GCSE Maths Circle theorems questions by topic

Naming the parts.

1. (a)Name the straight line that touches a circle at exactly one point.[1]
2. (b)Name the chord that passes through the centre.[1]
3. (c)Distinguish a sector from a segment in one sentence each.[2]

Centre and circumference. (Higher)

1. (a)An arc subtends 36° at the circumference. Find the angle it subtends at the centre.[2]
2. (b)A different arc subtends 140° at the centre. Find the angle at the circumference.[2]

Semicircles and segments. (Higher)

1. (a)AB is a diameter and C is on the circle with angle BAC = 42°. Find angle ACB and angle ABC.[2]
2. (b)Two angles stand in the same segment on chord PQ. One is 65°. Find the other.[2]

Cyclic quadrilaterals. (Higher)

1. (a)A cyclic quadrilateral has one angle of 102°. Find the angle opposite it.[2]
2. (b)A cyclic quadrilateral has angles 88°, 95°, x° and y° in order. Find x and y.[2]

A chord and a centre are named.

1. (a)A circle has centre O and chord AB. Point C lies on the major arc. Name the region between the chord AB and the minor arc.[1]
2. (b)Angle ACB = 26°. Work out angle AOB.[2]
3. (c)State the theorem you used.[1]

A diameter anchors a triangle.

1. (a)PQ is a diameter of a circle and R lies on the circumference. Write down angle PRQ, giving a reason.[2]
2. (b)Angle PQR = 41°. Work out angle QPR.[2]

An isosceles triangle from two radii. (Higher)

1. (a)Explain why triangle OAB (O the centre, A and B on the circle) is isosceles.[1]
2. (b)Angle AOB = 68°. Find the base angles of triangle OAB.[2]
3. (c)A different chord gives base angles of 60°. Find the angle at the centre and name the special triangle.[2]

Tangents from a point. (Higher)

1. (a)A circle has centre O and radius 5 cm. Point P is 13 cm from O, and PT is a tangent. Find PT.[3]
2. (b)Find angle OPT, to 1 decimal place.[2]

A three-theorem angle chase. (Higher)

1. (a)A, B, C and D lie on a circle, centre O. Angle AOB = 110°, and C and D both stand on the same arc AB. Find angle ACB and angle ADB.[3]
2. (b)Name the theorem that makes angle ACB and angle ADB equal.[2]

A cyclic quadrilateral with algebra. (Higher)

1. (a)A cyclic quadrilateral has opposite angles (2x + 10)° and (3x − 5)°. Form an equation in x.[2]
2. (b)Solve for x.[1]
3. (c)Find both angles.[2]

A cyclic quadrilateral carries algebra.

1. (a)ABCD is a cyclic quadrilateral with angle A = 2x and angle C = 3x − 30. Form an equation in x.[2]
2. (b)Solve your equation.[1]
3. (c)Work out angles A and C.[2]

The alternate segment theorem. (Higher)

1. (a)State the alternate segment theorem.[2]
2. (b)The angle between tangent TA and chord AB at A is 63°. C is on the circle in the alternate segment. Find angle ACB.[2]
3. (c)Given also that angle ABC = 43°, find angle BAC.[2]

Proving the semicircle result. (Higher)

1. (a)AB is a diameter, C is on the circle, and O is the centre. Describe the two triangles created by drawing OC.[2]
2. (b)Let angle ACO = x and angle OCB = y. Write down the two isosceles-triangle facts.[2]
3. (c)Prove that angle ACB = 90°.[2]

Diameter, triangle and tangent. (Higher)

1. (a)AB is a diameter of a circle and C lies on it. Angle BAC = 33°. Find angle ACB and angle ABC.[3]
2. (b)CT is the tangent to the circle at C. Find the angle between CT and chord CA.[3]

A chord 16 cm from a nearby centre. (Higher)

1. (a)A chord PQ is 16 cm long and its perpendicular distance from the centre O is 6 cm. Find the radius.[3]
2. (b)Find the angle POQ subtended at the centre, to 1 decimal place.[3]

Two tangents hug a circle. (Higher)

1. (a)Tangents PA and PB are drawn from an external point P. Explain why PA = PB.[2]
2. (b)Angle APB = 60°. Work out angle AOB, where O is the centre.[2]
3. (c)Work out angle OAB.[1]

A tangent and a chord agree with the centre. (Higher)

1. (a)A tangent at P meets chord PQ at 65°. Point R lies on the circle in the alternate segment. Work out angle PRQ.[2]
2. (b)Chord PQ subtends 130° at the centre. Show that this agrees with your answer to part (a).[3]
3. (c)State the name of the theorem linking parts (a) and (b).[1]

A four-point angle chase runs through a diameter. (Higher)

1. (a)Points A, B, C and D lie on a circle with AB a diameter. Write down angle ACB, giving a reason.[2]
2. (b)Given angle CAB = 34°, work out angle ABC.[2]
3. (c)Point D lies on the same segment as B with respect to chord AC. Write down angle ADC, giving a reason.[2]