Question 6 Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
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Solution
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 PRbisect each other Similarly, inΔABC, 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 SQand PR. Also, we know that diagonals of a parallelogram bisect each other. So, SQ and PR bisect each other