Question

Abcd is a parallelogram ac and bd are the diagonals intersect at o p and q are the point of trisection of the diagonal bd.Prove that cq is parallel to ap and also ac bisects pq

Answer 1

 

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

Construction: Join A to Q and P, Join C to Q and P

To Prove: CQIIAP and AC bisects PQ means OP = OQ

Proof: Diagonals AC and BD of a parallelogram bisect each other at O.

So, OB = OD and OC = OA

Now, P and Q trisect the diagonal BD where trisect means divide something in three equal parts.

So, DQ = PQ = BP

OB = OD  

BP +OP = DQ +OQ

OP = OQ (because BP is already equal to DQ)

Thus, in quadrilateral APCQ, diagonal AC and PQ are such that OP = OQ and OA = OC

So, the diagonal AC and PQ bisect each other.

And if the diagonal of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Therefore, APCQ is a parallelogram.

So, CQ || AP

Hence Proved