Question

if p and q are point of trisection pf the diagonal BD of a parallelogram ABCD, prove that CQ is parallel to AP

Answer 1

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)