Circle theorems — Paper 1 · IGCSE Mathematics 0580

23 D C NOT TO SCALE O 25o A B E In the diagram, DE is a diameter of the circle, centre O. AEB is the tangent at the point E. The line DCB cuts the circle at C. Angle DEC = 25°. (a) Write down the size of angle DCE. Answer (a) Angle DCE = [1] (b) Calculate the size of angle CDE. Answer (b) Angle CDE = [2] (c) Calculate the size of angle DBE. Answer (c) Angle DBE = [2]

5 marks

Mark scheme: 23 (a) 90 1 (b) 65 2ft M1 for 180 – 25 – their (a) [155 – their (a)] (c) 25 2ft ft. 90 – their (b) B1 for angle DEB = 90° used or 10 B1 for angle CEB = 65° seen

This question in 0580/11 May/June 2005

14 (a) Calculate the circumference of a circle of diameter 8 cm. For Examiner's Use Answer(a) cm [2] (b) Q NOT TO SCALE 29º A B AQB is a semi-circle. Angle QAB = 29°. Work out the size of angle ABQ. Answer(b) Angle ABQ = [2]

4 marks

Mark scheme: 14 (a) art 25.1 www 2 M1 for π × 8 or 2π × 8 ÷ 2 implied by answer of 25 (b) 61 (Can be on 2 M1 for 90 – 29 or 180 – 90 – 29 diagram) SC1 for angle Q = 90° soi

This question in 0580/11 Oct/Nov 2007

14 Examiner's S C T Use 54° NOT TO SCALE A O B A, B and C lie on a circle, centre O. BC is a diameter and SCT is a tangent at C. Angle ACB = 54°. Find (a) angle BCT, Answer(a) Angle BCT = [1] (b) angle COA, Answer(b) Angle COA = [1] (c) angle CAB, Answer(c) Angle CAB = [1] (d) angle ABC. Answer(d) Angle ABC = [1]

4 marks

Mark scheme: 14 (a) 90° 1 (b) 72° 1 (c) 90° 1 (d) 36° 1 Ft 180 − (54 + their (c)) 4 

This question in 0580/11 May/June 2010

18 R Examiner's Use NOT TO SCALE O 55° Q P P, Q and R lie on a circle, centre O. PR is a diameter and angle OQR = 55°. Find (a) angle PQR, Answer(a) Angle PQR = [1] (b) angle ROQ, Answer(b) Angle ROQ = [1] (c) angle OPQ. Answer(c) Angle OPQ = [1]

3 marks

Mark scheme: 18 (a) 90° 1 (b) 70° 1 (c) 35° 1ft ft their (b) ÷ 2 only 18 

This question in 0580/13 May/June 2010

16 B NOT TO SCALE O T A C U In the diagram, TAU is a tangent to the circle at A. AB is a diameter of the circle and AC = BC. Find (a) angle BCA, Answer(a) Angle BCA = [1] (b) angle ABC, Answer(b) Angle ABC = [1] (c) angle CAU. Answer(c) Angle CAU = [1]

3 marks

Mark scheme: 16 (a) 90 1 1 (b) 45 1ft ft (180 − their (a)) 2 (c) 45 1ft ft 90 − their (b)

This question in 0580/13 Oct/Nov 2010

17 (a) For C Examiner's Use A NOT TO SCALE B Points A, B and C lie on the circumference of the circle shown above. When angle BAC is 90° write down a statement about the line BC. Answer(a) [1] (b) NOT TO SCALE O 54° A D B C O is the centre of a circle and the line ABC is a tangent to the circle at B. D is a point on the circumference and angle BOD = 54°. Calculate angle DBC. Answer(b) Angle DBC = [3]

4 marks

Mark scheme: 17 (a) Diameter 1 (b) 27 3 M1 for (180 − 54) ÷ 2 M1 ind for 90 − their angle OBD.

This question in 0580/12 May/June 2011

16 (a) For C Examiner's Use 110° A B NOT TO SCALE D E In the diagram, AB is parallel to DE. Angle ABC = 110°. Find angle BDE. Answer(a) Angle BDE = [2] (b) NOT TO SCALE O y° B 50° z° t ° A T TA is a tangent at A to the circle, centre O. Angle OAB = 50°. Find the value of (i) y, Answer(b)(i) y = [1] (ii) z, Answer(b)(ii) z = [1] (iii) t. Answer(b)(iii) t = [1]

5 marks

Mark scheme: 16 (a) 70 2 B1 for angle ABD = 70° stated or seen on the diagram (b) (i) (y =) 80 1 (ii) (z =) 40 1 (iii) (t =) 10 1ft Follow through 90 − their y or 50 − their z IGCSE – October/November 2011 0580 11 2 2 ( )

