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in Quadrilaterals by (65.5k points)

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

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Data : P, Q, R and S are mid-points of AB, BC, CD and DA respectively in quadrilateral ABCD. 

To Prove: PR and SQ line segments bisect mutually. 

Construction: Join the diagonal AC. 

Proof: In ∆ADC, S and R are the mid-points of AD and DC. 

∴ SR || AC 

SR = \(\frac{1}{2}\)AC ………… (i) 

(Mid-point Theorem) 

Similarly, in ∆ABC, 

PQ || AC 

PQ = \(\frac{1}{2}\)AC …………. (ii) 

From (i) and (ii), 

SR = PQ and 

SR || PQ 

∴ PQRS is a parallelogram. PR and SQ are the diagonals of parallelogram PQRS. 

∴PR and SQ meet at O’.

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