Question

P and Q are the points of trisection of the diagonal AC of parallelogram ABCD. Prove that BQ // DP and BD bisects PQ.

Answer 1

Given: ABCD is a parallelogram. AC and BD are the diagonals of the parallelogram ABCD which intersect in O. The points P and Q trisects the diagonal BD.

To prove:

(i) CQ || AP

(ii) AC bisect PQ.

Proof:

We know that, the diagonals of a parallelogram bisect each other.

∴ AC and BD bisect each other at O.
⇒ OB = OD and OA = OC

Given, P and PQ trisects the diagonal BD.

∴ DQ = PQ = BP

OB = OD

BP = DQ

∴ OB – BP = OD – DQ

⇒ OP = OQ

Thus, in quadrilateral APCQ diagonals AC and PQ are such that OP = OQ and OA = OC, i.e., the diagonals AC and PQ bisect each other at O.
Hence, APCQ is a parallelogram. (If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram)

∴ CQ || AP (Opposite sides of parallelogram are parallel)

Cheers!

Answer 2

In triangles DPC and BQA,
DP = BQ (Given)
AB = CD (ABCD is a parallelogram)

Therefore, CP = AQ
Similarly, AP = CQ

Thus, APCQ is a parallelogram.
Hence, PQ and AC bisect each other.