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Circle Theorems

discia.org 10 questions · 28 marks
Name Date
  1. Either paper Higher2 marks

    \(A\), \(B\) and \(C\) are points on a circle, centre \(O\).

    130°xABCO
    (a)

    Calculate the size of the angle marked \(x\).

    (b)

    Give a reason for your answer.

    Show answer
    (a) \(65^\circ\)
    (b) The angle at the centre is twice the angle at the circumference
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  2. Either paper Higher2 marks

    \(A\), \(B\), \(C\) and \(D\) are points on a circle.

    35°xABCD
    (a)

    Write down the size of the angle marked \(x\).

    (b)

    Explain your answer.

    Show answer
    (a) \(35^\circ\)
    (b) Angles in the same segment are equal
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  3. Either paper Higher4 marks

    \(A\), \(B\) and \(C\) are points on a circle, centre \(O\).

    \(AB\) is a diameter of the circle.

    59°xABCO
    (a)

    State the size of angle \(ACB\).

    (b)

    Explain your answer.

    (c)

    Calculate the size of the angle marked \(x\).

    Show answer
    (a) \(90^\circ\)
    (b) The angle in a semicircle is 90 degrees
    (c) \(31^\circ\)
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  4. Either paper Higher4 marks

    \(A\), \(B\), \(C\) and \(D\) are points on a circle.

    78°101°xABCD
    (a)

    Work out the size of the angle marked \(x\).

    (b)

    Give a reason for your answer.

    (c)

    Work out the size of angle \(ADC\).

    Show answer
    (a) \(102^\circ\)
    (b) Opposite angles of a cyclic quadrilateral add up to 180 degrees
    (c) \(79^\circ\)
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  5. Either paper Higher4 marks

    \(A\) and \(B\) are points on a circle, centre \(O\).

    \(PA\) and \(PB\) are tangents to the circle.

    60°xABPO
    (a)

    Write down the size of angle \(OAP\).

    (b)

    Give a reason for your answer.

    (c)

    Calculate the size of the angle marked \(x\).

    Show answer
    (a) \(90^\circ\)
    (b) The angle between a tangent and a radius is 90 degrees
    (c) \(120^\circ\)
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  6. Either paper 3 marks

    \(A\), \(B\) and \(C\) are three points on a circle, centre \(O\).

    The radii \(OA\), \(OB\) and \(OC\) are shown. Angle \(OAB = 59^\circ\) and angle \(OCB = 28^\circ\).

    ABCODiagram NOT accurately drawn

    The diagram is not drawn accurately.

    Is \(AC\) a diameter of the circle?

    You must show your working.

    Show answer
    No: angle \(ABC = 59^\circ + 28^\circ = 87^\circ\), so \(AC\) is not a diameter
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  7. Either paper Higher1 mark

    \(A\), \(B\) and \(C\) are points on a circle.

    \(PAQ\) is a tangent to the circle at \(A\).

    64°mABCPAQ

    Sam is trying to work out the size of angle \(m\).

    Here is his working.

    Line 1: The angle between the tangent and \(AB\) is \(64^\circ\), from the diagram.

    Line 2: This is the angle at the centre, and angle \(ACB\) is the angle at the circumference on the same chord \(AB\), so angle \(ACB\) is half of it.

    Line 3: \(m = 64^\circ \div 2 = 32^\circ\), because the angle at the centre is twice the angle at the circumference.

    Make a criticism of his working.

    Show answer
    Line 2 is wrong: there is no centre in the diagram, and the angle between the tangent and \(AB\) is at \(A\) on the circumference, so the angle at the centre theorem cannot be used.
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  8. Either paper Higher3 marks

    \(PA\) and \(PB\) are tangents to a circle, touching it at \(A\) and at \(B\).

    58°xABP
    (a)

    Find the size of the angle marked \(x\).

    (b)

    Give a reason why triangle \(ABP\) is isosceles.

    Show answer
    (a) \(61^\circ\)
    (b) The two tangents drawn from a point to a circle are equal in length, so PA = PB
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  9. Either paper Higher2 marks

    \(T\), \(A\) and \(B\) are points on a circle.

    The straight line through \(T\) is a tangent to the circle.

    x70°TAB
    (a)

    Write down the size of the angle marked \(x\).

    (b)

    Explain your answer.

    Show answer
    (a) \(70^\circ\)
    (b) The angle between a tangent and a chord is equal to the angle in the alternate segment
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  10. Either paper Higher3 marks

    \(P\) and \(Q\) are points on a circle, centre \(O\).

    The tangent to the circle at \(Q\) passes through \(R\).

    Angle \(POQ = x^\circ\)

    x°PQOR

    Show that angle \(PQR = 180° - \frac{ x }{2}°\)

    You must give a reason for each stage of your working.

    Show answer
    Angle \(PQR = 180° - \frac{ x }{2}°\), by angle \(PQO = 90 - \frac{ x }{2}\) (isosceles triangle) and angle \(OQR = 90\) (tangent and radius) combined.
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discia.org Circle Theorems · sheet 1
Answers Circle Theorems · sheet 1
  1. (a) \(65^\circ\)
    (b) The angle at the centre is twice the angle at the circumference
  2. (a) \(35^\circ\)
    (b) Angles in the same segment are equal
  3. (a) \(90^\circ\)
    (b) The angle in a semicircle is 90 degrees
    (c) \(31^\circ\)
  4. (a) \(102^\circ\)
    (b) Opposite angles of a cyclic quadrilateral add up to 180 degrees
    (c) \(79^\circ\)
  5. (a) \(90^\circ\)
    (b) The angle between a tangent and a radius is 90 degrees
    (c) \(120^\circ\)
  6. No: angle \(ABC = 59^\circ + 28^\circ = 87^\circ\), so \(AC\) is not a diameter
  7. Line 2 is wrong: there is no centre in the diagram, and the angle between the tangent and \(AB\) is at \(A\) on the circumference, so the angle at the centre theorem cannot be used.
  8. (a) \(61^\circ\)
    (b) The two tangents drawn from a point to a circle are equal in length, so PA = PB
  9. (a) \(70^\circ\)
    (b) The angle between a tangent and a chord is equal to the angle in the alternate segment
  10. Angle \(PQR = 180° - \frac{ x }{2}°\), by angle \(PQO = 90 - \frac{ x }{2}\) (isosceles triangle) and angle \(OQR = 90\) (tangent and radius) combined.
discia.org Circle Theorems · sheet 1