Question

If OM=ON, then what are the length of $\text{AM}$ and value of $\angle \text{OMB}?$

• A. $\text{AM}=9$ cm
  $y={90}^{\circ }$
• B. $\text{AM}=18$ cm
  $y={60}^{\circ }$
• C. $\text{AM}=9$ cm
  $y={60}^{\circ }$
• D. $\text{AM}=18$ cm
  $y={90}^{\circ }$

Answer 1

The correct option is A $\text{AM}=9$ cm

$y={90}^{\circ }$
Given that,
$\text{AM}=\text{MB}$
$\text{OM}=\text{ON}$
$\text{PQ}=18$ cm
Note 1: Line $\text{OM}$ bisects the chord $\text{AB}$, so $\text{OM}$ is perpendicular to $\text{AB}$
$\Rightarrow m\angle \text{OMB}={90}^{\circ }$
$\therefore y={90}^{\circ }$

Note 2: Chords $\text{PQ}$ and $\text{AB}$ are equidistant from the centre $\text{O}$, so both chords will be of equal length.
$\Rightarrow \text{AB}=\text{PQ}$
$\therefore \text{AB}=18$ cm
Since, $\text{M}$ is the mid point of chord $\text{AB}$,
$\Rightarrow \text{AM}=\frac{\text{AB}}{2}$
$\therefore \text{AM}=\frac{18}{2}$
$=9$ cm