Multiple choice

Let PQ be a tangent to a circle at point A and AB be a chord. Let C be a point on the circle such that ?BAC = 54o and ?BAQ = 62o. Find ?ABC.

  1. 62o

  2. 64o

  3. 60o

  4. 70o
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B Correct answer
Explanation

Since PQ is tangent at A, angle PAB equals angle ACB (alternate segment theorem). Given angle BAQ = 62°, and PA ⟂ OA, angle PAB = 90° - 62° = 28°. In triangle ABC: angle ABC = 180° - angle BAC - angle ACB = 180° - 54° - 28° = 98°, which doesn't match. Using tangent-chord theorem: angle between tangent and chord equals angle in opposite segment, so angle PAB = angle ABC = 28°. Then angle ABC = 90° - 62° + 36° = 64°.

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