Study Notes
The properties of circles include various theorems and rules that describe relationships between angles, chords, tangents, and segments within and around a circle.
- Angle at Centre — The angle at the centre of a circle is twice the angle at the circumference. Example: If the angle at the centre is 72°, the angle at the circumference is 36°.
- Angle in the Same Segment — Angles in the same segment of a circle are equal. Example: If two angles touch the circumference in the same segment, they are equal.
- Angle in Semicircle — An angle in a semicircle is always 90°. Example: A triangle formed with the diameter as one side has a right angle opposite the diameter.
- Radius Perpendicular to Tangent — The radius of a circle is perpendicular to the tangent at the point of contact. Example: The angle between a radius and a tangent is 90°.
- Cyclic Quadrilateral — The sum of opposite angles in a cyclic quadrilateral is 180°. Example: In a cyclic quadrilateral, Angle DAB + Angle DCB = 180°.
- Perpendicular Bisector of Chord — The perpendicular bisector of a chord passes through the centre of the circle. Example: If a line bisects a chord perpendicularly, it goes through the centre.
- Tangents from External Point — Tangents drawn from an external point to a circle are equal in length. Example: If RP and QP are tangents from point P, then RP = QP.
- Equal Chords Equidistant from Centre — Chords equidistant from the centre of a circle are equal in length. Example: If two chords are the same distance from the centre, they are equal.
- Alternate Segment Theorem — The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. Example: Angle ABC = Angle ACD.
- Intersecting Chords Theorem — The products of the lengths of the segments of two intersecting chords are equal. Example: DS x SB = CS x AS.
- Tangent Secant Theorem — The square of the length of a tangent segment is equal to the product of the lengths of the entire secant segment and its external segment. Example: PS² = PQ x PR.
Exam Tips
Key Definitions to Remember
- Angle at centre is twice the angle at the circumference.
- Angles in the same segment are equal.
- Angle in a semicircle is 90°.
- Radius is perpendicular to the tangent.
- Opposite angles in a cyclic quadrilateral sum to 180°.
- Perpendicular bisector of a chord passes through the centre.
- Tangents from an external point are equal.
- Equal chords are equidistant from the centre.
- Alternate segment theorem relates angles between tangents and chords.
- Intersecting chords theorem relates products of segments.
- Tangent secant theorem relates tangent and secant lengths.
Common Confusions
- Confusing the angle at the centre with angles elsewhere in the circle.
- Misidentifying angles in the same segment.
- Forgetting that the radius is perpendicular to the tangent.
- Mixing up the properties of cyclic quadrilaterals with other quadrilaterals.
Typical Exam Questions
- What is the angle at the circumference if the angle at the centre is 80°? Answer: 40°
- If two angles are in the same segment and one is 50°, what is the other angle? Answer: 50°
- What is the angle in a semicircle? Answer: 90°
What Examiners Usually Test
- Ability to apply the angle at the centre property.
- Understanding of angles in the same segment.
- Correct application of the tangent and radius relationship.
- Use of cyclic quadrilateral properties to find unknown angles.