Question

In figure, if AB || DC and AC, PQ intersect each other at the point 0. Prove that OA . CQ = OC . AP.

Answer 1

$\bf\huge\underline\mathtt\orange{Question}$

In figure, if AB || DC and AC, PQ intersect each other at the point 0. Prove that OA . CQ = OC . AP.

$\bf\huge\underline\mathtt\green{Answer}$

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

To prove: OA • CQ = OC • AP

Proof: In ∆AOP and ∆COQ,

∠AOP = ∠COQ[vertically opposite angles]

∠APO = ∠CQO

[since AB || DC and PQ is transversal, so alternate angles]

Therefore, ∆AOP ~ ∆COQ

⠀⠀⠀⠀⠀⠀⠀⠀⠀[by AA similarity criterion]

Then, $\dfrac{OA}{OC}$ = $\dfrac{AP}{CQ}$

[since, corresponding sides are proportional]

=> OA • CQ = OC • AP ⠀⠀Hence Proved.