🔵 Circles & Trigonometry
This chapter covers circle properties, angle properties of circles, Pythagoras' theorem, trigonometric ratios (SOH CAH TOA), sine rule, cosine rule, area of triangle, and bearings. These topics carry very heavy marks in Paper 2 and require both conceptual understanding and computational accuracy.
Circle theorems are a favourite topic for examiners because they test geometric reasoning. Each theorem must be quoted precisely in your working. Trigonometry, on the other hand, is a computational skill — you need to know when to use SOH CAH TOA (right-angled triangles only) versus the sine rule or cosine rule (any triangle).
Circle Theorems
Circle theorems describe the relationships between angles, chords, and tangents in a circle. There are six main theorems you must know. In the exam, you must state the theorem name and apply it correctly. Many questions combine two or three theorems in a single problem.
The angle at the centre is twice the angle at the circumference subtended by the same arc
6 Essential Circle Theorems
Angle at centre = 2 x angle at circumference
The angle subtended by an arc at the centre of the circle is exactly twice the angle subtended at any point on the circumference, from the same arc.
Angle in a semicircle = 90 degrees
Any angle inscribed in a semicircle (i.e., subtended by a diameter at the circumference) is always a right angle.
Angles in same segment are equal
All angles subtended by the same arc (or chord) at points on the same side of the chord are equal.
Opposite angles of cyclic quadrilateral sum to 180 degrees
A cyclic quadrilateral has all four vertices on the circle. Its opposite angles are supplementary.
Tangent is perpendicular to radius
A tangent to a circle is perpendicular (90 degrees) to the radius drawn to the point of tangency.
Tangents from external point are equal
Two tangent lines drawn from the same external point to a circle have equal length.
⭕Circle Theorems
Essential circle properties tested in O-Level E-Math. Must know all theorems and their converses.
\angle \text{centre} = 2 \times \angle \text{circumference} \text{ (same arc)}\angle \text{in semicircle} = 90°\text{are equal}\text{sum} = 180°\text{Tangent} \perp \text{Radius at point of contact}\text{are equal in length}\angle \text{between tangent and chord} = \angle \text{in the alternate segment}📝Worked Example 1: Angle at Centre Theorem
In the circle with centre O, angle AOB = 124 degrees. C is a point on the major arc. Find angle ACB.
Angle at centre is twice the angle at the circumference
📝Worked Example 2: Cyclic Quadrilateral
ABCD is a cyclic quadrilateral. Angle A = 110 degrees and angle B = 85 degrees. Find angles C and D.
Opposite angles of a cyclic quadrilateral sum to 180°
📝Worked Example 3: Tangent-Radius Problem
A tangent touches a circle with centre O at point T. OT = 5 cm and the tangent meets a line from O at point P, where OP = 13 cm. Find PT.
Tangent is perpendicular to radius at the point of contact
📝Worked Example 3b: Angles in the Same Segment
Points A, B, C, and D lie on a circle. AC and BD intersect at E inside the circle. Angle AEB = 70 degrees and angle ABD = 40 degrees. Find angle ACD.
The angle at the centre is twice the angle at the circumference subtended by the same arc
📝Worked Example 3c: Alternate Segment Theorem
A tangent at point T makes angle 55 degrees with chord TA. B is a point on the circle on the opposite side of chord TA from the tangent. Find angle ABT.
Alternate segment theorem: both angles equal 55°
📝Worked Example 3d: Multi-Theorem Circle Problem
ABCD is a cyclic quadrilateral. The tangent at A makes angle 50 degrees with AB. Angle ADC = 115 degrees. Find (a) angle ACB, (b) angle ABC.
Opposite angles of a cyclic quadrilateral sum to 180°
Circle Theorem Exam Language
Examiners award marks for precise theorem names. Use these exact phrasings:
- "Angle at centre = 2 x angle at circumference"
- "Angle in a semicircle = 90 degrees"
- "Angles in the same segment are equal"
- "Opposite angles of a cyclic quadrilateral are supplementary"
- "Tangent perpendicular to radius"
- "Tangents from an external point are equal in length"
- "Alternate segment theorem"
Do NOT use vague language like "the angles are the same" or "they add up to 180". Be specific about WHICH theorem you are using.
What the full chapter covers
Circle Theorems
Pythagoras' Theorem & Trigonometric Ratios
Exact Trigonometric Values
Sine Rule & Cosine Rule
Bearings
3D Trigonometry Problems
Cosine Rule: Finding an Angle
Circle Theorem Exam Language
Memory Tricks for Exact Values
SOH CAH TOA — How to Remember
When to Use Sine Rule vs Cosine Rule
The Ambiguous Case of Sine Rule
Bearing Problem Checklist
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