Multiple choice

PA and PB are tangents to the circle with centre O. If $\angle APB = 60^{\circ} ,$ then $\angle OAB $ is

  1. $30^{\circ}$
  2. $120^{\circ}$
  3. $90^{\circ}$
  4. $15^{\circ}$
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A Correct answer
Explanation

In triangle OAP, angle OAP = 90 degrees. In triangle OAP, angle AOP = 180 - 90 - 30 = 60 degrees. Since triangle OAB is isosceles (OA=OB), angle OAB = (180 - angle AOB)/2. Angle AOB = 2 * angle AOP = 120 degrees. Angle OAB = (180 - 120)/2 = 30 degrees.

AI explanation

Tangents drawn from an external point are equal in length, so triangle OAP and OBP are congruent right triangles, making angle OAP equal to angle OBP. The line AO bisects angle APB, so angle OAP is 60 divided by 2 which is 30 degrees. Since OA and OB are radii, the angle between the tangent and the radius at the point of contact is 90 degrees. The result for angle OAB is 30 degrees.

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