This question in 0580/11 Oct/Nov 2011

15 (a) Find the value of x. For Examiner's Use NOT TO x° SCALE 51° Answer(a) x = [1] (b) EF is a diameter of the circle. Find the value of y. y° E 63° NOT TO SCALE F Answer(b) y = [1] (c) Find the value of z in this isosceles triangle. NOT TO 48° SCALE z° Answer(c) z = [1]

3 marks

Mark scheme: 15 (a) 51° 1 (b) 90° 1 (c) 66° 1

This question in 0580/11 May/June 2012

20 For A Examiner's Use B NOT TO 34° SCALE E 68° D C The points A, B, C, D and E lie on a circle with diameter BD. AE is parallel to BD. Angle BDE = 68° and angle DBC = 34°. (a) Give the reason why angle BCD is 90°. Answer(a) [1] (b) Find (i) angle BDC, Answer(b)(i) [1] (ii) angle DEA. Answer(b)(ii) [1] (c) Find the sum of the angles of the pentagon ABCDE. Answer(c) [2]

5 marks

Mark scheme: 20 (a) Angle (in a) semi-circle 1 (b) (i) 56 1 (ii) 112 1 (c) 540 cao 2 M1 for all attempts to sum all the angles or any correct method for the sum of angles of a pentagon.

This question in 0580/12 Oct/Nov 2012

20 B C NOT TO O SCALE A A, B and C are points on the circumference of a circle centre O. AC is a straight line. (a) Explain why angle ABC is 90°. Answer(a) … [1] (b) The diameter of the circle is 3 cm. Calculate the area of this circle. Answer(b) … cm2 [2] _____________________________________________________________________________________

3 marks

Mark scheme: 20 (a) Angle or triangle [in a] semi-circle 1 (b) 7.068 to 7.07 2 M1 for π × 1.52 seen 2 3  

This question in 0580/12 May/June 2013

20 For Examiner′s B Use NOT TO SCALE 17 cm C 6 cm A In the diagram, AB is a diameter of the circle and C is a point on the circumference. AB = 17 cm and AC = 6 cm. (a) Calculate the area of the circle. Answer(a) … cm2 [2] (b) (i) Explain why angle ACB = 90°. Answer(b)(i) … [1] (ii) Calculate BC. Answer(b)(ii) BC = … cm [3] _____________________________________________________________________________________

6 marks

Mark scheme: 20 (a) 226.98 to 227.01 2 M1 for π × (17 ÷ 2) 2 (b) (i) Angle or triangle [in a] semi-circle 1 (ii) 15.9 or 15.90 to 15.91 3 2 2 M2 for 17 − 6 or or 253 2 2 2 M1 for 17 = BC + 6 or better.

This question in 0580/13 May/June 2013

17 Q NOT TO 17 cm SCALE 9 cm P R O The diagram shows a circle, centre O. P, Q and R are points on the circumference. PQ = 17 cm and QR = 9 cm. (a) Explain why angle PQR is 90°. Answer(a) … … [1] (b) Calculate the length PR. Answer(b) PR = … cm [2] __________________________________________________________________________________________

3 marks

Mark scheme: 17 (a) Angle [in a] semi-circle 1 (b) 19.2 or 19.23 to 19.24 2 M1 for 172 + 92 16 21 8 25

This question in 0580/11 May/June 2014

21 B NOT TO SCALE O 63° A C The diagram shows a circle, centre O with diameter AB = 15 cm. AC is a tangent to the circle at A and angle AOC = 63°. (a) Calculate the area of the circle. Answer(a) … cm2 [2] (b) (i) Work out the size of angle ACO. Answer(b)(i) Angle ACO = … [2] (ii) Give one geometrical reason for your answer to part (b)(i). Answer(b)(ii) … … [1] __________________________________________________________________________________________

5 marks

Mark scheme: 21 (a) 177 or 176.7 to 176.74 2 M1 for π × 7.52 oe (b) (i) 27 2 B1 for angle CAO marked, or clearly used, as a right-angle (or 90°) or M1 for 180 – 90 – 63 oe (ii) one correct geometrical reason 1 [angle between a] radius [and a] tangent [is a] right-angle or [angles in a] triangle [add up to] 180° or [the] angles [have to] add to 180°

This question in 0580/12 May/June 2014

16 B NOT TO SCALE y° O A D 24° x° C E The diagram shows a circle with centre O. ED is a tangent to the circle at C. AB is parallel to ED and angle ACO = 24°. Find the value of (a) x, Answer(a) x = … [1] (b) y. Answer(b) y = … [2] __________________________________________________________________________________________

