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

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Given : ABCD is a quadrilateral in which EG and FH are the line segments joining the mid-points of opposite sides. 

To Prove : EG and FH bisect each other. 

Construction : Join EF, FG, GH, HE and AC. 

Proof : In ∆ABC, E and F are mid-points of AB and BC respectively. 

∴ EF = 1/ 2 AC and EF || AC …(i) 

In ∆ADC, H and G are mid-points of AD and CD respectively. 

∴ HG = 1 2 AC and HG || AC …(ii) 

From (i) and (ii), we get 

EF = HG and EF || HG 

∴ EFGH is a parallelogram. [∵ a quadrilateral is a parallelogram if its one pair of opposite sides is equal and parallel] 

Now, EG and FH are diagonals of the parallelogram EFGH. ∴ EG and FH bisect each other. 

[Diagonal of a parallelogram bisect each other] Proved.

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