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Mathematics 0580

Quick revision notes · Mathematics 0580 · Chapter 12 of 37

Circles

Circle vocabulary and every circle theorem, with the reasons examiners want.

5 sectionsWorked examples Common mistakes
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At a glance

Circle vocabulary and every circle theorem, with the reasons examiners want.

  1. Circle words
  2. Right angles in circles
  3. Angles at the centre and circumference
  4. Alternate segment theorem
  5. Writing reasons

1Circle words

  • Radius: centre to edge. Diameter: across through the centre (2 × radius).
  • Chord: a line joining two points on the circle. Tangent: a line touching the circle at one point.
  • Arc: part of the circumference. Sector: a 'pizza slice'. Segment: area between a chord and the arc.

Worked example

What is the line from the centre to the point where a tangent touches?

  1. It is a radius.

A radius (it meets the tangent at 90°)

2Right angles in circles

  • The angle in a semicircle is 90° (angle at the circumference standing on a diameter).
  • A tangent meets the radius at 90°.
  • Tangents from an external point are equal in length (look for an isosceles triangle or a kite).
  • The perpendicular from the centre to a chord bisects the chord.

Worked example

AB is a diameter and C is on the circle. Angle CAB = 35°. Find angle CBA.

  1. Angle ACB = 90° (angle in a semicircle).
  2. 180 − 90 − 35.

55°

3Angles at the centre and circumference

  • The angle at the centre is twice the angle at the circumference standing on the same arc.
  • Angles in the same segment (standing on the same chord, same side) are equal.
  • Opposite angles of a cyclic quadrilateral add to 180°.

Worked example

O is the centre. Angle AOC = 140°. B is on the major arc and D on the minor arc. Find angle ADC.

  1. Angle ABC = 70° (angle at the centre is twice the angle at the circumference).
  2. ABCD is cyclic, so angle ADC = 180 − 70.

110°

4Alternate segment theorem

  • The angle between a tangent and a chord equals the angle in the alternate segment (on the other side of the chord).
  • Spot it: a tangent, a chord from the point of contact, and a triangle drawn in the circle.

Worked example

A tangent at T makes 58° with chord TA. B is on the circle in the alternate segment. Find angle TBA.

  1. Alternate segment theorem.

58°

5Writing reasons

  • Name the theorem every time: 'angle in a semicircle', 'angle at the centre is twice the angle at the circumference', 'opposite angles of a cyclic quadrilateral add to 180°', 'tangent perpendicular to radius', 'alternate segment theorem'.
  • Radii are equal → isosceles triangles: use them to find base angles.

Where to start

  1. Mark radii and tangents on the diagram.
  2. Look for right angles first.
  3. Look for isosceles triangles.
  4. Then centre/circumference, same segment, cyclic quadrilateral, alternate segment.

Worked example

Tangents from P touch the circle at A and B. Angle APB = 50°. Find angle PAB.

  1. PA = PB (tangents from a point).
  2. Triangle PAB is isosceles: (180 − 50) ÷ 2.

65°

✗Common mistakes

Don’t writeWrite instead
Angle at circumference = 2 × angle at centre It's the other way: centre = 2 × circumference
Cyclic quadrilateral rules with the centre as a corner All four corners must be on the circle
Giving the angle without a reason Each step needs the theorem named
Assuming a line is a diameter Only if it goes through the centre

★In the exam

  • Diagrams are not to scale — use theorems, never measure.
  • Find the easy angles first (right angles, isosceles), then the theorems.
  • Learn the exact wording of each theorem for the 'give a reason' marks.