Right-angled triangles — Paper 4 · IGCSE Mathematics - International 0607

3 A NOT TO SCALE 30 m 37° 26° B D C In the diagram, BCD is a straight line. (a) Find AC. AC = … m [3] (b) Find BC. BC = … m [3] (c) Find CD. CD = … m [3] (d) Find the area of triangle ACD. … m2 [2]

11 marks

Mark scheme: 3(a) 49.8 or 49.84 to 49.85 3 30 M2 for oe sin37 30 or M1 for sin 37 = oe AC 3(b) 39.7 or 39.8 or 39.74 to 39.81… 3 30 M2 for or their (a) × cos 37 oe tan37 30 BC or M1 for tan 37 = or cos 37 = oe BC their (a) 3(c) 3 30 21.7 or 21.8 or 21.67 to 21.81 M2 for – their(b) tan 26 (their (a)) × sin(180 − (180 − 37) − 26) or oe sin26 30 or M1 for tan 26 their (a) CD or = oe sin26 sin(180 − (180 − 37) − 26) 3(d) 325 or 326 or 327 2 1 M1 for × their (c) × 30 oe or 325[.0] to 327.2 2

This question in 0607/41 May/June 2017

5 8 cm NOT TO SCALE 16 cm The diagram shows a solid sphere of radius 4 cm inside a hollow cone of radius 8 cm and height 16 cm. The sphere touches the interior of the cone. (a) Calculate the volume of the cone that is not occupied by the sphere. … cm3 [3] (b) Calculate the curved surface area of the cone. … cm2 [3] (c) 8 cm NOT TO SCALE O 4cm 16 cm V The centre, O, of the sphere is directly above the vertex, V, of the cone. Calculate the length OV. OV = … cm [4]

10 marks

Mark scheme: 5(a) 804 or 804.2 to 804.4 3 1 2 M1 for 3π× 8 × 16 4 3 M1 for π× 4 3 5(b) 450 or 449.5 to 449.6… 3 2 2 M2 for π× 8 × 8 + 16 or M1 for 8 2 + 16 2 or π× 8 ×their l 5(c) 8.94 or 8.944… 4 P is point of contact between slant edge and circle. B2 for PV = 8 nfww 8 16 or M1 for = oe 4 PV M1 for OV 2 = 4 2 + PV 2 OR B2 for l = 320 oe or M1 for l 2 = 82 + 16 2 8 l M1 for = soi 4 OV OR x is semi-vertical angle of cone 8 M1 for tan x = oe 16 4 M2 for sin x 4 or M1 for = sin x OV

This question in 0607/43 May/June 2017

7 A NOT TO SCALE 35° 8 cm B D 6 cm 9 cm C (a) Calculate AB. AB = … cm [3] (b) Calculate angle BCD. Angle BCD = … [3]

6 marks

Mark scheme: 7(a) 9.77 or 9.766… 3 8 M2 for oe cos35 8 or M1 for cos35 = oe AB 7(b) 60.6 or 60.61… 3 6 2 + 9 2 − 82 M2 for 2 × 6 × 9 or M1 for 82 = 6 2 + 9 2 −×2 6 × 9cos C

This question in 0607/43 May/June 2017

9 E 50° NOT TO SCALE 13 cm D 70° A B C 15 cm In the diagram, ABC is a straight line, AE = BE = 13 cm and BC = 15 cm. Angle EAB = 70°, angle EBD = 90° and angle BED = 50°. Calculate (a) the length of the perpendicular line from E to AB, … cm [2] (b) the length BD, BD = … cm [2] (c) the length CD, CD = … cm [4] (d) the area of the quadrilateral ACDE. … cm2 [3]

11 marks

Mark scheme: 9(a) 12.2 or 12.21 to 12.22 2 [ ] M1 for sin70 = oe 13 9(b) 15.5 or 15.49… 2 BD M1 for tan50 = oe 13 9(c) 5.32 or 5.316 to 5.319… 4 B1 for [angle DBC = ] 20 M1 for (theirBD)2 + 152 – 2 × their BD × 15cos(their DBC) A1 for 28.26 to 28.30… 9(d) art 195 3 M2 two of 5.0 × 13 × 13 × sin 40 oe 0.5 × 13 × their BD oe 0.5 × 15 × their BD × sin(their 20) or M1 for one of above

