Circle theorems I — Paper 4 · IGCSE Mathematics - International 0607

5 A NOT TO SCALE O 46° P D C B A, B, C and D lie on a circle, centre O. AP and BP are tangents to the circle. Angle APB = 46°. (a) Complete the statement. Angle OAP = 90° because … … [1] (b) Find the value of (i) angle AOB, Angle AOB = … [2] (ii) angle OAB, Angle OAB = … [2] (iii) angle ACB, Angle ACB = … [2] (iv) angle ADB. Angle ADB = … [2] (c) OB bisects angle ABC. Find angle OAC. Angle OAC = … [3]

12 marks

Mark scheme: 5(a) [Angle between] tangent [and] radius 1 / diameter [=90] oe 5(b)(i) 134 2 M1 for 360 – 90 – 90 – 46 oe 5(b)(ii) 23 2 M1 for (180 – their (i)) ÷ 2 oe 5(b)(iii) 67 2 FT (their (i)) ÷ 2 M1 for (their (i)) ÷ 2 oe 5(b)(iv) 113 2 FT 180 – their (iii) or (360 – their (i)) ÷ 2 M1 for 180 – their (iii) or (360 – their (i)) ÷ 2 oe 5(c) 44 3 M2 for 180 – 67 – 23 – 23 – 23 oe or 360 – 226 – 67 – 23 oe or B1 for angle OBC = 23 or 226 seen

This question in 0607/41 May/June 2017

8 B NOT TO A SCALE 36° O C 120° D E F A, B, C and D lie on a circle, centre O. DEF is a tangent to the circle at D. AOCF and BCE are straight lines. (a) Complete the statement. Angle ODE = 90° because … … [1] (b) Find the value of (i) angle AOD, Angle AOD = … [2] (ii) angle ODC, Angle ODC = … [2] (iii) angle ABC, Angle ABC = … [1] (iv) angle CFD, Angle CFD = … [1] (v) angle CAB. Angle CAB = … [1]

8 marks

Mark scheme: 8(a) [Angle between] tangent and 1 radius/diameter 8(b)(i) 108 2 M1 for ADO = 36 soi 8(b)(ii) 54 2 their(b)(i) 180 − 72 M1 for or 90 – 36 or 2 2 8(b)(iii) 90 1 8(b)(iv) 18 1 8(b)(v) 48 1

This question in 0607/42 May/June 2018

8 S A NOT TO SCALE B O D T C A, B, C and D lie on a circle, centre O. ST is a tangent to the circle at A. ODT is a straight line that bisects angle AOC. (a) Complete the statement. Angle OAT = … because … … [2] (b) DT = OC Find angle ABC. Angle ABC = … [4]

6 marks

Mark scheme: 8(a) 90 2 B1 for each Angle [between] tangent and radius oe 8(b) 60 4 B3 for angle AOT = 60 1 or M2 for sin(OAT ) = 2 1 or cos( AOT ) = oe 2 or M1 for OT = 2OA oe

This question in 0607/41 Oct/Nov 2018

5 NOT TO A SCALE 25°25° 20° C D O X B A, B, C and D lie on a circle, centre O. AX is a tangent to the circle at A and BX is a tangent to the circle at B. Angle OAB = 20 ° and angle DAX = 25 °. (a) Find the value of (i) angle AOB, Angle AOB = … [2] (ii) angle ACB, Angle ACB = … [1] (iii) angle ADB, Angle ADB = … [1] (iv) angle BAD, Angle BAD = … [1] (v) angle DBA, Angle DBA = … [1] (vi) angle AXB. Angle AXB = … [1] (b) What type of quadrilateral is ACBD? … [1]

8 marks

Mark scheme: 5(a)(i) 140 2 B1 for angle OBA = 20 soi 5(a)(ii) 70 1 FT 0.5 × their (i) 5(a)(iii) 110 1 FT 180 − their (ii) 5(a)(iv) 45 1 5(a) (v) 25 1 5(a)(vi) 40 1 5(b) Cyclic [quadrilateral] 1

