Multiple choice

In figure $A,B$ and $C$ are three points on a circle with centre $O$ such that $\angle{BOC}={30}^{o}$ and $\angle {AOB}={60}^{o}$. If $D$ is a point on the circle other than the arc $ABC$, find $\angle {ADC}$.

  1. $30^\circ$
  2. $90^\circ$
  3. $45^\circ$
  4. $60^\circ$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The central angle subtended by arc AC is angle AOC = angle AOB + angle BOC = 60 + 30 = 90 degrees. The angle subtended by an arc at the center is double the angle subtended at any point on the remaining part of the circle. Therefore, angle ADC = 1/2 * angle AOC = 1/2 * 90 = 45 degrees.

AI explanation

The total angle at the center is angle AOB + angle BOC = 60 degrees + 30 degrees = 90 degrees. The Inscribed Angle Theorem states that an angle subtended by an arc at the circumference is half the angle subtended at the center. Therefore, angle ADC = 90 degrees / 2 = 45 degrees.

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