1.

ABCD is a parallelogram. AC and BD are the diagonals intersect at O. P and Q are the points of tri section of the diagonal BD. Prove that CQ||AP and also AC bisects PQ (see figure).

Answer»


Given AC and BD are the diagonals of the parallelogram ABCD.
The POINTS P and Q trisects the diagonal BD
We know that, the diagonals of a parallelogram bisect each other.
∴ AC and BD bisect each other at O.
⇒ OA = OC and OB = OD
Given P and Q trisects the diagonal BD.
∴ BP = PQ = DQ → (1)
Consider, OB = OD
BP + OP = OQ + DQ
DQ + OP = OQ + DQ [From (1)]
∴ OP = OQ
In quadrilateral APCQ diagonals AC and PQ bisect each other
Hence, APCQ is a parallelogram. [Since, if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram]
∴ CQ || AP [OPPOSITE sides of parallelogram].

See FIGURE down in attachment.

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