1. 1
   A, B, C and D lie on a circle with centre O, and ABCD is a cyclic quadrilateral. Angle AOC (at the centre, on the same side as B) = 130°.

   (a)

   B lies on the major arc AC and D lies on the minor arc AC. What is the size of angle ABC (at the circumference, subtended by the same arc AC as the 130° angle AOC)?

   [1 mark]

   • A65°
   • B130°
   • C260°
   • D50°

   (b)

   What is the size of angle ADC?

   [1 mark]

   • A50°
   • B65°
   • C130°
   • D115°

   (c)

   Which circle theorem justifies the calculation of angle ABC in part (a)?

   [1 mark]

   • AThe alternate segment theorem
   • BAngles in the same segment are equal
   • CThe angle at the centre is twice the angle at the circumference subtended by the same arc
   • DAngle in a semicircle is 90°

   Total for question 1: 3 marks

2. 2
   Two chords PR and QS of a circle intersect at point X inside the circle. PX = 6 cm, XR = 8 cm and QX = 4 cm.

   (a)

   Find the length of XS.

   [2 marks]

   (b)

   Find the length of QS.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   From external point T, two tangents TA and TB touch a circle with centre O at points A and B. Angle ATB = 46°.

   (a)

   Find the size of angle AOB.

   [3 marks]

   (b)

   Find the size of angle OAB.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   In a circle with centre O, points A, B and C lie on the circumference, with B on the major arc AC. The tangent to the circle at A meets the tangent at C at external point E, with angle AEC = 40°, and OA = OC = 7 cm (the radius). D also lies on the circle, on the minor arc AC (opposite B), so that ABCD is a cyclic quadrilateral.

   (a)

   Find the size of angle AOC, the angle at the centre subtended by chord AC.

   [4 marks]

   (b)

   Using your answer to part (a), find the size of angle ABC.

   [4 marks]

   (c)

   Find the size of angle ADC, and verify that angle ABC and angle ADC obey the circle theorem for opposite angles of a cyclic quadrilateral.

   [5 marks]

   Total for question 4: 13 marks

End of questions