Question

in the figure below, $overleftrightarrow{ln}$ is tangent to circle $o$.

which of the following can be concluded from the information?

• $overline{on}$ is a radius of the circle.
• $\triangle mno$ is a right triangle.
• $angle onm$ is an obtuse angle.
• $angle omn$ is a central angle of circle $o$.

Explanation:

1. Recall the property of a tangent to a circle: A tangent to a circle is perpendicular to the radius at the point of tangency. So, \( \angle OMN = 90^\circ \) because \( LN \) is tangent to circle \( O \) at \( M \), and \( OM \) is the radius.
2. Analyze each option:

• Option 1: \( \overline{ON} \) is not a radius (radius is from center to circle, \( OM \) is radius, \( ON \) extends beyond the circle), so this is false.
• Option 2: In \( \triangle MNO \), \( \angle OMN = 90^\circ \) (from tangent - radius property), so \( \triangle MNO \) is a right triangle. This is true.
• Option 3: \( \angle ONM \) is acute (since \( \triangle MNO \) is right - angled at \( M \), the other two angles are acute), so it's not obtuse. This is false.
• Option 4: A central angle has its vertex at the center (\( O \)), \( \angle OMN \) has vertex at \( M \), so it's not a central angle. This is false.

Answer:

B. \( \triangle MNO \) is a right triangle.