OxfordAQA International GCSE Mathematics: Circle properties, circle theorems and geometrical proof (9260)

This guide covers references G8 and G9 in section 3.3.1 (Properties and constructions) of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1, for exams from May/June 2018 onwards. The circle vocabulary and properties in G8 are Core content. The circle theorems in G8 and the geometrical reasoning and proof in G9 are listed in the Extension column, so they are Extension only. All content can be assessed on the Extension papers (1E and 2E).

Use the hub for this course and the printable checklist to track your progress. When you have read this guide, move on to the revision notes and the practice set. Not sure where your geometry stands? Try a free 10-minute diagnostic first.

What you need to be able to do

Reference	What you must be able to do	Tier
G8	Identify and apply circle definitions and properties: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment	Core
G8	Apply the standard circle theorems about angles, radii, tangents and chords (the eight in this guide)	Extension only
G9	Use standard theorems to justify results in geometric contexts, and use them to prove related results	Extension only

Angle facts for triangles, parallel lines and polygons belong to other units. Here you may simply quote them: angles in a triangle add to 180°, angles on a straight line add to 180°, base angles of an isosceles triangle are equal, and alternate angles between parallel lines are equal.

Circle words and properties (G8, Core)

Term	Meaning
Centre	The fixed point, equidistant from every point on the circle
Radius	A straight line joining the centre to any point on the circle
Diameter	A chord passing through the centre; its length is 2 × radius
Circumference	The distance round the circle (also the name of the curve itself)
Chord	A straight line segment with both ends on the circle
Arc	Part of the circumference; the shorter one is the minor arc
Sector	The slice bounded by two radii and the arc between them
Segment	The region cut off by a chord, bounded by the chord and an arc
Tangent	A straight line that touches the circle at exactly one point

Two properties do most of the work in this unit. All radii are equal, so any triangle made from two radii and a chord is isosceles. The diameter is the longest chord.

Worked example 1. A and B lie on a circle with centre O. Angle AOB = 112°. Find angle OAB.

OA = OB (radii), so triangle OAB is isosceles
Angle OAB = (180° − 112°) ÷ 2 = 34°

The eight circle theorems (G8, Extension only)

For each theorem, learn the reason as a short sentence you can write in words. A diagram alone is rarely enough when a question says “give a reason”.

1. Angle at the centre: the angle subtended by an arc at the centre is twice the angle subtended at any point on the circumference.
2. Semicircle: the angle subtended at the circumference by a semicircle is 90°.
3. Same segment: angles in the same segment are equal.
4. Cyclic quadrilateral: opposite angles sum to 180°.
5. Tangent and radius: a tangent at any point on a circle is perpendicular to the radius at that point.
6. Equal tangents: tangents from an external point are equal in length.
7. Chord: the perpendicular from the centre to a chord bisects the chord.
8. Alternate segment: the angle between a tangent and a chord equals the angle in the alternate segment.

“Subtended by an arc” means the angle made at a point by lines drawn to the two ends of the arc. “Alternate segment” means the segment on the other side of the chord from the angle you started with.

Worked example 2 (centre). P, Q and R lie on a circle with centre O, and R is on the major arc PQ. Angle PRQ = 53°. Find angle OPQ.

Angle POQ = 2 × 53° = 106°   (angle at the centre is twice the angle at the circumference)
OP = OQ (radii), so angle OPQ = (180° − 106°) ÷ 2 = 37°

Worked example 3 (semicircle). C lies on a circle with diameter AB. Angle CAB = (4x + 2)° and angle ABC = (2x + 10)°. Find x and both angles.

Angle ACB = 90°   (angle in a semicircle)
(4x + 2) + (2x + 10) + 90 = 180
6x + 102 = 180, so x = 13
Angle CAB = 54° and angle ABC = 36°

Worked example 4 (same segment). P, Q, R and S lie in that order on a circle. The chords PR and QS meet at X. Angle QPR = 52° and angle PQS = 31°. Find angle QSR and angle SXR.

Angle QSR = 52°   (angles in the same segment, both on chord QR, are equal)
In triangle PXQ: angle PXQ = 180° − 52° − 31° = 97°
Angle SXR = 97°   (vertically opposite angles)

Worked example 5 (cyclic quadrilateral). ABCD is cyclic. Angle A = (2x + 15)° and angle C = (3x + 25)°. Angle B = (y + 40)° and angle D = (3y − 20)°. Find the four angles.

A + C = 180°:  5x + 40 = 180, so x = 28; A = 71°, C = 109°
B + D = 180°:  4y + 20 = 180, so y = 40; B = 80°, D = 100°

Opposite angles must be paired. Adjacent angles are not linked by this theorem.

