From an external point P, two tangents PQ and PR are drawn to a circle with centre O. If the angle between the tangents, ∠QPR, is 120°, then what is the measure of the central angle ∠POQ in degrees?

Introduction / Context:
This question checks understanding of the relationship between the angle formed by two tangents drawn from an external point to a circle and the central angle subtended by the points of contact. This is a standard geometry result used in many circle based questions.

Given Data / Assumptions:
- A circle with centre O is given.
- From an external point P, two tangents PQ and PR are drawn, touching the circle at Q and R respectively.
- The angle between the tangents, ∠QPR, is 120°.
- We need to find the central angle ∠POQ in degrees.

Concept / Approach:
The line segments OQ and OR are radii drawn to the points of tangency, so they are perpendicular to the tangents PQ and PR. Therefore, quadrilateral POQR has right angles at Q and R. A known theorem states that the angle between two tangents from an external point is supplementary to the central angle subtended by the chord joining the points of contact. In symbols:
∠QPR = 180° − ∠QOR.

Step-by-Step Solution:
Step 1: Recognize that OQ ⟂ PQ and OR ⟂ PR, since a radius to the point of tangency is perpendicular to the tangent. Step 2: Note that the lines OQ and OR form the central angle ∠QOR, which is the angle between the radii to the points of tangency. Step 3: Use the known relation between the angle between tangents and the corresponding central angle: ∠QPR = 180° − ∠QOR. Step 4: Substitute the given value: 120° = 180° − ∠QOR. Step 5: Rearrange to find ∠QOR: ∠QOR = 180° − 120° = 60°. Step 6: The central angle ∠POQ is the angle between OP and OQ, but OP is the line from centre to external point and does not change the central angle subtended by chord QR. The central angle associated with the points of contact is ∠QOR, so the required central angle is 60°.

Verification / Alternative Check:
In some presentations, the angle between tangents is described as the exterior angle of the triangle formed by joining the centre O to the points of contact Q and R and the point P. The interior angle at O then satisfies the same supplementary relation with ∠QPR, again leading to 60° as the central angle. This consistency supports the answer.

Why Other Options Are Wrong:
Option a, 90°, would make the angle between tangents equal to 180° − 90° = 90°, not the given 120°.
Option b, 45°, would lead to an angle between tangents of 135°, which does not match 120°.
Option c, 30°, would give ∠QPR = 150°, again not the given 120°.

Common Pitfalls:
A frequent error is to think that the angle between tangents equals the central angle, rather than being supplementary to it. Another common mistake is to forget that OQ and OR are perpendicular to the tangents, which is essential for visualizing the correct quadrilateral and angle relationships. Carefully recalling the direct formula ∠between tangents = 180° − ∠central helps avoid such confusion.

Final Answer:
The measure of the central angle is 60°.