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.
Step-by-step solution
Idea: This activity leads to an important result: the angle subtended by an arc at the centre is double the angle it subtends at any point on the rest of the circle. So all points outside the arc see it at the same angle.
- Sample first arc: draw the circle and chord AB so that ∠AOB = 80° (the angle swept from OA to OB through K).½ mark
- Join P, Q, R (on the circle, outside arc AKB) to A and B and measure: ∠APB ≈ 40°, ∠AQB ≈ 40°, ∠ARB ≈ 40°.1 mark
- Repeat with a different arc, say ∠AOB = 130°: now ∠APB ≈ ∠AQB ≈ ∠ARB ≈ 65°.½ mark
- What we notice: (1) the angles at P, Q, R are equal: they do not depend on where the point is, as long as it is on the circle outside the arc; (2) each is half the angle at the centre (40° = ½ × 80°, 65° = ½ × 130°).1 mark
Answer to write in the exam
First arc: ∠AOB = 80°
∠APB = 40°, ∠AQB = 40°, ∠ARB = 40°
Second arc: ∠AOB = 130°; ∠APB = ∠AQB = ∠ARB = 65°
∴ The angles at P, Q, R are equal, and each = ½ × angle subtended at the centre.
Common mistakes that cost marks
- Taking a point on arc AKB itself. The points P, Q, R must be on the circle outside the arc.
- Measuring the angle at P between PA and the tangent, or between PO and PA. The angle needed is ∠APB, between the lines to the ends of the arc.
- Expecting exact equality from a rough drawing; small differences (1° or 2°) are measuring errors.
How this can come in the exam
An arc of a circle subtends 110° at the centre. The angle it subtends at a point on the remaining part of the circle is
- 55°
- 110°
- 220°
- 70°
Show answer
(A) 55°
The angle at a point on the circle outside the arc is half the angle at the centre: ½ × 110° = 55°.
Try one yourself
Arc AB subtends 150° at the centre. Find the angle it subtends at two different points P and Q on the remaining part of the circle.
Show answer
∠APB = ∠AQB = ½ × 150° = 75° each.
More questions like this
- 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.
- Find x in the figure.