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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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Best answer

ABCD is a quadrilateral 

P,Q,R,S are mid points of sides AB,BC,CD and DA 

In ∆ABC, 

P and PQ are the mid points of AB and AC respectively 

So, by using mid point theorem,

PQ || AC and PO = \(\frac{1}{2}\) AC ...(i)

Similarly,

In ∆BCD,

RS || AC and RS = \(\frac{1}{2}\) AC ... (ii)

From equation (i) and (ii)

PQ || RS and PQ = RS 

Similarly, we have 

PS || QR and PS = QR 

Hence, 

PQRS is a parallelogram. 

Since, 

Diagonals of a parallelogram bisects each other 

Hence, 

PR and QS bisect each other 

Proved.

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