Multiple choice

Angles subtended by chords $AC$ and $BC$ at the centre $O$ of the circle are ${ 55 }^{ o }$ and ${ 155 }^{ o }$ respectively. What is the measure of $\angle ACB$?

  1. ${ 65 }^{ o }$
  2. ${ 75 }^{ o }$
  3. ${ 105 }^{ o }$
  4. ${ 135 }^{ o }$
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B Correct answer
Explanation

The angle subtended by a chord at the center is twice the angle at the circumference. However, this question describes angles subtended by chords AC and BC at the center. The angle ACB is half the angle subtended by the arc AB at the center. If the angles subtended by AC and BC are 55 and 155, the angle subtended by arc AB is 360 - (55+155) = 150. Thus, angle ACB = 150/2 = 75 degrees.

AI explanation

The total angle around the center O is 360 degrees, so the reflex angle AOC is 360 minus 155 minus 55, which equals 150 degrees. The interior angle ACB of triangle ABC is half of this reflex angle because it is an inscribed angle subtending the same arc. Thus, angle ACB is 150 divided by 2, equaling 75 degrees.

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