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4024/22 Oct/Nov 2023, Question 2: Combining Circle Theorems

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 8 marks

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The question, in short

For a regular 15-sided polygon, find one interior angle, then use a diagram of part of it to find angle x; finally, with AB a diameter and angle BAC = 24°, find angle ADC with reasons.

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

Part (ai) · 2 marks

Calculate the size of a single interior angle of a regular 15-sided polygon.

  1. 1Exterior angle = 360 ÷ 15 = 24°
  2. 2Interior angle = 180 − 24 = 156°

Answer156

Part (aii) · 2 marks

Using the diagram of the regular 15-sided polygon with angle x marked, determine the value of x.

  1. 1The dashed line cuts off an isosceles triangle with apex angle 156°
  2. 2Base angles = (180 − 156) ÷ 2 = 12°
  3. 3x = 156 − 12 = 144

Answer144

Part (b) · 4 marks

Given points A, B, C, D on a circle centre O, AB a diameter and angle BAC = 24°, find angle ADC, justifying each step.

  1. 1Angle ACB = 90° (angle in a semicircle)
  2. 2Angle ABC = 180 − 90 − 24 = 66° (angles in a triangle add to 180°)
  3. 3Angle ADC = 180 − 66 = 114° (opposite angles of a cyclic quadrilateral add to 180°)

Answerangle ADC = 114, with complete reasons, e.g. angle ACB = 90 (angle in semicircle), angle ABC = 66 (sum of angles in triangle), angle ADC = 114 (angles in opposite segments are supplementary); OR angle ADB = 90 (angle in semicircle), angle BDC = 24 (angles in same segment are equal), angle ADC = 114

Stuck on a step? Read the free lesson: Combining Circle Theorems

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Also searched as: 4024/22/O/N/23 · 4024_w23_qp_22 · 4024 w23 22 · question 2.
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