An arc of a circle subtends an angle of 70° at the centre. What is the measure of the angle subtended by the arc at a point on the circle?
Answer: The angle at a point on the circle (outside the arc) is half the angle at the centre: ½ × 70° = 35°.
Step-by-step solution
Idea: The angle subtended by an arc at the centre is double the angle it subtends at any point on the remaining part of the circle.
- Let the arc be AB with ∠AOB = 70° at the centre O, and P any point on the circle not on arc AB.
- ∠APB = ½∠AOB = ½ × 70° = 35° (angle at the centre is double the angle at the circle).1 mark
- (If P were on the arc AB itself, P would see the other, major arc, of 360° − 70° = 290°, and the angle would be 145°.)
The arc subtends 35° at a point on the remaining part of the circle.
Answer to write in the exam
∠AOB = 70° (angle subtended by arc AB at the centre)
∠APB = ½∠AOB (P on the remaining part of the circle)
∴ ∠APB = ½ × 70° = 35°
Common mistakes that cost marks
- Doubling instead of halving (140°). The angle at the centre is the larger one.
- Answering 70°: the angle at the circle is not equal to the angle at the centre.
- Taking the point on the arc itself; the rule is for points on the remaining part of the circle.
How this can come in the exam
MCQ (1 mark)
An arc subtends 42° at a point on the remaining part of a circle. The angle it subtends at the centre is
- 21°
- 42°
- 84°
- 138°
Show answer
(C) 84°
Angle at the centre = 2 × 42° = 84°.
Try one yourself
An arc subtends 156° at the centre. Find the angle it subtends at a point on the remaining part of the circle.
Show answer
½ × 156° = 78°.
More questions like this
- The diameter of a circle is 26 cm. A chord of length 24 cm is drawn in the circle. Find the distance from the centre of the circle to the chord.
- A circle has a radius of 15 cm. A chord is drawn. The distance from the centre of the circle to the chord is 9 cm. What is the length of the chord?
- Prove that the perpendicular bisector of a chord passes through the centre of the circle.
- The diameter of a circle is AB. Point C is on the circumference. What is the measure of the ∠ACB? Explain your reasoning.
- ABCD is a cyclic quadrilateral inscribed in a circle. If ∠A measures 75°, what is the measure of ∠C? If ∠B measures 110°, what is the measure of ∠D?