PAST PAPER QUESTIONS IGCSE/O LEVEL 0580/4024 MATHS TOPIC: ANGLE PROPERTIES OF CIRCLES @MathleteBySaad saad@mathletebysaad.com Difficulty level: Easy 18 C A O 38° B T A, B and C are points on the circumference of a circle centre O. t = 38°. O is the midpoint of BC and ABC Tangents are drawn from T to touch the circle at A and B. t . (a) Calculate BCA t = ............................. [1] Answer BCA t . (b) Calculate ATB t = .............................. [2] Answer ATB 9 Find the integers that satisfy 1 1 3x + 5 G 11. Answer .......................................... [2] © UCLES 2018 4024/11/M/J/18 [Turn over ' 17 222 A 53° 62° O x° t° D y° C T z° B The diagram shows a circle, centre O, that passes through A, B, C and D. The tangents at A and B meet at T. t = 53° . t = 62° and DAB ATB (a) Find x. Answer x = ..................................... [1] (b) Find y. Answer y = ..................................... [1] (c) Find z. Answer z = ..................................... [1] (d) Find t. Answer t = .................................... [1] © UCLES 2018 4024/12/M/J/18 [Turn over ' 7 10 3 E NOT TO SCALE D 24° F y° O 110° x° A C B Points A, B, C, D, E and F lie on the circle, centre O. Find the value of x and the value of y. x = .................................................. y = .................................................. [2] © UCLES 2020 0580/21/M/J/20 [Turn over 12 174 A B NOT TO SCALE O x° y° 62° T C In the diagram, AC is a diameter of the circle centre O. The tangents from T touch the circle at B and C. t = 62° . BCT (a) Find x. x = .................................................. [1] (b) Find y. y = ................................................ [2] © UCLES 2020 4024/11/O/N/20 10 195 In this Venn diagram, shade the region M l , N , P . ! M N P [1] 20 (4x - 87) ° NOT TO SCALE 2x° (x + 60) ° y° The diagram shows a cyclic quadrilateral. Find the value of y. y = ................................................ [4] © UCLES 2020 0580/21/O/N/20 7 613 A x° B NOT TO SCALE D O 44° C A, B and C are points on a circle, centre O. DA and DC are tangents. Angle ADC = 44° . Work out the value of x. x = ................................................. [3] © UCLES 2021 0580/22/M/J/21 [Turn over 9 7 19 (a) A 50° T P B NOT TO SCALE Q P, Q and T are points on a circle. ATB is a tangent to the circle at T and PT = PQ. Find angle TPQ. Angle TPQ = ..................................................... [2] (b) 3x° 68° 2x° NOT TO SCALE w° The diagram shows a cyclic quadrilateral with an exterior angle of 68°. Find the value of w and the value of x. w = ..................................................... x = ..................................................... [3] © UCLES 2021 0580/23/M/J/21 [Turn over Answer Key: 1. S18/qp11/q8 (E) 2. S18/qp12/q22 (E) 3. S20/qp21/q10 (0580) (E) 4. W20/qp11/q17 (E) 5. W20/qp21/q20 (0580) (E) 6. S21/qp22/q13 (0580) (E) 7. S21/qp23/q19 (0580) (E) Difficulty level: Intermediate 19 1 C B y° t° z° T 43° A x° 68° O D In the diagram, A, B, C and D lie on the circle, centre O. AD is a diameter. The tangent to the circle at B meets the line DA produced at T. t = 68° and CAO t = 43°. AOB (a) Find x. Answer x = ........................................ [1] (b) Find y. Answer y = ........................................ [1] (c) Find z. Answer z = ........................................ [1] (d) Find t. Answer t = ......................................... [1] © UCLES 2015 4024/11/M/J/15 [Turn over 17 2 24 B C y° 34° t° A D x° O z° E In the diagram, A, B, C, D and E lie on the circle, centre O. BOE is a straight line. ˆ = 34° . DAB (a) Find x. Answer x = .................................... [1] (b) Find y. Answer y = .................................... [1] (c) Find z. Answer z = .................................... [1] (d) Find t. Answer t = .................................... [1] © UCLES 2016 4024/12/O/N/16 [Turn over 12 317 P T 125º O X S Q 35º R In the diagram, P, Q, R, S and T lie on the circle. QT is a diameter of the circle, centre O. X is the point of intersection of PS and QT. t = 125° and PStQ = 35°. PXT (a) Complete the following statement with a geometrical reason. t = 35° because ..................................................................................................................... [1] PTQ t . (b) Find PQT t = .............................. [1] Answer PQT t . (c) Find SPQ t = ............................... [1] Answer SPQ t . (d) Find SRQ t = ............................... [1] Answer SRQ © UCLES 2017 4024/12/M/J/17 16 421 C B D NOT TO SCALE y° z° x° O 53° t° E A In the diagram, the points A, B, C, D and E lie on the circle, centre O. The points B, O and E lie on a straight line. t = 53° . AB is parallel to ED and DEO (a) Find x. x = ................................................... [1] (b) Find y. y = ................................................... [1] (c) Find z. z = ................................................... [1] (d) Find t. t = ................................................... [1] © UCLES 2019 4024/11/O/N/19 15 520 A B NOT TO SCALE O D 34° 69° C In the diagram, the points A, B, C, and D lie on the circle, centre O. ' t = 69° and DCtA = 34° . ACB t . (a) Find ABD t = ................................................... [1] ABD t . (b) Find AOD t = ................................................... [1] AOD t . (c) Find DAB t = ................................................... [1] DAB © UCLES 2019 4024/12/O/N/19 [Turn over 17 22 6 NOT TO SCALE D E 54° z° C O y° 48° x° A B In the diagram, A, B, C, D and E lie on the circle, centre O. t = 48° , DEB t = 54° . AOB (a) Find x. x = ................................................ [2] (b) Find y. y = ................................................ [1] (c) Find z. z = ................................................ [1] © UCLES 2020 4024/12/O/N/20 [Turn over Answer Key: 1. S15/qp11/q19 (M) 2. W16/qp12/q24 (M) 3. S17/qp12/q17 (M) 4. W19/qp11/q21 (M) 5. W19/qp12/q20 (M) 6. W20/qp12/q22 (M) Difficulty level: Hard 13 (a) V Q P U R 38° Circle I Circle II 106° S T The points P, T, U and V lie on Circle I, and the points P, Q, R, S and T lie on Circle II. UPQ and VPR are straight lines. VT̂ U = 38˚ and TŜR = 106˚. Find (i) VP̂ U, [1] (ii) QT̂ R, [1] (iii) TP̂ R, [1] (iv) UP̂ T. [1] (b) A y° B x° O C A and B are points on the circle, centre O. CA is a tangent at A and BOC is a straight line. AĈ B = xº and BÂ O = yº. Find an expression for y in terms of x. © UCLES 2010 [3] 4024/21/O/N/10 [Turn over 7 2 (b) S P L x° Q y° R M TwocirclesintersectatLandM. R and P are on the circumference of one circle. S and Q are on the circumference of the other circle. PLQ andRLSarestraightlines. t =x°andMLQ t = y°. PLR t =x°. (i) Completetheproofthat SMQ Statement Reason t =SLQ t x°= PLR ............................................................................................. t =SMQ t =x° SLQ ............................................................................................. [2] t =y°. (ii) Provethat PRM Statement Reason [2] (iii) Completethefollowingstatement,givingyourreasons. ThetrianglesPRM andQSM are............................................. Reasons...................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ............................................................................................................................................... [3] © UCLES 2015 4024/21/M/J/15 [Turn over 19 3 (b) B A 60° O C E D A, B, C, D and E are points on the circumference of a circle, centre O. t = 60c. AE = ED = DC = CB and AOB t . (i) Find ECD Show all your working. Answer t = ........................... [2] ECD Answer .................................. cm [2] (ii) The radius of the circle is 12 cm. Calculate the length of the minor arc AB. Use r = 3.14 . Question 25 is printed on the next page © UCLES 2015 4024/11/O/N/15 [Turn over 13 419 B 40° A P R Q C D In the diagram, the two circles intersect at P and Q. The line AB is a tangent to the circles at A and B. AD and BC are diameters. BD intersects the larger circle at R. t = 40° . DBC t . (a) Find CPR t = ............................................. [1] Answer CPR t . (b) Find CQR t = ............................................. [1] Answer CQR t . (c) Find ABD t = ............................................. [1] Answer ABD t . (d) Find ADB t = ............................................. [1] Answer ADB © UCLES 2016 4024/11/M/J/16 [Turn over 16 5 8 A B E O y° D F X G C The diagram shows two circles each with centre O. A, B, C and D are points on the circumference of the large circle. E, F and G are points on the circumference of the small circle. CGD and CFB are tangents to the small circle. Lines AEOC and FG intersect at 90° at X. t = y° . GOX (a) Find each of these angles, as simply as possible, in terms of y. Give reasons for your answers. (i) t GEO t = ............................ because .............................................................................. Answer GEO ............................................................................................................................................... [2] (ii) t GCX t = ............................ because .............................................................................. Answer GCX ............................................................................................................................................... [2] (iii) t DAB t = ............................ because .............................................................................. Answer DAB ............................................................................................................................................... [2] © UCLES 2017 4024/22/O/N/17 14 69 (a) B A 109° C NOT TO SCALE O 35° 28° D E A, B, C, D and E lie on the circle, centre O. Angle AEB = 35°, angle ODE = 28° and angle ACD = 109°. (i) Work out the following angles, giving reasons for your answers. (a) Angle EBD = ............................. because ........................................................................... .............................................................................................................................................. .............................................................................................................................................. [3] (b) Angle EAD = ............................. because ........................................................................... .............................................................................................................................................. [2] (ii) Work out angle BEO. Angle BEO = ................................................ [3] © UCLES 2018 0580/42/M/J/18 15 (b) In a regular polygon, the interior angle is 11 times the exterior angle. (i) Work out the number of sides of this polygon. ................................................. [3] (ii) Find the sum of the interior angles of this polygon. ................................................. [2] © UCLES 2018 0580/42/M/J/18 14 7 18 C B x° 62° 127° D y° A z° 40° E In the diagram, A, B, C, D and E lie on the circle. AB is parallel to ED. t = 62°, CDE t = 127° and BEC t = 40°. ABE (a) Find x. Answer x = .................................... [1] (b) Find y. Answer y = .................................... [1] (c) Find z. Answer z = .................................... [1] © UCLES 2018 4024/12/O/N/18 9 8 13 D C NOT TO SCALE 138° O 12° E F B A A, B, C, D and E are points on the circumference of a circle, centre O. Tangent AF touches the circle at A. O is the midpoint of AD. t = 138° and BEO t = 12° . DOE t . (a) Find EAD t = ................................................... [1] EAD t . (b) Find EAF t = ................................................... [1] EAF t . (c) Find BEA t = ................................................... [1] BEA t . (d) Find BCD t = ................................................... [2] BCD © UCLES 2019 4024/11/M/J/19 [Turn over 7 9 20 D C NOT TO SCALE 20° O 131° B T A A, B, C and D lie on the circle, centre O. TA is a tangent to the circle at A. Angle ABC = 131° and angle ADB = 20°. Find (a) angle ADC, Angle ADC = ................................................. [1] (b) angle AOC, Angle AOC = ................................................. [1] (c) angle BAT, Angle BAT = ................................................. [1] (d) angle OAB. Angle OAB = ................................................. [1] © UCLES 2020 0580/22/M/J/20 [Turn over Answer Key: 1. W10/qp21/q3(a) (H) 2. S15/qp21/q3(b) (H) 3. W15/qp11/q24(b)(i)(H) 4. S16/qp11/q19 (H) 5. W17/qp22/q8(a) (H) 6. S18/qp42/q9 (0580) (H) 7. W18/qp12/q18 (H) 8. S19/qp11/q13 (H) 9. S20/qp22/q20 (0580) (H)
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