Circle theorems Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Circle theorems questions. See exactly how to solve problems on angle at the centre is twice the angle at the circumference, angle in a semicircle, angles in the same segment, opposite angles of a cyclic quadrilateral.

angle at the centre is twice the angle at the circumferenceangle in a semicircleangles in the same segmentopposite angles of a cyclic quadrilateraltangent perpendicular to the radiustwo tangents from a point
GCSE Higher70 questionsStep-by-step solutions
Question 1
2 markseasy
AA, BB and CC are points on a circle with centre OO. CC lies on the major arc ABAB. Angle AOB=140∘AOB = 140^\circ. Work out the size of angle ACBACB.

Worked solution

  1. Identify which angle is at the centre and which is at the circumference.

    ∠AOB=140∘ (centre),∠ACB (circumference)\angle AOB = 140^\circ \text{ (centre)}, \quad \angle ACB \text{ (circumference)}

    Both angles stand on the same chord ABAB. OO is the centre, so ∠AOB=140∘\angle AOB = 140^\circ is the angle at the centre; CC is on the circumference, so ∠ACB\angle ACB is the angle at the circumference.

  2. Halve the angle at the centre.

    ∠ACB=12×140∘=70∘\angle ACB = \frac{1}{2} \times 140^\circ = 70^\circ

    The angle at the centre is twice the angle at the circumference, so the angle at the circumference is half the angle at the centre. Half of 140∘140^\circ is 70∘70^\circ.

  3. State the size of the angle ACB.

    ∠ACB=70∘\angle ACB = 70^\circ

    The angle ACBACB is 70∘70^\circ.

Answer
∠ACB=70∘\angle ACB = 70^\circ
Question 2
2 markseasy
AA, BB and CC are points on a circle with centre OO. CC lies on the major arc ABAB. Angle AOB=110∘AOB = 110^\circ. Work out the size of angle ACBACB.

Worked solution

  1. Identify which angle is at the centre and which is at the circumference.

    ∠AOB=110∘ (centre),∠ACB (circumference)\angle AOB = 110^\circ \text{ (centre)}, \quad \angle ACB \text{ (circumference)}

    Both angles stand on the same chord ABAB. OO is the centre, so ∠AOB=110∘\angle AOB = 110^\circ is the angle at the centre; CC is on the circumference, so ∠ACB\angle ACB is the angle at the circumference.

  2. Halve the angle at the centre.

    ∠ACB=12×110∘=55∘\angle ACB = \frac{1}{2} \times 110^\circ = 55^\circ

    The angle at the centre is twice the angle at the circumference, so the angle at the circumference is half the angle at the centre. Half of 110∘110^\circ is 55∘55^\circ.

  3. State the size of the angle ACB.

    ∠ACB=55∘\angle ACB = 55^\circ

    The angle ACBACB is 55∘55^\circ.

Answer
∠ACB=55∘\angle ACB = 55^\circ
Question 3
2 markseasy
AA, BB and CC are points on a circle with centre OO. CC lies on the major arc ABAB. Angle ACB=35∘ACB = 35^\circ. Work out the size of angle AOBAOB.

Worked solution

  1. Identify which angle is at the centre and which is at the circumference.

    ∠ACB=35∘ (circumference),∠AOB (centre)\angle ACB = 35^\circ \text{ (circumference)}, \quad \angle AOB \text{ (centre)}

    CC is on the circumference and OO is the centre, and both angles stand on the same chord ABAB. So ∠ACB=35∘\angle ACB = 35^\circ is the angle at the circumference.

  2. Double the angle at the circumference.

    ∠AOB=2×35∘=70∘\angle AOB = 2 \times 35^\circ = 70^\circ

    The angle at the centre is twice the angle at the circumference, so ∠AOB=2×35∘=70∘\angle AOB = 2 \times 35^\circ = 70^\circ.

  3. State the size of the angle AOB.

    ∠AOB=70∘\angle AOB = 70^\circ

    The angle AOBAOB is 70∘70^\circ.

Answer
∠AOB=70∘\angle AOB = 70^\circ
Question 4
2 markseasy
AA, BB and CC are points on a circle with centre OO. CC lies on the major arc ABAB. Angle ACB=56∘ACB = 56^\circ. Work out the size of angle AOBAOB.

Worked solution

  1. Identify which angle is at the centre and which is at the circumference.

    ∠ACB=56∘ (circumference),∠AOB (centre)\angle ACB = 56^\circ \text{ (circumference)}, \quad \angle AOB \text{ (centre)}

    CC is on the circumference and OO is the centre, and both angles stand on the same chord ABAB. So ∠ACB=56∘\angle ACB = 56^\circ is the angle at the circumference.

  2. Double the angle at the circumference.

    ∠AOB=2×56∘=112∘\angle AOB = 2 \times 56^\circ = 112^\circ

    The angle at the centre is twice the angle at the circumference, so ∠AOB=2×56∘=112∘\angle AOB = 2 \times 56^\circ = 112^\circ.

  3. State the size of the angle AOB.

    ∠AOB=112∘\angle AOB = 112^\circ

    The angle AOBAOB is 112∘112^\circ.

Answer
∠AOB=112∘\angle AOB = 112^\circ
Question 5
2 markseasy
ABAB is a diameter of a circle. CC is a point on the circle. Angle CAB=32∘CAB = 32^\circ. Work out the size of angle ABCABC.

Worked solution

  1. Use the angle in a semicircle.

    ∠ACB=90∘\angle ACB = 90^\circ

    ABAB is a diameter, and CC is on the circle, so angle ACBACB is the angle in a semicircle. The angle in a semicircle is a right angle, so ∠ACB=90∘\angle ACB = 90^\circ.

  2. Use the angle sum of triangle ABC.

    ∠ABC=180∘−90∘−32∘=58∘\angle ABC = 180^\circ - 90^\circ - 32^\circ = 58^\circ

    The angles of triangle ABCABC add to 180∘180^\circ. Two of them are 90∘90^\circ and 32∘32^\circ, so the third is 58∘58^\circ.

  3. State the size of the angle ABC.

    ∠ABC=58∘\angle ABC = 58^\circ

    The angle ABCABC is 58∘58^\circ.

Answer
∠ABC=58∘\angle ABC = 58^\circ

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