Question : Let P and Q be two points on a circle with centre O. If two tangents of the circle through P and Q meet at A with $\angle PAQ=48^{\circ}$, then $\angle APQ$ is:
Option 1: 96°
Option 2: 48°
Option 3: 66°
Option 4: 124°
Correct Answer: 66°
Solution :
We have,
$\angle PAQ = 48^{\circ}$
$\angle APQ=\angle AQP$
AP = AQ (tangents from the same exterior point are equal).
In $\triangle APQ$,
$\angle APQ + \angle AQP + \angle PAQ = 180^{\circ}$
⇒ $2 \angle APQ + 48^{\circ} = 180^{\circ}$
⇒ $2 \angle APQ = 180^{\circ} - 48^{\circ} = 132^{\circ}$
⇒ $\angle APQ = \frac{132^{\circ}}{2} = 66^{\circ}$
Hence, the correct answer is 66°.
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