Multiple choice

If PA and PB are two tangents to a circle with centre O such that ∠APB = 80°, then ∠AOP equals

  1. 40°

  2. 50°

  3. 60°

  4. 70°
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a circle, the line from the center to the external point bisects the angle between the tangents. Thus, angle APO = 80/2 = 40 degrees. In triangle OAP, angle OAP = 90 degrees (tangent is perpendicular to radius). Therefore, angle AOP = 180 - 90 - 40 = 50 degrees.

AI explanation

The radius is perpendicular to the tangent at the point of contact, making triangles OAP and OBP right-angled, and the radii OA and OB are equal, making angles OAP and OBP equal to 90 degrees. In triangle OAP, the sum of the angles is 180 degrees, so angle AOP = 180 - 90 - (80 / 2). Thus, angle AOP equals 50 degrees.

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