This question in 0607/41 May/June 2018

9 B 9.1 cm NOT TO SCALE 8.2 cm A 11 cm C (a) Show that angle BAC = 47.0° , correct to 1 decimal place. [3] (b) Use the sine rule to find angle ABC. Angle ABC = … [3] (c) Find the area of triangle ABC. … cm2 [2] (d) Find the length of the perpendicular from B to AC. … cm [2]

10 marks

Mark scheme: 9(a) 112 + 9.12 − 8.2 2 M2 M1 for 8.2 2 = 112 + 9.12 − 2 × 11 × 9.1 × cos[ ] [cos A] = 2 × 11 × 9.1 46.98 to 46.99 A1 9(b) 11 M2 8.2 11 [sin B = ] × sin 47.0 M1 for = 8.2 sin 47 sin B 78.8 or 78.74 to 78.84 A1 If 0 scored then SC1 for correct answer from cosine rule or other method 9(c) 36.6 or 36.54 to 36.60… 2 M1 for 0.5 × 9.1 × 11 × sin47.0 or M1 for 0.5 × 9.1 × 8.2 × sin( their (b)) or M1 for 0.5 × 8.2 × 11 × sin (180 – 47 – their (b)) 9(d) 6.65 or 6.66 or 6.647 to 6.656… 2 M1 for 9.1 × sin47.0 oe or their (c) ÷ (0.5 × 11)

This question in 0607/42 May/June 2018

11 North D C 30° 60° NOT TO 102 m SCALE 110 m B A The diagram shows two fields on horizontal ground. A is due south of D and C is due east of D. (a) Calculate DC. DC = … m [3] (b) Calculate AB. AB = … m [3] (c) Calculate the total area of the fields. … m2 [3] (d) Calculate the bearing of A from B. … [4]

13 marks

Mark scheme: 11(a) 118 or 117.7 to 117.8 3 102 M2 for oe cos 30 102 or M1 for = cos 30 oe DC or 102 = DC × cos30 oe 11(b) 106 or 106.2... 3 M1 for 1102 + 1022 – 2 × 110 × 102 × cos 60 A1 for 11 284 11(c) 7860 or 7858 to 7870 3 M1 for 0.5 × 102 × their DC × sin30 oe (3000 or 3010 or 3001 to 3009) M1 for 0.5 × 102 × 110 × sin60 oe (4860 or 4858...) 11(d) 236 or 236.2 to 236.4... 4 B2 for 56.3 or 56.4 or 56.25 to 56.44... 102 sin 60 or M2 for oe their AB sin 60 sin BAD or M1 for = oe theirAB 102 and M1 for 180 + their angle BAD oe

This question in 0607/42 May/June 2019

11 F NOT TO SCALE B 41° 5.5 m 6.2 m A C D The diagram shows four points A, B, C and D on horizontal ground. There is a vertical flagpole, FB, held in place by straight wires AF, CF and DF. BCD is a straight line, AB = 5.5 m, BC = 6.2 m and angle FAB = 41°. (a) Show that FB = 4.781 m, correct to 3 decimal places. [2] (b) Calculate angle FCB. Angle FCB = … [2] (c) Angle CDF = 18°. Show that CD = 8.514, correct to 3 decimal places. [3] (d) Angle ABC = 78°. Find AD. AD = … m [3] (e) Find the area of triangle ABD. … m2 [2]

12 marks

Mark scheme: 11(a) FB M1 sin49 sin41 tan 41 = oe e.g. = 5.5 5.5 FB = 4.7810.. [= 4.781] A1 11(b) 37.6 or 37.63 to 37.64 2 4.781 M1 for tan[ FCB ] = oe 6.2 11(c) 4.781 M2 4.781 [CD = ] − 6.2 oe M1 for tan18 = oe tan18 BD 8.5144... A1 11(d) 14.6 or 14.58 to 14.60 3 B2 for 212.7... or M1 for [ AD2 =] 5.52 + (8.514 + 6.2)2 − 2 × 5.5 × (8.514 + 6.2) × cos78 11(e) 39.5 or 39.6 or 39.54 to 39.6[0] 2 M1 for 0.5 × 5.5 × (8.514 + 6.2) × sin 78

