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Maths · Circle Theorems

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Maths · AQA Paper 1

Circle Theorems

Angles at the Centre and in a Semicircle (Higher tier)

Using the angle at the centre and the angle in a semicircle, with reasons and radii.

  • Circle theorem A rule about angles and lines in a circle.
  • Chord A straight line joining two points on a circle.
  • Arc A part of the circumference.
A circle with the angle AOB at the centre equal to 100 degrees and the angle ACB at the circumference equal to 50 degrees.

Same Segment and Cyclic Quadrilaterals (Higher tier)

Using equal angles in the same segment and the opposite angles of a cyclic quadrilateral.

  • Segment The part of a circle cut off by a chord.
  • Same segment The same side of a chord.
  • Cyclic quadrilateral A quadrilateral with all four corners on a circle.
A circle with a chord AB and two points C and D on the same side, with the angle ACB equal to 40 degrees and the angle ADB marked x.

Tangents and Chords (Higher tier)

Using the tangent and radius, equal tangents, and the perpendicular from the centre to a chord.

  • Tangent A straight line that touches a circle at one point.
  • Chord A straight line joining two points on a circle.
  • Point of contact The point where a tangent touches the circle.
A circle with centre O and two tangents from an outside point T touching at A and B, with the angle AOB equal to 110 degrees and the angle ATB marked x.

The Alternate Segment Theorem (Higher tier)

Finding angles between a tangent and a chord using the alternate segment theorem.

  • Alternate segment The segment on the other side of a chord from a given angle.
  • Tangent A straight line that touches a circle at one point.
  • Chord A straight line joining two points on a circle.
A circle with a tangent TAS at A, a chord AB, and a point C in the alternate segment, with the angle between the tangent and the chord equal to 55 degrees and the angle ACB marked x.

Circle Theorem Proofs and Problems (Higher tier)

Proving circle theorems, solving multi-step problems, and finding the equation of a tangent.

  • Proof A series of steps, with reasons, that shows a result is always true.
  • Theorem A result that has been proved.
  • Exterior angle An angle between a side extended and the next side.
A circle with centre O and points A, B, C and D, used to prove that the angle at the centre is twice the angle at the circumference.