Multiple choice

In fig 9.4, AB is a chord of the circle and AOC is its diameter such that ${\angle ACB= 50 ^0}$. If AT is the tangent to the circle at the point A, then ${\angle BAT}$ is equal to

  1. ${65 ^0}$
  2. ${60^ 0}$
  3. ${50 ^0}$
  4. ${40^0}$
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C Correct answer
Explanation

Angle in a semicircle is 90 degrees, so angle ABC = 90. In triangle ABC, angle BAC = 180 - 90 - 50 = 40. Angle between tangent AT and chord AB is equal to the angle in the alternate segment, which is angle ACB = 50.

AI explanation

Using the property of circles, the angle in a semicircle is 90 degrees, which makes angle ABC equal to 90 degrees. In triangle ABC, the sum of angles is 180 degrees, so angle BAC equals 180 minus 90 minus 50, which is 40 degrees. Using the alternate segment theorem, the angle between the tangent AT and chord AB equals the angle in the alternate segment, angle ACB. Therefore, angle BAT is 50 degrees.

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