In the given figure, $M$ is the mid-point of the side $C D$ of the parallelogram $A B C D .$ What is $O N : O B ?$

3 : 2

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2 : 1

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3 : 1

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5 : 2

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Solution

The correct option is B $2 : 1$
In $△$ s $A O B$ and $C O M$
$∠ A O B = ∠ C O M$ (vertically opposite angles)
$∠ O A B = ∠ O C M$ (Alternate angles)
Thus, $△ A O B \sim △ C O M$ (AA rule)
$\frac{A B}{M C} = \frac{A O}{O C} = \frac{O B}{O M} \Rightarrow \frac{2}{1} = \frac{A O}{O C} = \frac{O B}{O M}$ (Since, M is the mid point of CD and AB = CD)
Now In $△$ s $A O N$ and $C O B$
$∠ A O N = ∠ C O B$ (Vertically opposite angles)
$∠ A N O = ∠ O B C$ (Alternate angles)
Thus, $△ A O N \sim △ C O B$ (AA rule)
Hence, $\frac{A O}{O C} = \frac{O N}{O B} \Rightarrow \frac{O N}{O B} = 2 \Rightarrow O N : O B = 2 : 1.$

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