OxfordAQA International GCSE Mathematics: Circle properties, theorems and proof (9260) – Revision Notes

These are condensed notes for references G8 and G9 in section 3.3.1 of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1, for exams from May/June 2018 onwards. Circle vocabulary and properties are Core. The circle theorems and the geometrical reasoning and proof are Extension only, and all content can be assessed on the Extension papers. For full explanations and worked proofs, read the study guide. When you are ready to test yourself, use the practice set, the course hub, the printable checklist and the free 10-minute diagnostics.

Circle words (G8, Core)

Word	Remember
Radius	Centre to circumference; all radii are equal
Diameter	Chord through the centre; 2 × radius; longest chord
Chord	Straight line joining two points on the circle
Arc	Part of the circumference
Sector	Two radii and an arc
Segment	A chord and an arc
Tangent	Touches the circle at one point only

Two radii and a chord always make an isosceles triangle. If the angle at the centre is c, each base angle is (180° − c) ÷ 2.

The eight theorems (G8, Extension only)

No.	Theorem (reason to write)	Picture to hold in mind
1	The angle subtended by an arc at the centre is twice the angle at the circumference	Centre angle on the same arc, doubled
2	The angle in a semicircle is 90°	Diameter as the base of the triangle
3	Angles in the same segment are equal	A “bow tie” from one chord
4	Opposite angles of a cyclic quadrilateral sum to 180°	Four points on the circle
5	A tangent is perpendicular to the radius at the point of contact	Right angle at the contact point
6	Tangents from an external point are equal in length	Kite with two right angles
7	The perpendicular from the centre to a chord bisects the chord	Right triangle with half the chord
8	Alternate segment: angle between tangent and chord equals the angle in the alternate segment	Tangent angle matches the far side

Method for an angle question

1. Mark every radius and every right angle you can justify.
2. Look for the centre: if one angle is at O and one is on the circle on the same arc, use theorem 1.
3. Look for a diameter: that gives 90° by theorem 2.
4. Look for four points on a circle forming a quadrilateral: theorem 4.
5. Look for two angles looking at the same chord from the same side: theorem 3.
6. If a tangent appears, think theorems 5, 6 and 8.
7. Write each angle with its reason, in the order you found it.

Method for a length question

1. Join the centre to the point you need (a tangent contact or a chord end).
2. The right angle comes from theorem 5 or 7.
3. Use Pythagoras in that right-angled triangle.
4. Use theorem 6 for the second tangent, or double the half-chord for the whole chord.

Useful triples in circle problems: 5-12-13, 3-4-5 and 8-15-17 (and multiples). If your numbers do not fit one, check which side is the hypotenuse. In a tangent triangle, OT is the hypotenuse. In a chord triangle, the radius is.

Small worked reminders

Centre angle. Angle ACB = 35° on the major arc, so angle AOB = 70°. If OA and OB are radii, angle OAB = 55°.

Cyclic quadrilateral. Angle A = 2x + 10 and angle C = x + 50: 3x + 60 = 180, so x = 40; A = 90°, C = 90°.

Chord. Radius 13 cm, chord 24 cm: half-chord 12 cm, so the distance from the centre is √(169 − 144) = 5 cm.

Tangent. Radius 7 cm, OT = 25 cm: TA = √(625 − 49) = 24 cm.

Alternate segment. The tangent makes 56° with chord AB, so the angle in the segment on the other side of AB is also 56°.

Geometrical proof (G9, Extension only)

Every proof line is a statement plus a reason. Plan it like this.

Step	What to do
Define	Give a name to an angle (“let angle OAB = a”) or mark equal radii
Reason	Quote one theorem per line
Link	Show how the lines combine (sum to 180°, double, subtract)
Conclude	Write the result in the form the question asked for

Standard proof patterns:

• Congruent triangles in a circle. Two radii give one pair of equal sides. A tangent contact gives a right angle. A shared line (such as OT) gives a common side. Right angle, hypotenuse, side, or three sides, proves congruence.
• Isosceles triangles from radii. Base angles are equal, which links the centre angle to the circumference angle.
• Chain through the centre. Angle at the centre = 2 × angle on the circle, so two angles on the same arc must be equal.

