Question

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

Answer 1

Proof of the given Relation:

According to the given details

$AC$ and $PQ$ intersect each other at the point $O$ and $AB\parallel DC$.

From $△AOP$ and$△COQ$,

$\angle AOP=\angle COQ$ [Since they are vertically opposite angles]

$\angle APO=\angle CQO$ [since, $AB\parallel DC$ and $PQ$ is transversal, the angles are alternate angles]

∴$△AOP~△COQ$ [using AAA similarity criterion]

Then, since, the corresponding sides are proportional we know that

$\frac{OA}{OC}=\frac{AP}{CQ}$

$OA.CQ=OC.AP$

Hence Proved.