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Paper 2
Circle Theorems (Advanced)
Both
Learn all circle theorems and their names.
Key Facts
- Angle at centre = 2 × angle at circumference.
- Angle in a semicircle = 90°.
- Alternate segment theorem for tangents.
- Opposite angles in cyclic quadrilateral sum to 180°.
- Tangent perpendicular to radius.
Topics Covered
Alternate Segment Theorem
What you need to know
- •Angle between tangent and chord equals angle in alternate segment.
- •Key for finding missing angles in complex diagrams.
Exam Tips
- Mark the tangent and chord clearly.
Tangent Properties
What you need to know
- •Tangent is perpendicular to radius at point of contact.
- •Two tangents from external point are equal length.
- •Angle between tangent and radius is 90°.
Exam Tips
- Draw radius to tangent point to create right angle.
Circle Theorem Proofs
What you need to know
- •Use properties like angles in same segment.
- •Link cyclic quadrilateral to opposite angles sum to 180°.
- •State theorem names in proofs.
Exam Tips
- Quote the theorem name for proof marks.
Key Terms
Alternate segment
Segment on opposite side of chord from tangent.
Cyclic quadrilateral
Quadrilateral with all vertices on a circle.
Tangent
Line that touches circle at exactly one point.
Chord
Line segment with both endpoints on circle.
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Common Exam Questions
Angle at centre is 140°. Find angle at circumference on same arc.
2 markseasyPaper 2
Model Answer
70°
What examiners want to see
- ✓Use centre = 2 × circumference.
A tangent meets a chord. The angle between them is 50°. Find the angle in the alternate segment.
2 marksmediumPaper 2
Model Answer
50° (alternate segment theorem)
What examiners want to see
- ✓Apply alternate segment theorem.
Prove that the angle in a semicircle is 90°.
4 markshardPaper 2
Model Answer
Angle at centre on diameter is 180°. Angle at circumference = 180°/2 = 90°.
What examiners want to see
- ✓Use angle at centre theorem.
- ✓Clear logical steps.
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