Multiple choice

A circle centre O, passes through A, B and C. AT is the tangent to the circle at A. CBT is a straight line. Given that $\angle ABO = 68^o$ and $\angle OBC = 20^o$ the magnitude of $\angle ATB$ is

  1. $60^o$
  2. $64^o$
  3. $65^o$
  4. $70^o$
  5. $66^o$
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E Correct answer
AI explanation

Angle ABO is 68 degrees and angle OBC is 20 degrees, so angle ABC is 88 degrees. In triangle OAB, OA and OB are radii, so angle OAB equals angle ABO which is 68 degrees, making the central angle AOB equal to 180 minus 68 minus 68 degrees or 44 degrees. The angle between the tangent AT and chord AB equals half the central angle AOB, so angle TAB equals 44 divided by 2 which is 22 degrees. In triangle ABT, the angles sum to 180 degrees, so angle ATB equals 180 minus 88 minus 22 degrees, giving angle ATB as 70 degrees.

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