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0580/32 Feb/Mar 2021, Question 9: Choosing the Right Trigonometric Ratio

Cambridge IGCSE Mathematics 0580 · Paper 32 (Core) · Calculator · 13 marks

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The question, in short

On a 1 cm² grid, construct a rectangle meeting given perimeter and area conditions, then solve four separate geometry problems involving angles, a circle, a polygon and a triangle.

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

Part (a) · 2 marks

Draw a rectangle on the grid with perimeter 22 cm and area 24 cm².

  1. 1Perimeter 22: length + width = 11
  2. 2Area 24: length × width = 24
  3. 38 + 3 = 11 and 8 × 3 = 24
  4. 4Draw a rectangle 8 cm by 3 cm

Answer8cm by 3cm rectangle drawn

Part (b) · 4 marks

Find x using the marked angles and state two geometric facts that justify the method.

  1. 1Fourth interior angle = 360 − 298 = 62° (angles at a point add to 360°)
  2. 2x = 360 − 127 − 94 − 62 (angles in a quadrilateral add to 360°)
  3. 3x = 77

Answer77; angles at a point add to 360° and angles in a quadrilateral add to 360°

Part (c) · 1 mark

Construct a tangent line touching the circle at point P.

  1. 1Draw the radius from the centre to P
  2. 2Draw a ruled line through P at 90° to this radius; this is the tangent

AnswerRuled tangent drawn

Part (d) · 1 mark

Given an exterior angle of 24°, determine how many sides the regular polygon has.

  1. 1Number of sides = 360 ÷ 24 = 15

Answer15

Part (e) · 5 marks

Using the right-angled triangle setup with AD = 13.6 cm, AB = 7.4 cm and angle DCB = 41°, calculate x.

  1. 1BD² = 13.6² − 7.4² = 184.96 − 54.76 = 130.2
  2. 2BD = √130.2 = 11.41…
  3. 3sin 41 = BD/x
  4. 4x = 11.41… ÷ sin 41
  5. 5x = 17.39… ≈ 17.4

Answer17.4 or 17.39…

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