicular
diagonals of PQRS are equal
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Solution
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rhombus, if diagonals of PQRS are equal.
Explanation:
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 = 1/2 QS` [By mid-point theorem] ...(i)
Similarly, in ΔPSR, `BC = 1/2 PR` [By mid-point theorem] ...(ii)
As BC = DC ...[Since, ABCD is a rhombus]
∴ `1/2 QS = 1/2 PR` ...[From equations (i) and (ii)]
⇒ QS = PR
Hence, diagonals of PQRS are equal.
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