Question

In figure, if AB || DC and AC and PQ intersect each other at the point O, prove that OA . CQ = OC . AP

Answer 1

According to the question,

AC and PQ intersect each other at the point O and AB || DC.

From ∆AOP and ∆COQ,

∠AOP = ∠COQ   ...[Since they are vertically opposite angles]

∠APO = ∠CQO   ...[Since, AB || DC and PQ is transversal, the angles are alternate angles]

∴ ∆AOP ∼ ∆COQ  ...[Using AAA similarity criterion]

Then, since, corresponding sides are proportional

We have,

`("OA")/("OC") = ("AP")/("CQ")`

OA × CQ = OC × AP

Hence proved.