Multiple choice

A, B and C are three consecutive points on the arc of a semi-circle, such that the angles subtended by the chords AB and AC at the centre of the circle are 60 and 100 degrees, respectively. Find the measure of angle BAC.

  1. 30o

  2. 50o

  3. 25o

  4. 20o

  5. 15o
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D Correct answer
Explanation

The angles subtended by chords AB and AC at the center are 60 and 100 degrees. The angle subtended by chord BC at the center is 100 - 60 = 40 degrees. The angle subtended by a chord at the circumference is half the angle subtended at the center. Angle BAC is the angle subtended by chord BC at the circumference, which is 40 / 2 = 20 degrees.

AI explanation

The semi-circle arc spans 180 degrees, and the chords AB and AC subtend central angles of 60 and 100 degrees. Because B lies between A and C on the semi-circle, the central angle for chord BC is 100 - 60 = 40 degrees. The inscribed angle BAC subtends the arc BC, so by the Inscribed Angle Theorem, angle BAC is half of the central angle for arc BC. This makes angle BAC equal to 40 / 2 = 20 degrees.

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