This question in 0607/42 Oct/Nov 2019

9 A 58° NOT TO SCALE 14 cm 12 cm O B N C A, B and C are points on the circle, centre O. ON is perpendicular to BC. AB = 14 cm, AC = 12 cm and angle BAC = 58°. (a) Show that BC = 12.73 cm, correct to 2 decimal places. [3] (b) Explain why angle BON = 58°. … … [1] (c) Calculate OB, the radius of the circle. OB = … cm [3] (d) Calculate the area of the shaded segment. … cm2 [3]

10 marks

Mark scheme: 9(a) 142 + 122 – 2 × 14 × 12 × cos58 M1 12.725 to 12.726 A2 or A1 for 161.9... 9(b) Angle at centre = 2 × angle at 1 circumference oe 9(c) 7.49 or 7.5[0] or 7.51 or 7.487 to 7.506 3 6.365 M2 for oe sin58 6.365 or M1 for sin 58 = oe OB 9(d) 31.3 to 31.9 nfww 3 116 M2 for × π × (their (c))2 360 1 − × (their (c))2 × sin116 oe 2 116 or M1 for × π × (their (c))2 oe 360 1 or × (their (c))2 × sin116 oe 2

This question in 0607/43 Oct/Nov 2019

5 X B T 47° 65° NOT TO O SCALE A C D A, B, C and D lie on a circle, centre O. AD = CD and XBT is a tangent to the circle at B. TCD is a straight line. Angle XBA = 47° and angle TBC = 65° . Find the value of (a) angle OBX, Angle OBX = … [1] (b) angle AOB, Angle AOB = … [2] (c) angle CAO, Angle CAO = … [2] (d) angle CDA, Angle CDA = … [2] (e) angle DAC, Angle DAC = … [2] (f) angle CTB. Angle CTB = … [2]

11 marks

Mark scheme: 5(a) 90 1 5(b) 94 2 M1 for 180 – 43 – 43 or 2 × 47 5(c) 22 2 B1 for CAB = 65 or ACB = 47 5(d) 112 2 M1 for 180 – their ABC 5(e) 34 2 FT their (d) M1 for (180 – their(d))/2 5(f) 16 2 B1 for BCT = 99

This question in 0607/41 Oct/Nov 2020

2 (a) Find the size of one interior angle of a regular polygon with 45 sides. … [3] (b) B A 35° NOT TO 75° C SCALE T D In the diagram, A, B, C and D lie on the circle. TA is a tangent to the circle at A. Angle TAD = 75˚ and angle DAC = 35˚. Find (i) angle ACD, Angle ACD = … [1] (ii) angle ABC. Angle ABC = … [2] (c) C E NOT TO SCALE A D B In the diagram, DE is parallel to BC. (i) Complete the statement. Triangle ADE is … to triangle ABC. [1] (ii) AE = 6 cm, EC = 3 cm and DB = 2 cm. Calculate the length of AD. AD = … cm [3] (iii) The area of triangle ADE is 9 cm2. Calculate the area of triangle ABC. … cm2 [2]

12 marks

Mark scheme: 2(a) 172 3 360 180 × (45 − 2) M2 for 180 − or for 45 45 360 or M1 for (implied by 8) 45 or for 180 × (45 – 2) (implied by 7740) 2(b)(i) 75 1 2(b)(ii) 110 2 B1 for angle CAT = 110 or angle CDA = 70 or M1 for 180 – their angle CDA. 2(c)(i) similar 1 2(c)(ii) 4 3 9 AD + 2 M2 for = oe 6 AD AD 6 or M1 for = oe AB 9 2(c)(iii) 20.25 2 2 2  3   2  M1 for   or   oe seen  2   3 

This question in 0607/43 Oct/Nov 2020

9 A NOT TO SCALE x° D O P C B A, B and C lie on a circle, centre O. AP and BP are tangents to the circle. AB intersects OP at D and angle O AB = x° . (a) Write down the size of angle OBP. Angle OBP = … [1] (b) Find, in terms of x, (i) angle AOD, Angle AOD = … [1] (ii) angle ACB, Angle ACB = … [1] (iii) angle APB. Angle APB = … [1] (c) Write down the mathematical name of quadrilateral AOBP. … [1] (d) Write down (i) two triangles that are congruent, … [1] (ii) two triangles that are similar but not congruent. … [1]

