Question

happy women's Day to all my near and dear .... always be happy and keep smiling... p and q are the trisection of the diagonal bd of a parallelogram abcd.prove that cq is parallel to ap.also prove that ac 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)

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