Lemmara logoLemmaraWorked solution

0580/31 May/Jun 2023, Question 7: Combining Circle Theorems

Cambridge IGCSE Mathematics 0580 · Paper 31 (Core) · Calculator · 14 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Circle with centre O, tangent AC at B, straight lines BOD and AOEF, angle OAB 28 degrees; name lines, justify angles of 62 degrees, and calculate several angles; then state regular polygon properties and find a decagon's interior angle.

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

Part (aia) · 1 mark

State the mathematical name for the straight line BOD through the centre.

  1. 1BOD is a chord through the centre: diameter

AnswerDiameter

Part (aib) · 1 mark

State the mathematical name for the line ABC touching the circle at B.

  1. 1ABC touches the circle at B: tangent

AnswerTangent

Part (aii) · 2 marks

Give two geometrical reasons explaining why angle AOB equals 62 degrees.

  1. 1Angle OBA = 90° because the angle between a tangent and a radius is 90°
  2. 2Angle AOB = 180 − 90 − 28 = 62° because angles in a triangle add to 180°

AnswerAngle between tangent and radius = 90, and angles in a triangle add to 180

Part (aiii) · 1 mark

Give the geometrical reason why angle DOE also equals 62 degrees.

  1. 1Angle DOE = angle AOB = 62° because vertically opposite angles are equal

AnswerVertically opposite angles are equal

Part (aiva) · 1 mark

Calculate the size of angle DEB.

  1. 1BD is a diameter, so angle DEB is an angle in a semicircle
  2. 2Angle DEB = 90°

Answer90

Part (aivb) · 2 marks

Calculate the size of angle ODE.

  1. 1OD = OE (radii), so triangle ODE is isosceles
  2. 2Angle ODE = (180 − 62) ÷ 2 = 59°

Answer59

Part (aivc) · 2 marks

Calculate the size of angle BEF.

  1. 1Angle OED = angle ODE = 59°
  2. 2Angle OEB = 90 − 59 = 31°
  3. 3AOEF is straight, so angle BEF = 180 − 31 = 149°

Answer149

Part (b) · 2 marks

State two geometrical properties that together prove a polygon is regular.

  1. 1All sides are equal
  2. 2All interior angles are equal

AnswerEqual sides and equal angles

Part (c) · 2 marks

Calculate the interior angle of a regular polygon with 10 sides.

  1. 1Exterior angle = 360 ÷ 10 = 36°
  2. 2Interior angle = 180 − 36 = 144°

Answer144

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

Practise more combining circle theorems questions

Real 0580 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on combining circle theorems

Also searched as: 0580/31/M/J/23 · 0580_s23_qp_31 · 0580 s23 31 · question 7.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.