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. \(60^{\circ}\)
• B. \(70^{\circ}\)
• C. \(80^{\circ}\)
• D. \(90^{\circ}\)

Hint:

For any external point \(T\) and center \(O\), the angle between the tangents (\(\angle PTQ\)) and the angle subtended by the radii at the center (\(\angle POQ\)) are supplementary.
Simply compute: \(\angle PTQ = 180^{\circ} - 120^{\circ} = 60^{\circ}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Tangents to a circle from an external point are perpendicular to the radii at the point of contact. Therefore, \(\angle OPT = \angle OQT = 90^{\circ}\).
Step 2: Key Formula or Approach:
In quadrilateral \(OPTQ\), the sum of all interior angles is \(360^{\circ}\).
Step 3: Detailed Explanation:
\[ \angle OPT + \angle PTQ + \angle TQO + \angle POQ = 360^{\circ} \]
Substituting the known values:
\[ 90^{\circ} + \angle PTQ + 90^{\circ} + 120^{\circ} = 360^{\circ} \]
\[ \angle PTQ + 300^{\circ} = 360^{\circ} \]
\[ \angle PTQ = 360^{\circ} - 300^{\circ} = 60^{\circ} \]
Step 4: Final Answer:
The measure of \(\angle PTQ\) is \(60^{\circ}\).