question_answer

Diagonals AC and BD of a trapezium ABCD with \[AB||DC\]intersect each other at the point O, then \[\frac{OA}{OC}=\frac{OB}{OD}\]

A) True

B) False

C) Cannot say

D) Partially True/False m

Correct Answer: A

Solution :

In  \[\Delta AOB\] and \[\Delta OCD\],

\[\angle AOB=\angle COD\]         [vertically opposite angle]

\[\angle ABO=\angle CDO\]         [\[AB\,|\,|\,CD\], alternate angle]

\[\angle BAO=\angle OCD\]         [\[AB\,|\,|\,CD\], alternate angle]

\[\Delta OAB\tilde{\ }\Delta OCD\]           (AAA similarity)

Then, \[\frac{OA}{OC}=\frac{OB}{OD}\]

[corresponding sides are proportional].