Grade 11 Circle Geometry (Euclidean Geometry)

Grade 11 Euclidean geometry: circle theorems for chords, angles at the centre, cyclic quadrilaterals and tangents, with reasons, proofs and worked examples.

Euclidean geometry and measurement make up about 50 of the 150 marks in Grade 11 Paper 2, and most of the geometry is circle theorems. You use the theorems about chords, angles subtended by arcs, cyclic quadrilaterals and tangents, and you need to be able to prove some of them.

Every statement in a geometry answer needs a reason in the accepted wording, such as '∠ at centre = 2 × ∠ at circumference' or 'opp ∠s of cyclic quad'. Grade 12 uses these theorems again alongside proportion and similar triangles.

Key concepts

  • A line from the centre perpendicular to a chord bisects the chord (and the converse)
  • The angle at the centre is twice the angle at the circumference
  • Angles in the same segment are equal; the angle in a semicircle is 90°
  • Opposite angles of a cyclic quadrilateral are supplementary; the exterior angle equals the interior opposite angle
  • A tangent is perpendicular to the radius at the point of contact
  • Tangents from the same point are equal; the tan-chord angle equals the angle in the alternate segment

Worked examples

An angle at the centre of a circle is 110°. Find the angle at the circumference on the same arc.

∠ at centre = 2 × ∠ at circumference, so the angle at the circumference is 110° ÷ 2 = 55°.

In cyclic quadrilateral ABCD, Â = 85°. Find Ĉ.

Opposite angles of a cyclic quadrilateral are supplementary: Ĉ = 180° − 85° = 95°.

O is the centre of a circle with radius 10 cm. OM is perpendicular to chord AB and OM = 6 cm. Find AB.

The line from the centre perpendicular to a chord bisects it. In △OMA, AM = √(10² − 6²) = 8 cm, so AB = 16 cm.

PA and PB are tangents to a circle from P. If P̂AB = 65°, find P̂.

Tangents from the same point are equal, so PA = PB and △PAB is isosceles: P̂BA = 65°. P̂ = 180° − 2(65°) = 50°.

Common mistakes

  • Giving a statement without its reason.
  • Using the cyclic quadrilateral theorem on a quadrilateral that isn't cyclic.
  • Confusing the angle at the centre with an angle at the circumference.
  • Assuming a line is a diameter or a tangent because of how it looks.

Exam tips

  • Mark radii as equal and look for isosceles triangles: they unlock many questions.
  • Learn the theorems in the exact wording your teacher uses; examiners accept the standard abbreviations.
  • Know which theorems you must be able to prove: your textbook lists them.

Frequently asked questions

Which circle theorems must I prove in Grade 11?

CAPS lists the proofs of: the line from the centre perpendicular to a chord bisects it; the angle at the centre is twice the angle at the circumference; opposite angles of a cyclic quadrilateral are supplementary; and the tan-chord theorem. Check your textbook for the exact list.

How do I spot a cyclic quadrilateral?

All four vertices lie on the circle. To prove a quadrilateral is cyclic, show opposite angles are supplementary, an exterior angle equals the interior opposite angle, or two angles on the same side are equal.

How many marks is geometry in Grade 11?

Euclidean geometry and measurement together are about 50 of the 150 marks in Paper 2.

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