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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 such that AC ⊥ BD. Prove that PQRS is a rectangle.
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Solution
Given, P, Q, R and S are mid-points of the sides AB, BC, CD and DA, respectively.
Also, AC is perpendicular to BD
∠COD = ∠AOD = ∠AOB = ∠COB = 90°
In ΔADC, by mid-point theorem,
SR || AC and SR = `1/2` AC
In ΔABC, by mid-point theorem,
PQ || AC and PQ = `1/2` AC
∴ PQ || SR and SR = PQ = `1/2` AC
Similarly, SP || RQ and SP = RQ = `1/2` BD
Now, in quadrilateral EOFR,
OE || FR, OF || ER
∠EOF = ∠ERF = 90°
Hence, PQRS is a rectangle.
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