GCSE Maths Circle Theorems: lesson and practice questions

By the end of this chapter you will be able to

• Use the words chord, tangent, arc, sector and segment correctly, and spot isosceles triangles made from radii
• Know, use and prove that the angle in a semicircle is 90°
• Know, use and prove that the angle at the centre is twice the angle at the circumference
• Know and use that angles in the same segment are equal, and that opposite angles of a cyclic quadrilateral add up to 180°
• Know and use that a tangent is perpendicular to the radius, and that two tangents from a point are equal in length
• Know, use and prove the alternate segment theorem
• Know and use that the perpendicular from the centre to a chord bisects the chord, with Pythagoras and trigonometry
• Combine theorems with isosceles triangles and algebra, give reasons in standard wording, and write short proofs

You should already know

• Angle facts, isosceles triangles and angle sums (Chapter 17)
• Circles: radius, diameter, circumference (Chapter 18)
• Pythagoras' theorem and basic trigonometry (Chapter 20)
• Solving linear equations (Chapter 9)

Where this leads

• Equation of a circle and the tangent to a circle (Chapter 25)
• Geometric proof in vector geometry (Chapter 22)
• Further circle geometry in A-level mathematics