A chord divides a circle into two arcs. The angle subtended by the major arc at the circumference is 110° . What is the central angle subtended by the major arc?

Correct — C. The angle an arc subtends at the centre is twice the angle it subtends at a point on the rest of the circle.

Angle of the major arc at the circumference = 110°
Central angle = 2 × 110° = 220° → option (c)

Check: the minor arc then has 360° − 220° = 140° at the centre and 70° at the circumference, and 110° + 70° = 180°, as opposite angles of a cyclic quadrilateral must be.

Why the others are wrong

• (a)
  55° — 55° halves 110° instead of doubling it. The centre sees an arc at twice the angle a point on the circle does, not half.
• (b)
  110° — 110° copies the angle at the circumference without doubling it. A major arc's central angle is also always more than 180°, which 110° is not.
• (d)
  270° — 270° would need an angle of 270° ÷ 2 = 135° at the circumference. The given angle is 110°, which doubles to 220°.

The inscribed angle theorem: the angle an arc subtends at the centre is double the angle it subtends at any point on the remaining part of the circle.

For a major arc the central angle is a reflex angle, more than 180°. The point that looks at a major arc lies on the minor arc, and the angle there is obtuse, like the 110° here.

Key facts

• Angle at the centre = 2 × angle at the circumference, for the same arc.
• A major arc subtends a reflex angle (more than 180°) at the centre.
• The central angles on the two sides of a chord add up to 360°.
• Opposite angles of a cyclic quadrilateral add up to 180°.

Common traps

• Halving 110° instead of doubling it, which gives 55°.
• Stopping at 110°, the angle at the circumference, as if the centre saw the arc at the same angle.

No directly related past PYQ was found.