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