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0580/42 Oct/Nov 2024, Question 4: Tangent Properties of a Circle

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 15 marks

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The question, in short

Use quadrilateral angle ratio and circle tangent-chord geometry to find angles, then calculate arc length and major sector area with radius 6.75 cm and OT 11.5 cm.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 2 marks

Find w given the four angles sum to 360° and w relates to the other three in ratio 3:5.

  1. 1Angles of a quadrilateral add to 360°
  2. 2w = 3/(3 + 5) × 360 = 135

Answer135

Part (bi) · 2 marks

Find angle BAM using tangent-chord angle 45° and angle AMB 105°.

  1. 1Angle ABD = angle ACD = 49° (angles in the same segment)
  2. 2In triangle ABM: angle BAM = 180 − 105 − 49
  3. 3Angle BAM = 26°

Answer26

Part (biia) · 2 marks

Find angle BAD using angle ACD 49° and cyclic quadrilateral properties.

  1. 1Angle QAD = angle ACD = 49° (alternate segment theorem)
  2. 2Angle BAD = 180 − 45 − 49 = 86°

Answer86

Part (biib) · 1 mark

State a geometrical reason why BD cannot be the diameter, using angle AMB 105°.

  1. 1Angle ABD = angle ACD = 49° (angles in the same segment)
  2. 2Angle ADB = angle PAB = 45° (alternate segment theorem)
  3. 3Angle BAD = 180 − 49 − 45 = 86°, which is not 90°
  4. 4If BD were a diameter, angle BAD would be 90°: Angle in a semicircle = 90°

AnswerAngle in a semicircle = 90°

Part (ci) · 3 marks

Show angle AOC equals 108.12° using right-angled triangles OAT and OCT with OA 6.75 cm and OT 11.5 cm.

  1. 1Angle OAT = 90° (tangent ⟂ radius)
  2. 2cos AOT = 6.75/11.5, so angle AOT = 54.058…°
  3. 3Angle AOC = 2 × 54.058… = 108.117… = 108.12° (2 d.p.)

Answer108.117…

Part (cii) · 2 marks

Calculate the minor arc ABC length using radius 6.75 cm and angle AOC 108.12°.

  1. 1Angle OAT = 90° (tangent meets radius).
  2. 2cos AOT = 6.75/11.5, so angle AOT = 54.06°
  3. 3Angle AOC = 2 × 54.06 = 108.12°
  4. 4Arc ABC = 108.12/360 × 2 × π × 6.75
  5. 5= 12.73… ≈ 12.7 cm

Answer12.7 or 12.73 to 12.74

Part (ciii) · 3 marks

Calculate the major sector OCDA area using radius 6.75 cm and the reflex angle at O.

  1. 1Reflex angle AOC = 360 − 108.12 = 251.88°
  2. 2Area = 251.88/360 × π × 6.75²
  3. 3= 100.1… ≈ 100 cm²

Answer100 or 100.1 to 100.2

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