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Question

The line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

A
True
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B
False
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Solution

The correct option is A True
Let ABCD is a quadrilateral and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively, i.e., AS = SD, AP = BP, BQ = CQ and CR = DR. We have to show that PR and SQ bisect each other i.e., SO = OQ and PO = OR.

Now, in Δ ADC, S and R are mid-point of AD and CD.
We know that, the line segment joining the mid-points of two sides of a triangle is parallel to the third side. (By mid-point theorem)
SR || AC and SR = 12 AC ....(i)
Diagonals of a parallelogram bisect each other.
So, SQ and PR bisect each other
Similarly, in Δ ABC, P and Q are mid - point of AB and BC.
PQ || AC and PQ =
12 AC ( By mid-point theorem) ……(ii)
From equations (i) and (ii) we get,
PQ || RS and PQ || SR = 12 AC
Therefore, Quadrilateral PQRS is a parallelogram whose diagonals are SQ and PR. Also, we know that diagonals of a parallelogram bisect each other. So, SQ and PR bisect each other.

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