Question

Question 5
In figure, if ABIIDC and AC, PQ intersect each other at the point O. Prove that OA.CQ = OC.AP.

Answer 1

Given AC and PQ intersect each other at the point O and AB II DC.
To prove that OA.CQ = OC. AP
Proof
In $\Delta AOP$ and $\Delta COQ$,
$\angle AOP=\angle COQ$ [vertically opposite angles]
$\angle APO=\angle CQO$ [since, AB II DC and PQ is transversal, so alternate angles]
$\therefore \Delta AOP\sim \Delta COQ$ [by AAA similarity criterion]
Then, $\frac{OA}{OC}=\frac{AP}{CQ}$ [since, corresponding sides are proportional]
$\Rightarrow$ OA.CQ = OC.AP