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

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

Answers (1)

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

To prove:- OA.CQ = OC.AP

Proof:-

In\; \Delta AOP \; and \; \Delta COQ

\angle AOP = \angle COQ  \text{(vertically opposite angles)}

Since AB || OC and PQ is transversal

\therefore \angle APO = \angle CQO  \text{(alternate angles)}

As we know that if the two angles of one triangle are equal to the two angles of another triangle, then the two triangles are similar by AA similarity criterion

\therefore \Delta AOP\sim \Delta COQ

Then

\frac{OA}{OC}=\frac{AP}{CQ}    \text{[Corresponding sides are proportional]}

By cross multiply we get

OA.CQ = AP.OC

Hence proved.

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