Summary
Circle theorems help in understanding the properties of circles and solving problems involving angles, segments, and tangents.
- Chord of a Circle Theorem — A line drawn from the centre of a circle perpendicular to a chord will bisect the chord into two equal parts. Example: If chords AB and CD are equal in length and the perpendicular distance from the centre to AB is 4cm, the distance to CD is also 4cm.
- Radius Tangent Theorem — The angle between a tangent and the radius at the point of contact is always a right angle. Example: If a circle has a radius of 6cm and a tangent from an external point is 8cm, the distance from the centre to the point is 10cm.
- Two Tangents Theorem — The two tangents drawn from an external point to a circle are equal in length. Example: If the distance from the centre to the point is 26cm and the radius is 24cm, the tangents are each 10cm long.
- Angle in a Semicircle Theorem — The angle subtended by a diameter on the circumference is always a right angle. Example: If the diameter is 14cm and one side of the triangle is 9cm, the other side is 5cm.
- Angle at the Centre Theorem — The angle at the centre of the circle is twice the angle subtended by the same arc. Example: If the angle at the centre is 120°, the angle at the circumference is 60°.
- Angle Subtended by the Same Arc — Angles subtended by an arc on the circle are equal in size. Example: If one angle is 50°, another angle subtended by the same arc is also 50°.
- Alternate Segment Theorem — The angle between a tangent and a chord is equal to the angle subtended by the same chord in the alternate segment. Example: If the angle between the tangent and chord is 50°, the angle in the alternate segment is also 50°.
- Opposite Angles of a Cyclic Quadrilateral — The opposite angles of a cyclic quadrilateral are supplementary, adding to 180°. Example: If one angle is 75°, the opposite angle is 105°.
Exam Tips
Key Definitions to Remember
- A chord is a line segment with both endpoints on the circle.
- A tangent is a line that touches the circle at exactly one point.
- A cyclic quadrilateral is a four-sided figure with all vertices on the circle.
Common Confusions
- Confusing the angle at the centre with the angle at the circumference.
- Forgetting that tangents from the same external point are equal.
Typical Exam Questions
- What is the angle between a tangent and a radius? 90°
- If two chords are equal, what can be said about their perpendicular distances from the centre? They are equal.
- How do you find the length of a tangent from an external point? Use the Pythagorean theorem with the radius and distance from the centre.
What Examiners Usually Test
- Understanding and application of circle theorems.
- Ability to calculate unknown angles using circle properties.
- Problem-solving involving tangents and segments.