Question

Question 9
Points P and Q have been taken on opposite sides AB and CD, respectively of a parallelogram ABCD such that AP=CQ. Show that AC and PQ bisect each other.

Answer 1

Given ABCD is a parallelogram and AP=CQ

To show AC and PQ bisect each other.

Proof in $\Delta AMP$ and $\Delta MCQ$ [alternate interior angles]

AP = CQ [ given]

And $\angle APM=\angle CQM$ [alternate interior angles]

$\therefore \Delta AMP\cong \Delta CMQ$ [ By ASA congruence rule]

$\Rightarrow AM=CM$ [ By CPCT rule]

And PM = MQ [ By CPTC rule]

Hence, AC and PQ bisect each other.