question_answer

In the given figure, M is the mid-point of the side CD of the parallelogram ABCD. What is ON : OB?

A) 3 : 2     

B)  2 : 1   

C)  3 : 1    

D)  5 : 2

Correct Answer: B

Solution :

In \[\Delta DMN\] and \[\Delta BMC,\]

DM = MC                                 [Given]

\[\angle 1=\angle 2\]                   [vertically opposite]

\[\angle 3=\angle 4+\angle 9\]      [Alternate interior angles]

\[\Delta DMN=\Delta BME\] (as a) [Alternate interior angle]

DN = BC = AD

So,       \[AN=2BC\] \[\Rightarrow \] \[\frac{AN}{BC}=\frac{2}{1}\]          ?(i)

In \[\Delta OAN\] and \[\Delta OBC,\]

\[\angle 5=\angle 6\]       [Vertically opposite]

\[\angle 7=\angle 8\]             [Alternate interior angle]

\[\angle 9=\angle 10\]      [Rest angle]

\[\therefore \]      \[\Delta OAN\sim \Delta OBC\]

So, the sides will be in same ratio

\[\frac{AN}{BC}=\frac{ON}{OB}\]

\[\Rightarrow \]   \[\frac{2}{1}=\frac{ON}{OB}\]              [from Eq. (i)]