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Question
P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD in which AC = BD. Prove that PQRS is a rhombus.
Solution
Given: In a quadrilateral ABCD, P, Q, R and S are the mid-points of sides AB, BC, CD and DA, respectively.
Also, AC = BD
To prove: PQRS is a rhombus.
Proof: In ΔADC, S and R are the mid-points of AD and DC respectively.
Then, by mid-point theorem.
SR || AC and SR = `1/2` AC ...(i)
In ΔABC, P and Q are the mid-points of AB and BC respectively.
Then, by mid-point theorem.
PQ || AC and PQ = `1/2` AC ...(ii)
From equations (i) and (ii),
SR = PQ = `1/2` AC ...(iii)
Similarly, in ΔBCD,
RQ || BD and RQ = `1/2` BD ...(iv)
And in ΔBAD,
SP || BD and SP = `1/2` BD ...(v)
From equations (iv) and (v),
SP = RQ = `1/2` BD = `1/2` AC [Given, AC = BD] ...(vi)
From equations (iii) and (vi),
SR = PQ = SP = RQ
It shows that all sides of a quadrilateral PQRS are equal.
Hence, PQRS is a rhombus.
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