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

To find \( m\angle OPR \), let us analyze the given problem with the help of the image and the properties of circles.

Given:

• \( PQ \) is a tangent to the circle at point \( P \).
• \( O \) is the center of the circle.
• \(\angle POR = 65^{\circ}\).

Since \(PQ\) is a tangent at \(P\) and \(OP\) is the radius, the angle between the tangent and the radius at the point of contact is \(90^{\circ}\). Thus,

\(\angle OPR = 90^{\circ}\).

Now, in the triangle \( \triangle OPR \), we apply the angle sum property of a triangle which states:

\(\angle POR + \angle OPR + \angle ORP = 180^{\circ}\)

Substituting the known values:

\(65^{\circ} + 90^{\circ} + \angle ORP = 180^{\circ}\)

Solving for \( \angle ORP \):

\(\angle ORP = 180^{\circ} - 65^{\circ} - 90^{\circ} = 25^{\circ}\)

Therefore, the measure \( m\angle OPR \) which we found using calculations is:

55.0°\(.\)

Thus, the correct answer is \( 55.0^{\circ} \).