Question

\( TP \) and \( TQ \) are two tangents drawn from an external point \( T \) to a circle whose center is \( O \) such that \( \angle POQ = 120^\circ \). Then the value of \( \angle OTP \) is:

• A. 40°
• B. 30°
• C. 50°
• D. 60°

Hint:

The angle between two tangents drawn from a common external point is half the angle subtended by the chord at the center of the circle.

Answer 1

The Correct Option is C

We are given two tangents \( TP \) and \( TQ \) drawn from an external point \( T \) to the circle, and we know that the angle \( \angle POQ = 120^\circ \). The key property here is that the angle between two tangents from a common external point to a circle is equal to half the angle subtended by the chord joining the points of tangency at the center of the circle. Thus, the angle \( \angle OTP \) is: \[ \angle OTP = \frac{1}{2} \times \angle POQ = \frac{1}{2} \times 120^\circ = 60^\circ. \] Therefore, the value of \( \angle OTP \) is \( \boxed{60^\circ} \).