Worked example 6 (tangents). A circle has centre O and radius 5 cm. The tangents from T touch the circle at A and B, and OT = 13 cm. Find TA, then the area of the kite OATB.

Angle OAT = 90°   (tangent is perpendicular to the radius)
TA² = 13² − 5² = 144, so TA = 12 cm
TB = TA = 12 cm   (tangents from an external point)
Kite = 2 triangles, each ½ × 5 × 12 = 30
Area = 60 cm²

Worked example 7 (chord). A chord AB is 18 cm long and is 12 cm from the centre O. Find the radius.

Let M be the foot of the perpendicular from O. AM = 9 cm   (perpendicular from the centre bisects the chord)
OA² = 12² + 9² = 225, so radius = 15 cm

The same triangle works backwards: radius 25 cm and distance 7 cm from the centre give a half-chord of √(625 − 49) = 24 cm, so the chord is 48 cm.

Worked example 8 (alternate segment). A tangent touches a circle at A. B and C are on the circle on opposite sides of A, and the tangent makes 72° with AB and 41° with AC. Find the angles of triangle ABC.

Angle ACB = 72°   (alternate segment theorem, angle between tangent and AB)
Angle ABC = 41°   (alternate segment theorem, angle between tangent and AC)
Angle BAC = 180° − 72° − 41° = 67°   (angles on a straight line at A: 72 + 67 + 41 = 180)

Geometrical reasoning and proof (G9, Extension only)

G9 asks you to use standard theorems to justify results. A proof is a chain: each line is a statement followed by the reason, and the last line is the thing you were asked to prove.

Habits that earn credit:

1. Name the angles with three letters, or define a letter such as “let angle OAB = a”.
2. Give a reason for every line, in words from the list of theorems above.
3. Use the radii. Joining the centre to points on the circle makes isosceles triangles.
4. Finish by stating the result exactly as the question words it.

Proof A: the angle at the centre is twice the angle at the circumference. Take A, B and C on a circle with centre O, with O inside triangle ABC. Join CO and extend it to D on the circle.

OA = OC, so angle OAC = angle OCA = a      (isosceles triangle)
Exterior angle AOD = a + a = 2a            (exterior angle of triangle OAC)
Similarly OB = OC, so angle BOD = 2b where angle OCB = b
Angle AOB = 2a + 2b = 2(a + b) = 2 × angle ACB

Proof B: opposite angles of a cyclic quadrilateral sum to 180°. In cyclic quadrilateral ABCD, let angle BAD = p. The arc BCD subtends 2p at the centre. The other angle at the centre, on arc BAD, is 360° − 2p. The angle at C is half of that.

Angle BCD = (360° − 2p) ÷ 2 = 180° − p
So angle BAD + angle BCD = p + (180° − p) = 180°

Proof C: the alternate segment theorem. Tangent PAQ touches the circle at A, and AD is a diameter. Let B be on the circle with angle QAB = t.

Angle DAQ = 90°   (tangent perpendicular to radius), so angle DAB = 90° − t
Angle ABD = 90°   (angle in a semicircle)
Angle ADB = 180° − 90° − (90° − t) = t
Angle ADB = angle ACB for any C in the segment   (same segment)
So the angle between the tangent and chord AB equals the angle in the alternate segment

Proof D: tangents from a point are equal. Tangents TA and TB, centre O. Angles OAT and OBT are both 90°. OA = OB and OT is common. The triangles OAT and OBT are congruent (right angle, hypotenuse, side), so TA = TB.

Proof E: the line from the centre to the midpoint of a chord is perpendicular to the chord. Let M be the midpoint of chord AB, with centre O.

OA = OB (radii), AM = BM (M is the midpoint), OM is common
Triangles OAM and OBM are congruent (three sides)
So angle OMA = angle OMB, and they lie on a straight line, so each is 90°

A G9 question may also place a circle in a larger figure, for example a quadrilateral whose vertices all lie on a circle. List which theorems are available, then pick the ones that connect the angles you need.

Common errors

• Using the centre theorem when the angle is not at the centre, or forgetting that the angle at the centre is on the same arc.
• Pairing adjacent angles in a cyclic quadrilateral.
• Treating a tangent as a chord: the radius to the point of contact is at 90° to the tangent, not along it.
• Quoting “alternate segment” for the segment next to the angle. The angle you want is on the opposite side of the chord.
• Giving an answer with no reason in a “give a reason” part.

Official syllabus

This guide follows the OxfordAQA International GCSE Mathematics (9260) Specification, Version 5.1, for exams in May/June 2018 onwards, published by OxfordAQA, section 3.3.1, references G8 and G9.