question_answer

ABCD is a trapezium in which AB is parallel to DC. If the diagonals intersect at C, then which one of the following is correct?

A) \[\frac{OA}{OC}=\frac{OB}{OD}\]

B) \[\frac{AD}{BC}=\frac{AB}{DC}\]

C) \[\frac{OB}{OD}=\frac{BC}{CD}\]

D) \[\frac{OA}{OC}=\frac{DA}{DC}\]

Correct Answer: A

Solution :

 Consider \[\Delta \,OAB\]and \[\Delta \,OCD\]. \[\angle DOC=\angle BOA\] (Opposite angle) \[\angle OCD=\angle OAB\] (Alternate angle) \[\angle ODC=\angle OBA\] (Alternate angles) Hence \[\Delta \,OAB\] and \[\Delta \,OCD\] are similar. \[\therefore \]  \[\frac{OC}{OA}=\frac{OD}{OB}=\frac{DC}{AB}\] or            \[\frac{OA}{OC}=\frac{OB}{OD}\]