7 marks

Mark scheme: 9(a) 90 1 9(b)(i) 90 – x oe 1 9(b)(ii) 90 – x oe 1 9(b)(iii) 2x 1 9(c) Kite 1 9(d)(i) OAD and OBD or OAP and OBP 1 or ADP and BDP 9(d)(ii) One from one of pairs in part (i) 1 and One from one of other pairs in part (i)

This question in 0607/43 Oct/Nov 2021

3 NOT TO SCALE A Q x° P B ( x - 30)° O D C A, B, C and D lie on a circle centre O. PQA is a tangent to the circle. QBC and PBOD are straight lines. Angle BQA = x° and angle ODA = ( x - 30) °. Find, in terms of x, expressions for each of the following angles. Give each answer in its simplest form. (a) angle BOA Angle BOA = … [1] (b) angle QBO Angle QBO = … [3] (c) angle CDB Angle CDB = … [3]

7 marks

Mark scheme: 3(a) 2x – 60 or 2(x – 30) final answer 1 3(b) 330 – 3x or 3(110 – x) final answer 3 M2 for 360 – 90 – x – (their (a)) oe or B1 for OAQ = 90 3(c) 240 – 3x or 3(80 – x) final answer 3 M2 for 180 –90 – (180 – their (b)) oe or B1 for DCB = 90

This question in 0607/42 May/June 2022

10 (a) A NOT TO E SCALE 54° O F D 62° B C G A, B, C, D and E are points on the circle centre O. FBG is a tangent to the circle at B. Angle ABF = 62° and angle BED = 54° . Find (i) angle AEB, Angle AEB = … [1] (ii) angle BAD, Angle BAD = … [1] (iii) angle EAD, Angle EAD = … [1] (iv) angle BCD, Angle BCD = … [1] (v) angle FBD. Angle FBD = … [1] (b) A NOT TO 6 cm SCALE O 120° P B PA and PB are tangents to the circle centre O. The radius of the circle is 6 cm and angle AOB = 120° . The shaded area = ( a 3 - b r) cm 2 . Find the value of a and the value of b. a = … b = … [5]

10 marks

Mark scheme: 10(a)(i) 62 1 10(a)(ii) 54 1 10(a)(iii) 36 1 FT 90 – their BAD 10(a)(iv) 126 1 10(a)(v) 126 1 10(b) [a =] 36 5 B3 for 36 3 [b =] 12 or M1 for [AP =] 6tan 60 oe (6 3 ) (10.392...) 1 M1 for 2 × × 6 × their 6 3 oe (62.35...) 2 B2 for 12π 120 2 or M1 for  π  6 (37.699...) 360

This question in 0607/43 May/June 2022

4 A NOT TO SCALE D X O 52° P C B A, B, C and D lie on a circle, centre O. AP and BP are tangents to the circle. AC and BD intersect at X. Angle APB = 52° . (a) Complete the statement. Angle OAP = 90° because … … [1] (b) Find (i) angle AOB, Angle AOB = … [1] (ii) angle OAB, Angle OAB = … [1] (iii) angle ACB. Angle ACB = … [1] (c) ABCD is a trapezium with AB parallel to DC. (i) Write down a triangle that is similar to triangle ABX. Triangle … [1] (ii) The length CD = 4 cm and the length AB = 12 cm . Find the ratio area CDX : area ABX. area CDX : area ABX = … : … [1]

6 marks

Mark scheme: 4(a) Tangent [and] radius or diameter[= 90] 1 4(b)(i) 128 1 4(b)(ii) 26 1 180 − their ( i ) FT 2 4(b)(iii) 64 1 their ( i ) FT 2 4(c)(i) CDX 1 4(c)(ii) 1 : 9 oe 1

This question in 0607/41 Oct/Nov 2022

3 A NOT TO SCALE B x°x° C T O D A, B, C and D lie on a circle, centre O. CODT is a straight line. AT is a tangent to the circle at A. Angle DAT = x°. (a) Complete the statement. Angle CAD = 90° because … [1] (b) Find, in terms of x, (i) angle ACD Angle ACD = … [1] (ii) angle AOD Angle AOD = … [1] (iii) angle AOC Angle AOC = … [1] (iv) angle ADO Angle ADO = … [1] (v) angle ABC. Angle ABC = … [1] (c) Given that angle DTA = y°, find y in terms of x. y = … [1]

