Multiple choice

The points $A,\,B\;and\;C$ be on a circle in such a way that the $\angle ABC=52^{\circ}$ and $\angle ACB=78^{\circ}$. The measure of the angle subtended at the centre by the arc $BC$ will be

  1. $26^{\circ}$
  2. $50^{\circ}$
  3. $100^{\circ}$
  4. $115^{\circ}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In triangle ABC, angle A = 180 - (52 + 78) = 50 degrees. The angle subtended by arc BC at the center is twice the angle subtended at the circumference (angle A). Angle = 2 * 50 = 100 degrees.

AI explanation

The sum of angles in triangle ABC equals 180 degrees. Subtracting the given angles from 180 gives angle BAC as 180 - 52 - 78, which equals 50 degrees. The arc BC is subtended by the inscribed angle BAC at the remaining part of the circle. According to the Inscribed Angle Theorem, the central angle is twice the inscribed angle, so the angle subtended at the center is 2 multiplied by 50 degrees, resulting in 100 degrees.

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