Question

If TP and TQ are two tangents to a circle with centre O from an external point T so that \(\angle POQ = 120^\circ\), then \(\angle PTQ\) is equal to :

• A. 60°
• B. 70°
• C. 80°
• D. 90°

Hint:

Remember: \(\angle PTQ + \angle POQ = 180^\circ\). They are always supplementary.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In a circle, the angle between two tangents from an external point is supplementary to the angle subtended by the line segments joining the points of contact to the centre.
Step 2: Key Formula or Approach:
In quadrilateral \( OPTQ \): \[ \angle OPT + \angle PTQ + \angle TQO + \angle QOP = 360^\circ \] Since radii are perpendicular to tangents, \( \angle OPT = \angle OQT = 90^\circ \).
Step 3: Detailed Explanation:
1. In quadrilateral \( OPTQ \): \[ 90^\circ + \angle PTQ + 90^\circ + 120^\circ = 360^\circ \] 2. Simplify the sum: \[ 300^\circ + \angle PTQ = 360^\circ \] 3. Solve for \( \angle PTQ \): \[ \angle PTQ = 360^\circ - 300^\circ = 60^\circ \]
Step 4: Final Answer:
The measure of \(\angle PTQ\) is 60°.