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IGCSE Maths · Edexcel 4XMA1 · Unit 2 Higher · Circle properties

Intersecting Chords and Secants — Practice Questions

12 exam-style questions (35 marks), building from warm-up to grade 8–9. Show full working: Edexcel gives method marks even when the final answer is wrong.

Section A — Warm-up (grades 6–7)
  1. (2 marks)Chords AB and CD meet at P. AP = 4 cm, PB = 6 cm and CP = 3 cm. Find PD.
    ABCDP463PD
  2. (2 marks)Chords AB and CD meet at P. AP = 5 cm, PB = x cm, CP = 4 cm and PD = 10 cm. Find x.
  3. (3 marks)Two lines from P meet the circle at A and B, and at C and D. PA = 4 cm, AB = 6 cm and PC = 5 cm. Find CD.
    ABCDP465CD
  4. (2 marks)PT is a tangent to the circle. A line from P meets the circle at A and B. PA = 3 cm and AB = 9 cm. Find PT.
    ABTP39PT
Section B — Core (grades 7–8)
  1. (3 marks)Chords AB and CD meet at P. AP = x cm, PB = (x + 5) cm, CP = 4 cm and PD = 6 cm. Find x.
  2. (2 marks)PT is a tangent of length 8 cm. A line from P meets the circle at A and then B, with PA = 4 cm. Find AB.
  3. (3 marks)Two lines from P meet the circle at A and B, and at C and D. PA = x cm, AB = 7 cm, PC = 6 cm and CD = 4 cm. Find x.
  4. (4 marks)A diameter CD of a circle is perpendicular to a chord AB and meets it at P. AB = 16 cm and PC = 4 cm, where C is the end of the diameter nearer to AB. Find the radius of the circle.
Section C — Stretch (grades 8–9)
  1. (3 marks)A, B, C and D lie on a circle with centre O. ABCD is a cyclic quadrilateral with angle ABC = 104°, and B lies on the minor arc AC. Find (a) angle ADC (b) angle AOC. Give reasons.
  2. (3 marks)TA is a tangent to the circle at A. B and C are points on the circle. The angle between TA and the chord AB is 58°, and angle BAC = 70°. C is in the alternate segment. Find angle ABC. Give reasons.
  3. (4 marks)Chords AB and CD meet at P inside a circle. By considering triangles APC and DPB, prove that AP × PB = CP × PD.
  4. (4 marks)P is 13 cm from the centre O of a circle of radius 5 cm. PT is a tangent. The line PO meets the circle at A and B, with A nearer to P.
    (a) Find PT. (b) Verify that PT² = PA × PB.
When you’re doneRevise the method in the notes, then mark your answers with the worked solutions.
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