question_answer

\[ABCD\] is a trapezium such that \[\left. BC \right\|AD\] and \[\text{AB}=\text{4cm}\]. If the diagonals AC and BD intersect at O such that \[\frac{AO}{OC}=\frac{OB}{OD}=\frac{1}{2},\]then CD=

A) 7 cm

B) 8 cm

C) 9 cm

D) 6 cm

Correct Answer: B

Solution :

[b] In \[\Delta AOB\] and \[\Delta COD,\] we have

\[\angle AOB=\angle COD\] (Vertically opposite angles)

\[\frac{AO}{OC}=\frac{OB}{OD}=\frac{1}{2}\]                      (Given)

\[\therefore \,\Delta AOB\tilde{\ }\Delta COD\](By SAS similarity criterion)

\[\therefore \,\,\,\,\,\,\,\,\,\,\,\frac{AB}{CD}=\frac{AO}{OC}\,\,\,\,\,\,\,\Rightarrow \,\,\,\,\,\,\frac{4}{CD}=\frac{1}{2}\,\,\,\,\,\,\,\Rightarrow \,\,\,CD=8cm.\]