Two tangents AP and AQ are drawn to a circle with centre O from an external point A, where P and Q are points on the circle. If \(\angle PAQ=64^\circ\), what is the measure of \(\angle OPQ\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 32°
A tangent is perpendicular to the radius at the point of contact, so in quadrilateral APOQ the angles at P and Q are \(90^\circ\) each. OP and OQ are radii, so \(\triangle OPQ\) is isosceles.
\(\angle POQ=360^\circ-90^\circ-90^\circ-64^\circ=116^\circ\)
In \(\triangle OPQ\), OP \(=\) OQ, so \(\angle OPQ=\angle OQP\)
\(\angle OPQ=\displaystyle\frac{180^\circ-116^\circ}{2}=32^\circ\)
