Question

\(PQ\) is tangent to a circle with centre \(O\). If \(\angle POR = 65^{\circ}\), then \(m\angle OPR\) is

• A. \(65^{\circ}\)
• B. \(58.5^{\circ}\)
• C. \(57.5^{\circ}\)
• D. \(45^{\circ}\)

Hint:

In circle geometry, always look for isosceles triangles formed by radii. They are extremely common for calculating unknown angles.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In a circle, any triangle formed by two radii (\(OP\) and \(OR\)) and a chord (\(PR\)) is an isosceles triangle because \(OP = OR\).
Step 2: Key Formula or Approach:
In \(\triangle POR\), \(OP = OR\) (radii).
Therefore, \(\angle OPR = \angle ORP\) (angles opposite to equal sides).
The sum of angles in a triangle is \(180^{\circ}\).
Step 3: Detailed Explanation:
In \(\triangle POR\):
\[ \angle POR + \angle OPR + \angle ORP = 180^{\circ} \]
Substitute \(\angle POR = 65^{\circ}\) and let \(\angle OPR = \angle ORP = x\):
\[ 65^{\circ} + x + x = 180^{\circ} \]
\[ 65^{\circ} + 2x = 180^{\circ} \]
\[ 2x = 180^{\circ} - 65^{\circ} \]
\[ 2x = 115^{\circ} \]
\[ x = \frac{115^{\circ}}{2} = 57.5^{\circ} \]
So, \(m\angle OPR = 57.5^{\circ}\).
Step 4: Final Answer:
The measure of \(\angle OPR\) is \(57.5^{\circ}\).