Multiple choice

$TP$ and $TQ$ are two tangents to a circle $C(0, r)$ at points $P$ and $Q$ respectively. If $\angle POQ = 70^{\circ} $. Then find $\angle PTQ.$

  1. $90$ $^{\circ}$
  2. $70$ $^{\circ}$
  3. $60$ $^{\circ}$
  4. $110$ $^{\circ}$
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D Correct answer
Explanation

In the quadrilateral OQTP, the angles at P and Q are 90 degrees because tangents are perpendicular to the radius. The sum of angles in a quadrilateral is 360 degrees. Thus, angle PTQ = 360 - 90 - 90 - 70 = 110 degrees.

AI explanation

According to the properties of tangents drawn from an external point to a circle, the angle between the two tangents and the angle subtended at the center by the chord of contact are supplementary. Adding the given central angle of 70 degrees to the unknown angle PTQ yields 180 degrees. Solving 180 minus 70 gives the measure of angle PTQ as 110 degrees.

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