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Chapter 18 Practice Paper: Circles and circle theorems — question paper

QuizQuipPractice Series
QQ/GCSE-MATHS/CH18

GCSE (9–1) · GCSE Mathematics (Foundation with Higher stretch)

Chapter 18 Practice Paper

Circles and circle theorems

Duration: 40 minutes
Total: 30 marks
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READ THESE INSTRUCTIONS FIRST

  • Answer all questions.
  • Section A contains ten multiple-choice questions. Choose one answer, A, B, C or D, for each.
  • Section B contains structured questions. Write your answers in the spaces provided.
  • Give a reason for every angle you find, quoting the circle theorem used.
  • The number of marks is given in brackets [ ] at the end of each question or part question.
  • The total mark for this paper is 30.

Original practice material written for the MOE GCSE Mathematics syllabus. It is not an official MOE or school examination paper. Chapter 18: circle vocabulary, the angle at the centre, angles in the same segment, the semicircle result, cyclic quadrilaterals, tangent properties (perpendicular radius, equal tangents), chord–radius bisection and the alternate segment theorem, with short isosceles-triangle proofs.

For Examiner’s Use
Section A/ 10
Section B/ 20
Total/ 30

Section A: Multiple-choice questions[10 marks]

Choose one answer for each question. Every correct answer scores one mark.

  1. 1

    At the point where a tangent meets a circle, the angle between the tangent and the radius is

    1. Aalways 90°
    2. Bhalf the centre angle
    3. Cequal to the arc
    4. Dtwice the radius
    ( )
  2. 2

    With 70° measured at the middle of a circle, the same arc shows at the rim an angle of

    1. A140°
    2. B290°
    3. C35°
    4. D70°
    ( )
  3. 3

    Two angles standing on the same chord, in the same segment, are

    1. Asupplementary
    2. Bequal
    3. Ccomplementary
    4. Dboth 90°
    ( )
  4. 4

    A cyclic quadrilateral has one angle of 121°. The opposite angle is

    1. A239°
    2. B121°
    3. C60.5°
    4. D59°
    ( )
  5. 5

    A triangle built on a diameter, with its third corner on the circle, is

    1. Aright-angled
    2. Bequilateral
    3. Cisosceles only
    4. Dimpossible
    ( )
  6. 6

    One tangent from point P measures 9 cm. The other tangent from P measures

    1. A18 cm
    2. B4.5 cm
    3. C9 cm
    4. Dcannot be found
    ( )
  7. 7

    A chord’s ends joined to the centre trap a 40° angle there. Each base angle is

    1. A140°
    2. B70°
    3. C20°
    4. D50°
    ( )
  8. 8

    Tangent meets chord at 38°; the angle standing in the opposite segment is

    1. A19°
    2. B76°
    3. C142°
    4. D38°
    ( )
  9. 9

    The angle a diameter subtends anywhere on the circumference is

    1. A90°
    2. B45°
    3. C180°
    4. D360°
    ( )
  10. 10

    A chord is 24 cm long and sits 5 cm from the centre. The radius is

    1. A29 cm
    2. B√(576 − 25) cm
    3. C13 cm
    4. D11 cm
    ( )

Section B: Structured questions[20 marks]

Show all your working and quote a theorem for each angle.

  1. 11

    Centre and rim.

    1. (a)

      An angle at the centre measures 128°. Find the angle the same arc subtends at the circumference.

      [2]
    2. (b)

      State the theorem used.

      [2]
  2. 12

    A cyclic quadrilateral with unknowns.

    1. (a)

      The opposite corners of a cyclic quadrilateral hold (x + 20)° and (2x + 40)°. Write down the equation for x.

      [2]
    2. (b)

      Find x and both angles.

      [2]
  3. 13

    Two tangents.

    1. (a)

      A circle has centre O and radius 10 cm. Point P lies 26 cm from O. Find the length of the tangent from P to the circle.

      [2]
    2. (b)

      Find the angle between the tangent and the line OP, to 1 decimal place.

      [2]
  4. 14

    Sitting on a diameter.

    1. (a)

      A triangle is drawn inside a semicircle so that one side is the diameter AB. Given angle BAC = 34°, find angle ACB.

      [2]
    2. (b)

      Find angle ABC.

      [2]
  5. 15

    A chord close to the centre. (Higher)

    1. (a)

      A chord of length 12 cm lies 8 cm from the centre of a circle. Find the radius.

      [2]
    2. (b)

      How large is angle POQ, to 1 decimal place?

      [2]

— End of paper —