A circle with centre O is drawn, and A, B, C, D are points on the circle (see the figure). Measure the angles subtended by arc AKB and arc CLD at the centre O. If the angle at the centre is less than 180°, it is a minor arc. If the angle at the centre is greater than 180°, it is a major arc. State whether arcs AKB and CLD are minor arcs or major arcs.
Step-by-step solution
Idea: The angle subtended by an arc at the centre is the angle swept by the radius as it moves along that arc from one end to the other. For arc CLD the sweep goes the long way round, through L, so it is a reflex angle: 360° minus the ordinary angle COD.
- Arc AKB: move the radius from OA to OB through K. With a protractor, ∠AOB (on the side of K) ≈ 100°.½ mark
- 100° < 180°, so arc AKB is a minor arc.½ mark
- Arc CLD: the radius moves from OC to OD through L, the long way round. Measure the ordinary angle COD on the other side (through K): about 160°. The angle swept through L is 360° − 160° ≈ 200°.½ mark
- 200° > 180°, so arc CLD is a major arc.½ mark
Check: The two arcs between C and D together make the full circle: the arc through K (about 160°) and the arc through L (about 200°) add up to 360° ✓.
Answer to write in the exam
Angle subtended by arc AKB at O = ∠AOB ≈ 100° < 180°
∴ Arc AKB is a minor arc.
Angle subtended by arc CLD at O = reflex ∠COD = 360° − 160° ≈ 200° > 180°
∴ Arc CLD is a major arc.
Common mistakes that cost marks
- Measuring ∠COD the short way (about 160°) and calling arc CLD minor. The angle for arc CLD is the one swept through L, which is the reflex angle.
- Reading the wrong scale of the protractor (80° instead of 100°).
- Deciding by how long the chord looks. Minor and major depend on the angle at the centre.
How this can come in the exam
An arc of a circle subtends an angle of 210° at the centre. The arc is
- a minor arc
- a major arc
- a semicircle
- a diameter
Show answer
(B) a major arc
210° is more than 180°, so the arc is a major arc.
Try one yourself
The minute hand of a clock moves from 12 to 4. What angle does its tip sweep at the centre, and is the arc traced minor or major?
Show answer
Each hour mark is 360° ÷ 12 = 30°, so 4 marks give 120°, which is less than 180°: a minor arc.
More questions like this
- Draw a circle and a chord AB. Fix an arc AKB formed by AB and a point K between A, B on the circle. Measure the angle subtended at the centre by arc AKB. Take three points P, Q, R on the circle outside arc AKB. Measure the angles subtended by arc AKB at points P, Q, R. What do you notice?
Repeat this activity for a different arc AKB. - The angle subtended by an arc at the centre of the circle is double the angle subtended by the arc at any point on the circle outside the arc.
- The angle subtended by a diameter at any point on the circle is 90°.
- In a circle with centre O, the central angle AOB is 60°. If the radius of the circle is 12 cm, what is the length of the chord AB?
- Let A and B be two points on a circle with centre O.