Proof of cyclic quadrilateral: if angle BAD = p, the centre angle on the opposite arc is 2p, and the other centre angle is 360° − 2p, so angle BCD = 180° − p. Learn this one, and also the proof that tangents from a point are equal: right angles at A and B, OA = OB and OT common, so the triangles are congruent.

Short reasons to reuse

• “Radii of the same circle are equal, so the triangle is isosceles.”
• “Angle at the centre is twice the angle at the circumference.”
• “Angle in a semicircle is a right angle.”
• “Angles in the same segment are equal.”
• “Opposite angles in a cyclic quadrilateral sum to 180°.”
• “A tangent is perpendicular to the radius at the point of contact.”
• “Tangents from an external point are equal in length.”
• “The perpendicular from the centre bisects the chord.”
• “Alternate segment theorem.”
• “Angles in a triangle sum to 180°” (and the other basic angle facts, quoted as given).

Must-know distinctions

Often confused	Difference
Chord and tangent	A chord joins two points on the circle; a tangent meets it once
Segment and sector	Segment from a chord; sector from two radii
Centre theorem and same-segment	Centre theorem compares centre and circumference; same-segment compares two circumference angles
Opposite and adjacent angles in a cyclic quadrilateral	Only opposite angles sum to 180°
Semicircle and half circle	The 90° theorem needs the angle on the circumference, with the diameter as one side

Quick self-test

1. What is the name of the longest chord of a circle?
2. An arc makes an angle of 98° at the centre. What angle does the same arc make at the circumference?
3. Point C lies on a circle with diameter AB, and angle CAB = 27°. Find angle ABC.
4. ABCD is a cyclic quadrilateral and angle DAB = 88°. Find angle BCD.
5. A tangent from T touches a circle with centre O at A. The radius is 7 cm and OT = 25 cm. Find TA.
6. A circle has radius 13 cm. A chord is 24 cm long. How far is the chord from the centre?
7. The angle between a tangent and chord AB is 56°. Name the theorem that gives another angle equal to 56°, and say where it is.
8. Tangents TA and TB meet at T, and angle ATB = 40°. Find angle TAB.
9. What reason justifies that triangle OAB, with A and B on the circle and centre O, is isosceles?
10. In cyclic quadrilateral ABCD, angle ABC = 116°. Side AB is extended to E. Find angle ADC, and say why angle CBE equals it.

Answers

1. The diameter.
2. 49° (angle at the centre is twice the angle at the circumference).
3. 63° (angle ACB = 90° in a semicircle, then 180° − 90° − 27°).
4. 92° (opposite angles of a cyclic quadrilateral sum to 180°).
5. 24 cm (angle OAT = 90°, then 25² − 7² = 576).
6. 5 cm (half-chord 12 cm, then 13² − 12² = 25).
7. The alternate segment theorem: the angle in the segment on the opposite side of the chord from the angle.
8. 70° (TA = TB, so triangle TAB is isosceles, and (180° − 40°) ÷ 2 = 70°).
9. OA and OB are both radii, so they are equal.
10. Angle ADC = 64° (opposite angles sum to 180°). Angle CBE = 180° − 116° = 64° (angles on a straight line), so the exterior angle equals the opposite interior angle.

Where marks are usually lost

• Writing an angle with no reason when the question says “give reasons”, or giving a vague reason such as “circle rule”.
• Using the wrong pair of angles in a cyclic quadrilateral (adjacent instead of opposite).
• Applying the centre theorem to an angle that is not on the same arc as the centre angle.
• Reading the alternate segment as the nearer segment.
• Treating OT as a leg instead of the hypotenuse in tangent problems.
• Forgetting to double the half-chord, or to halve a full chord, in chord problems.
• Skipping the line that defines a letter, so a proof becomes hard to follow.
• Ending a proof with the working but not the stated conclusion.
• Rounding too early in a length or angle question that needs a decimal.

Official syllabus

These notes follow 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.