1. 1
   A circle with centre O has points A, B and C on its circumference, where AC is a diameter. A tangent line touches the circle at point D.

   (a)

   Points A, B and C lie on a circle with centre O, where AC is a diameter. If angle ABC is the angle subtended at B by the diameter AC, what is the size of angle ABC?

   [1 mark]

   • A$90°$
   • B$180°$
   • C$45°$
   • DCannot be determined

   (b)

   A tangent to the circle touches at point D, and OD is a radius drawn to that point. What is the size of the angle between the tangent and OD?

   [1 mark]

   • A$180°$
   • B$45°$
   • C$0°$
   • D$90°$

   (c)

   Given angle BAC = $35°$ in triangle ABC (where AC is a diameter), find angle BCA.

   [1 mark]

   • A$35°$
   • B$55°$
   • C$90°$
   • D$145°$

   Total for question 1: 3 marks

2. 2
   Points P, Q, R and S lie on a circle with centre O. The angle at the centre, angle POR, is $128°$, and PQRS forms a cyclic quadrilateral.

   (a)

   Points P, Q and R lie on a circle with centre O. The angle at the centre, angle POR, is $128°$. Find angle PQR, the angle at the circumference subtended by the same arc PR, giving a geometric reason.

   [2 marks]

   (b)

   PQRS is a cyclic quadrilateral. Given that angle PQR = $64°$ (from part (a)), find angle PSR, giving a geometric reason.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A circle with centre O contains two chords, EF and GH, and a tangent from an external point J touching the circle at two points.

   (a)

   Two chords, EF and GH, of a circle with centre O, are both known to be equidistant from O. Given EF = 6 cm, find the length of GH, giving a geometric reason.

   [3 marks]

   (b)

   From an external point J, two tangents are drawn to the circle, touching at points K and L. Given JK = 9 cm, find JL, and state the two geometric facts used to justify your answer.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A circle has a tangent touching at point T, with chord TW drawn from T, and point V on the circle in the alternate segment. A further point U also lies on the circle, forming cyclic quadrilateral TWVU.

   (a)

   A tangent touches a circle at point T. A chord TW is drawn from T, and angle between the tangent and chord TW is $58°$. A point V lies on the circle in the alternate segment. Using the alternate segment theorem, find angle TVW, giving a full geometric reason.

   [4 marks]

   (b)

   TWVU is a cyclic quadrilateral inscribed in the same circle, with vertices in the order T, W, V, U. Angle TVW $= 58°$ (from part (a)) and angle WTU $= 95°$. Find angle WVU, and hence find angle TVU, showing all reasoning.

   [4 marks]

   (c)

   Explain, using circle theorem vocabulary, why angle TVW (found in part (a)) and the angle at the centre subtended by the same arc TW would be related by a factor of two, and calculate that centre angle.

   [5 marks]

   Total for question 4: 13 marks

End of questions