# Let M and N be the midpoints of the sides BC and AD respectively in a quadrilateral ABCD and also let P and Q be the midpoints of its diagonals AC and BD respectively. Prove that MN bisects PQ.

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### 1 Answer

In triangle ABD , N is mid point of AD and Q is mid point of BD.

Thus by mid point theorem , QN is parallel to AB. (i)

In triangle ABC , P is mid point of AC and M is mid point of BC.

Thus by mid point theorem , PM is parallel to AB. (ii)

Thus from (i) and (ii)

NQ is parallel to PM (iii)

In triangle ADC , N is mid point of AD and P is mid point of AC.

Thus by mid point theorem , NP is parallel to DC. (iv)

and

In triangle BDC , M is mid point of BC and Q is mid point of BD.

Thus by mid point theorem , MQ is parallel to DC. (v)

from (iv) and (v)

NP is parallel to QM (vi)

From (iii) and (vi) , we have NQMP is parallelogram.

In Parallelogram ,diagonal bisect each other ,therefore MN will bisect PQ.

QED.