3 marks

Mark scheme: 16 (a) 66 1 (b) 42 2FT FT their (a) – 24, only if their (a) > 24 or B1 for either of these, may be on diagram, angle OAC = 24 or angle BAC = their (a)

This question in 0580/11 Oct/Nov 2014

21 C 5 cm B NOT TO SCALE 8 cm 32° A A, B and C lie on a circle with diameter AB. Angle CAB = 32°, AC = 8 cm and BC = 5 cm. (a) Work out the size of angle CBA. Answer(a) Angle CBA = … [2] (b) Work out the length of AB. Answer(b) AB = … cm [2] __________________________________________________________________________________________

4 marks

Mark scheme: 21 (a) 58 2 B1 for ACB = 90o soi as angle at C or 8 M1 for tan 5 2 + 5 2 (b) 9.43 to 9.44 2 M1 for [AB 2 = ] 8 5 8 or sin 32 = or cos 32 = oe AB AB

This question in 0580/13 Oct/Nov 2014

16 (a) The angle in a semi-circle is 90°. Use the circle drawn below, with its centre marked, to show this. [1] (b) The line AB is one side of an equilateral triangle ABC. Using a straight edge and compasses only, construct triangle ABC. A B [2]

3 marks

Mark scheme: 16 (a) Angle in semi-circle drawn with 1 diameter through centre (b) Equilateral triangle with correct 2 M1 for clear evidence of constructed 60° angles arcs. or arcs crossing equal in length to AB or an accurate diagram with no/incorrect arcs 10 5

This question in 0580/12 Feb/March 2017

20 B NOT TO SCALE C 42° A A, B and C are points on the circumference of a circle with diameter AB. A tangent is drawn at A. Find (a) angle BAC, Angle BAC = … [1] (b) angle ABC. Angle ABC = … [2]

3 marks

Mark scheme: 20(a) 48 1 20(b) 42 2FT FT ‘90 − their (a)’ provided their (a) < 90 B1 for angle BCA = 90 or marked as a right angle

This question in 0580/12 May/June 2017

15 B C 65° NOT TO SCALE O 46° x° D A y° E The diagram shows a circle, centre O, with diameter AC. A, B, C, D and E lie on the circumference of the circle. (a) Find the value of x. Give a reason for your answer. x = … because … [2] (b) Find the value of y. Give a reason for your answer. y = … because … [2]

4 marks

Mark scheme: 15(a) 25 1 Angle in a semicircle = 90° 1 15(b) 46 1 Base angles of an isosceles triangle are equal 1

This question in 0580/12 Feb/March 2023

20 C x° B NOT TO SCALE O 32° A The diagram shows a circle, centre O, diameter AB. A, B and C lie on the circumference of the circle. (a) Write down the mathematical name of the line AC. … [1] (b) Find the value of x. Give a geometrical reason for your answer. x = … because … [2]

3 marks

Mark scheme: 20(a) Chord 1 20(b) 58 1 Angle in a semicircle = 90° 1

This question in 0580/13 May/June 2023

17 A B NOT TO O SCALE C A, B and C are points on a circle, centre O. (a) Draw a tangent to the circle at point A. [1] (b) The circumference of the circle is 22.3 cm. Calculate the radius of the circle. … cm [2] (c) Give a geometrical reason why angle BCA is 90°. … [1]

4 marks

Mark scheme: 17(a) Ruled tangent drawn at A 1 17(b) 3.55 or 3.548 to 3.549… 2 M1 for [r =] 22.3 ÷ 2π 17(c) Angle in a semicircle is 90º oe 1

This question in 0580/12 Oct/Nov 2023

18 NOT TO SCALE P 25° O x° y° Q R S T The diagram shows a circle, centre O. P and S are points on the circle. POR is a straight line. QRST is a tangent to the circle at S. (a) Find the value of x. Give a geometrical reason for your answer. x = … because … [2] (b) Find the value of y. y = … [3]

5 marks

Mark scheme: 18(a) 25 2 B1 for either base angles of an isosceles triangle are equal 18(b) 140 3 M2 for [ORS = ]180 – (90 + their 25 + 25) oe or B1 for OSR= 90 or ROS = 50

This question in 0580/13 May/June 2025

24 NOT TO SCALE A O 36° B C B is a point on the circle, centre O. ABC is a tangent to the circle at B and angle OAB = 36°. Work out the size of angle AOB. Angle AOB = … [1]

1 marks

Mark scheme: 24 54 1

This question in 0580/12 Oct/Nov 2025