Circle Theorems

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Lesson notes

Angles at Centre & Circumference

  • The angle at the centre is twice the angle at the circumference subtended by the same arc.
  • Both angles are formed from the same two points on the circumference.
  • The theorem applies even when the triangle parts overlap or form a diamond shape.
  • Use the reflex angle at the centre when the angle at the circumference is obtuse.
  • Common mistake: confusing this theorem with opposite angles in a cyclic quadrilateral.

Angle at Centre & Circumference

Angle at Centre & Circumference

Angle in a Semicircle

  • The angle in a semicircle is 90°.
  • This is a special case of the centre-circumference theorem (angle on diameter = 180°, halved).
  • The right angle is opposite the diameter.
  • All three vertices must lie on the circumference, with one side as the diameter.
  • Often used with Pythagoras' theorem to find lengths.

Angle in a Semicircle

Angle in a Semicircle

Theorems with Chords

  • A radius that is perpendicular to a chord bisects the chord.
  • Conversely, the perpendicular bisector of a chord passes through the centre.
  • Equal chords are equidistant from the centre.
  • Chords and radii often form isosceles triangles.

Radius Bisects a Chord

Radius Bisects a Chord

Theorems with Tangents

  • A radius and a tangent meet at right angles (90°).
  • Tangents from an external point are equal in length.
  • Two tangents from the same point form a kite with two right angles.
  • Use Pythagoras or trigonometry in the right triangles formed.

Tangents from an External Point

Tangents from an External Point

Angles in Cyclic Quadrilaterals

  • A cyclic quadrilateral has all four vertices on the circumference.
  • Opposite angles in a cyclic quadrilateral add up to 180°.
  • This theorem does not apply to quadrilaterals with vertices not all on the circle.
  • Mark all angles on the diagram to help find relationships.

Cyclic Quadrilateral

Cyclic Quadrilateral

Angles in the Same Segment

  • Angles in the same segment are equal.
  • They are subtended by the same chord.
  • Look for a 'bowtie' shape with two angles on the same side of the chord.
  • The theorem works both ways: angles at either end of the chord are also equal.

Angles in the Same Segment

Angles in the Same Segment

The Alternate Segment Theorem

  • The angle between a chord and a tangent equals the angle in the alternate segment.
  • The alternate segment is the region on the opposite side of the chord from the angle.
  • Identify a cyclic triangle with one vertex touching the tangent.
  • The equal angle is inside the triangle opposite the side that forms the first angle.

Alternate Segment Theorem

Alternate Segment Theorem

General Tips

  • Always give a reason for each angle found (quote the circle theorem or angle fact).
  • Look for isosceles triangles formed by radii and chords.
  • Add radii and right angles to the diagram to reveal relationships.
  • Use properties of triangles, quadrilaterals, and parallel lines alongside circle theorems.

Slides

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Practice questions

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  1. 1.In a circle, the angle at the centre is twice the angle at the circumference. If the angle at the circumference is 35°, what is the angle at the centre?

    Easy
    Angle at centre = 70°70°r = 4 cm
    • A70°
    • B35°
    • C17.5°
    • D140°

  2. 2.What is the angle in a semicircle?

    Easy
    • A90°
    • B180°
    • C45°
    • D60°

  3. 3.A radius and a tangent meet at what angle?

    Easy
    • A90°
    • B180°
    • C45°
    • D0°

  4. 4.Opposite angles in a cyclic quadrilateral add up to:

    Easy
    • A180°
    • B90°
    • C360°
    • D270°

  5. 5.In the diagram, A, B, C, D lie on a circle with centre O. EA is a tangent at A. Angle EAB = 61°, angle BAC = 55°. Find angle BAO.

    Medium
    Circle with tangent EAr = 4 cm
    • A29°
    • B61°
    • C55°
    • D35°

  6. 6.A, B, C, D are points on a circle, centre O. DOB is a straight line. Angle DAC = 58°. Find angle CDB.

    Medium
    • A58°
    • B32°
    • C29°
    • D122°

  7. 7.P, Q, R are points on a circle, centre O. PO is parallel to QR and angle POQ = 48°. Find angle OPR.

    Medium
    • A24°
    • B48°
    • C12°
    • D36°

  8. 8.K, L, M are points on a circle. KS is a tangent at K. KM is a diameter and triangle KLM is isosceles. Find z.

    Hard
    • A45°
    • B90°
    • C60°
    • D30°

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