The diagonals AC and BD of a parallelogram ABCD intersect at O. If P is the mid-point of AD, prove that (i) PO || AB (ii) PO = CD.

(i) Given, ABCD is a parallelogram in which diagonals AC and BD intersect each other at O, P is the midpoint of AD. Join OP. In parallelogram, diagonals bisect each other, ∴ BO = OD. Here, O is the mid-point of BD. In △ABD, P and O are midpoints of AD and BD respectively, PO || AB and PO = AB (By midpoint theorem) ......(i) Hence, proved that PO || AB. (ii) ABCD is a parallelogram. ∴ AB = CD .......(ii) Using both (i) and (ii) we get, PO = AB = CD. Hence, proved that PO = CD.

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Mathematics

The diagonals AC and BD of a parallelogram ABCD intersect at O. If P is the mid-point of AD, prove that (i) PO || AB (ii) PO = CD.

Mid-point Theorem

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