7 marks

Mark scheme: 3(a) Angle in semicircle 1 3(b)(i) x 1 3(b)(ii) 2x oe 1 FT 2 × their (i) 3(b)(iii) 180 – 2x oe 1 FT 180 – their (ii) 3(b)(iv) 90 – x oe 1 3(b)(v) 90 + x oe 1 FT 180 – their (iv) 3(c) [y =] 90 – 2x oe 1 FT 90 – their (b)(ii) 4 In parts (b), (c) and (d), marks can only be earned with an increasing curve

This question in 0607/42 Feb/March 2023

10 B 78° NOT TO SCALE O A C 20° E D A, B, C and D lie on a circle, centre O. DE is a tangent to the circle at D. ACE is a straight line. Find (a) angle AOC Angle AOC = … [1] (b) angle OAC Angle OAC = … [2] (c) angle ADC Angle ADC = … [1] (d) angle CAD. Angle CAD = … [3]

7 marks

Mark scheme: 10(a) 156 1 10(b) 12 2 180  their ( a ) M1 for 2 10(c) 102 1 10(d) 29 3 M2 for x x their102  20  180 oe or M1 for indicating angle CDE = angle DAC

This question in 0607/43 May/June 2023

10 (a) Q NOT TO SCALE O R 135° P S P, Q, R and S are points on the circle centre O. Find angle PQR. Angle PQR = … [1] (b) C B D NOT TO SCALE O F 62° A E A, B, C, D and E are points on the circle centre O. AC is parallel to ED. Find the obtuse angle BOD. Angle BOD = … [2] (c) B NOT TO A C SCALE O 7.5 cm E D ABCDE is a regular pentagon. A, B, C, D and E are points on the circle centre O. The length of the perpendicular from O to ED is 7.5 cm. (i) Show that the length of one side of the pentagon is 10.9 cm correct to 3 significant figures. [4] (ii) Calculate the shaded area. … cm2 [4]

11 marks

Mark scheme: 10(a) 45 1 10(b) 124 2 B1 for BED = 62 10(c)(i) [EOD=] 72 or [ODE=] 54 B1 seen or implied by 36 2  7.5  tan36 oe M2 halfside 7.5 M1 for tan(36) = or tan54 = 7.5 7.5 halfside or 2  tan54 10.89…. A1 no errors or omissions 10(c)(ii) 13.1 or 13.11 to 13.13… 4 2 2 2  10.9  M1 r = 7.5 + or better or trig method    2  for r 1 1 2 M1  10.9  7.5 or 2  their r  their sin72 oe 2 their 72 2 M1   their r oe 360

This question in 0607/42 Oct/Nov 2023

5 A NOT TO B SCALE 61° 22° D O P C A, B, C and D lie on a circle, centre O. AP is a tangent to the circle at A. OP is perpendicular to AC and AD is parallel to BC. Angle ABC = 61° and angle PAD = 22° . (a) Write down the mathematical name of the cyclic quadrilateral ABCD. … [1] (b) Complete the statement. Angle OAP = 90° because … … [1] (c) Find (i) angle ADC Angle ADC = … [1] (ii) angle ACD Angle ACD = … [1] (iii) angle ACB Angle ACB = … [2] (iv) angle OCA. Angle OCA = … [2]

8 marks

Mark scheme: 5(a) trapezium 1 5(b) Angle between tangent and radius oe 1 5(c)(i) 119 1 5(c)(ii) 22 1 5(c)(iii) 39 2 M1 for 180 – their (i) – their (ii) 5(c)(iv) 29 2 B1 for 61 or 122 at centre (may be on diagram)

This question in 0607/41 May/June 2024

6 P NOT TO SCALE Q 113° R S P, Q, R and S are points on the circle. Find angle PQR. Angle PQR = … [1]

1 marks

Mark scheme: 6 67 1

This question in 0607/42 Feb/March 2025