The diameter of a circle is AB. Point C is on the circumference. What is the measure of the ∠ACB? Explain your reasoning.
Answer: ∠ACB = 90°. The arc AB not containing C subtends a straight angle (180°) at the centre O, so it subtends ½ × 180° = 90° at C.
Step-by-step solution
Given: AB is a diameter of a circle with centre O; C is on the circle
To find: ∠ACB, with reasons
To find: ∠ACB, with reasons
Idea: The angle in a semicircle is a right angle: it follows from “angle at the centre = 2 × angle at the circle”, because a diameter makes a straight angle at the centre.
- O is the midpoint of AB, so A, O, B are on a line and the arc AB not containing C subtends ∠AOB = 180° at the centre.1 mark
- C lies on the circle outside that arc, so ∠ACB = ½∠AOB = ½ × 180° = 90°.1 mark
- Another way: join OC. OA = OC = OB, so ∠OAC = ∠OCA = a and ∠OBC = ∠OCB = b. In ΔABC, a + b + (a + b) = 180°, so ∠ACB = a + b = 90°.
∠ACB = 90°, because the diameter subtends 180° at the centre and therefore half of that, 90°, at any point C on the circle.
Answer to write in the exam
AB is a diameter ⇒ ∠AOB = 180° (straight angle)
∠ACB = ½∠AOB (angle at the centre is double the angle at the circle)
∴ ∠ACB = ½ × 180° = 90° (angle in a semicircle)
Common mistakes that cost marks
- Answering 180°: that is the angle at the centre, not at C.
- Giving no reason. Write “angle in a semicircle” or show the half-of-180° step.
- Thinking the answer depends on where C is. It is 90° for every point C on the circle (other than A and B).
How this can come in the exam
MCQ (1 mark)
AB is a diameter of a circle and C is on the circle. If ∠CAB = 2x and ∠CBA = 3x, then x is
- 18°
- 30°
- 36°
- 45°
Show answer
(A) 18°
∠ACB = 90°, so 2x + 3x = 90° and x = 18°.
Try one yourself
AB is a diameter of length 65 mm and C is a point on the circle with AC = 33 mm. Find BC.
Show answer
∠ACB = 90° (angle in a semicircle), so BC = √(652 − 332) = √(4225 − 1089) = √3136 = 56 mm.
More questions like this
- 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?
- Quadrilateral PQRS is inscribed in a circle. If ∠P = (2x + 10)° and ∠R = (3x − 20)°, find the value of x and the measures of ∠P and ∠R.
- The distance of a chord of length 16 cm from the centre of a circle is 6 cm. Find the radius of the circle.
- A cyclic quadrilateral has sides 5, 5, 12, 12 units. Find its area.
- Consider a cyclic quadrilateral. Without drawing its circumcircle, how can we find out whether the centre of the circumcircle lies inside the quadrilateral or outside? What is the best way of finding out?