IGCSE Maths Circle theorems topical questions

O is the centre of a circle. A, B and C lie on the circumference, and angle AOB = 110°. C is on the major arc AB. (Extended)

1. (a)Find angle ACB.[2]
2. (b)Give the reason for your answer.[1]
3. (c)Another point D also lies on the major arc. Write down angle ADB.[1]

AB is a diameter of a circle and C lies on the circumference.

1. (a)Write down angle ACB and your reason.[2]
2. (b)Given angle CAB = 37°, find angle ABC.[2]

TP is a tangent to a circle with centre O, touching at T. P is outside the circle and angle TPO = 28°.

1. (a)Write down angle OTP.[1]
2. (b)Find angle TOP.[2]
3. (c)Give the reason for your answer to (a).[1]

ABCD is a cyclic quadrilateral with angle A = 82° and angle B = 118°. (Extended)

1. (a)Find angle C.[2]
2. (b)Find angle D.[2]
3. (c)Check your answers using the total angle sum.[1]

AT is the tangent at A to a circle with centre O and diameter AB. Angle BAT = 58°.

1. (a)Write down angle OAT, with a reason.[1]
2. (b)Find angle OAB (the angle between AO and AB).[2]
3. (c)C lies on the circle. Write down angle ACB and find angle ABC.[2]

ABCD is a cyclic quadrilateral in which angle A = 3x and angle C = 2x + 20. (Extended)

1. (a)Form an equation in x and solve it.[3]
2. (b)Find angles A and C.[2]

The tangent at P meets chord PQ at an angle of 48°. R lies in the alternate segment and S in the other segment of the circle. (Extended)

1. (a)Find angle PRQ.[2]
2. (b)T lies in the same segment as R. Write down angle PTQ.[1]
3. (c)Find angle PSQ.[2]

From P, tangents PX and PY are drawn to a circle with centre O, and angle XPY = 64°. (Extended)

1. (a)Write down angles OXP and OYP, with a reason.[1]
2. (b)Find angle XOY.[2]
3. (c)Find angle XOP.[2]

A chord of length 16 cm sits 6 cm from the centre of a circle. (Extended)

1. (a)State the length of each half of the chord, with a reason.[2]
2. (b)Find the radius of the circle.[3]

O is the centre of a circle and angle AOB = 116°. C lies on the major arc AB and D on the minor arc. (Extended)

1. (a)Find angle ACB.[2]
2. (b)Find angle ADB.[2]
3. (c)Explain why angle ACB + angle ADB = 180°.[2]

TA and TB are tangents from T to a circle with centre O, and angle ATB = 48°. (Extended)

1. (a)Find angle AOB.[3]
2. (b)Find angle TAB in triangle ATB.[2]
3. (c)Hence find angle OAB.[1]

A chord of length 24 cm sits 5 cm from the centre of a circle. (Extended)

1. (a)Find the radius of the circle.[3]
2. (b)A second chord of the same circle has length 10 cm. Find its distance from the centre.[3]

O is the centre of a circle. A and B lie on the circle and the reflex angle AOB = 224°. C lies on the minor arc AB and D on the major arc. (Extended)

1. (a)Find angle ACB.[3]
2. (b)Find the non-reflex angle AOB.[1]
3. (c)Find angle ADB.[2]

The tangents at C and D meet at T, and chord CD subtends angle DOC = 120° at the centre O. (Extended)

1. (a)Find angle DTC.[3]
2. (b)B lies on the major arc CD. Find angle DBC.[2]
3. (c)A lies on the minor arc CD. Find angle DAC.[2]