O-Level E-Math and SEC G3 Mathematics K310
G3: Properties of circles
Match each circle configuration to the relevant theorem and state the theorem used.
Core notes
Circle geometry becomes manageable when the centre, radii, chords, tangents and relevant arcs are marked. Match the configuration to one theorem and state that theorem explicitly.
Symmetry and tangent properties
Equal chords lie the same perpendicular distance from the centre, and the perpendicular bisector of a chord passes through the centre. Tangents from one external point are equal. The line from that point to the centre bisects the angle between the tangents.
Angle properties
An angle in a semicircle is . A tangent is perpendicular to the radius at the point of contact. For the same arc, the angle at the centre is twice the angle at the circumference, angles in the same segment are equal, and opposite angles of a cyclic quadrilateral sum to .
Formulae and methods
| Relationship | Use |
|---|---|
| Relate centre and circumference angles standing on the same arc. | |
| Use opposite angles in a cyclic quadrilateral. |
Worked examples
Example 1: The angle at the centre subtending arc is . Find the angle at the circumference subtending the same arc.
- Confirm both angles stand on arc .
- Use the centre-angle theorem.
- Calculate .
Answer: The angle at the circumference is .
Example 2: Two tangents from point touch a circle at and . If , find .
- Recognise that both tangents start from the same external point.
- Apply the equal-tangents theorem.
- Set .
Answer: .
Chapter checkpoint
- Mark every radius and equal tangent length that follows from the diagram.
- Identify the arc or segment subtended by the angles being compared.
- State the circle theorem before substituting angle values.
Exam traps and retrieval check
Avoid these traps
- Using a theorem for angles that do not stand on the same arc.
- Treating any quadrilateral in a circle diagram as cyclic without evidence.
- Stating a numerical angle with no theorem or reason.
Check from memory
What angle does a radius make with a tangent at the point of contact?
.
How are tangents from one external point related?
They are equal in length.
What do opposite angles of a cyclic quadrilateral sum to?
.
Official K310 coverage
This chapter maps to 2 official outcomes: G3.1, G3.2. Check the official 2027 K310 syllabus for exact assessable wording and paper details.