This question in 0607/43 May/June 2019

7 North North 10 km B 150° NOT TO A SCALE 12 km 18 km C 21 km D The diagram shows four villages A, B, C and D and five straight roads connecting them. B is 10 km due east of A. C is 12 km from B on a bearing of 150°. D is 21 km from C and 18 km from A. (a) Calculate the distance AC and show that your answer rounds to 19.08 km, correct to 2 decimal places. [4] (b) Using the sine rule, calculate angle ACB and show that your answer rounds to 27.0°, correct to 1 decimal place. [3] (c) Calculate the bearing of D from C. … [4] (d) A straight path, BP, connects B to the closest point, P, on AC. Calculate the length of this path. … km [2] (e) The area within triangle ABC is grassland. Calculate the area of this grassland. … km2 [2]

15 marks

Mark scheme: 7(a) [Angle ABC = ] 120 B1 10 2 + 12 2 − 2 × 10 × 12cos(their ABC ) M1 19.078 to 19.079 A2 A1 for 364 7(b) 10sin120 M2 19.08 10 M1 for = oe 19.08 sin120 sin ACB 26.99... A1 M1 only and A1 imply M2 A1 7(c) 249.8 to 250[.0] 4 212 + 19.082 − 182 M2 for [cos ACD = ] oe 2 × 21 × 19.08 or M1 for 18 2 = 212 + 19.08 2 −×2 21 × 19.08 × cos( ACD ) M1 for 360 – (30 + 27 + their ACD) oe 7(d) 5.45 or 5.446 to 5.448 2 M1 for 12sin27.0 oe 7(e) 52[.0] or 51.94 to 51.99... 2 1 M1 for × 19.08 × their (d) 2 1 or for × 10 × 12 × sin120 oe 2 1 or for × 19.08 × 12 × sin 27 2

This question in 0607/41 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

8 North B 5.37 km NOT TO SCALE North A C 48° 6.13 km 6.42 km D The diagram shows four points A, B, C and D on horizontal ground. B is due North of C and C is due East of A. (a) Find the bearing of (i) D from A, … [1] (ii) A from D. … [1] (b) Calculate angle ABC. Angle ABC = … [2] (c) Calculate the area of quadrilateral ABCD. … km 2 [3] (d) Calculate CD. CD = … km [3] (e) Angle ACD is acute. Find the bearing of D from C. … [4]

14 marks

Mark scheme: 8(a)(i) 138 1 8(a)(ii) 318 1 FT their (i) + 180 8(b) 48.8 or 48.78… 2 6.13 M1 for tan[ x = ] 5.37 8(c) 31.1 or 31.08… 3 M2 for 6. 13 × 5. 37 1 + × 6. 13 × 6. 42 × sin48 2 2 6.13 × 5. 37 or M1 for or 2 1 × 6. 13 × 6.42 × sin48 2 8(d) 5.11 or 5.111… 3 B2 for 26.1… or M1 for 6.132 + 6.42 2 − 2 × 6.13 × 6.42 × cos48 8(e) 201 or 200.9 to 201.1… 4 B3 for 69[.0] or 68.89 to 68.90 or M2 for sin 48 sin C = × 6.42, [C = 69.0] their (d) sin C sin 48 or M1 for = 6.42 their (d)

This question in 0607/42 May/June 2020

7 The diagram shows a radio in the shape of a prism. This diagram shows the base of the radio. E F A D G B C H I ABC is an equilateral triangle. The circles have their centres at A, B and C and each has a radius of 5 cm. DE, FG and HI are tangents to the circles. (a) Show that AB = 8.66 cm, correct to 3 significant figures. [3] (b) Calculate the area of the base of the radio. … cm2 [4] (c) The height of the radio is 12 cm. Calculate the volume of the radio. … cm3 [1]

8 marks

Mark scheme: 7(a) 2 × 5 × cos 30 M2 x or M1 for = cos 30 oe 5 8.660... A1 7(b) 241 or 240.9... to 241.2... 4 M1 for 3 × 8.66 × 5 120 M1 for 3 × × π × 52 360 M1 for × 8.662 × sin 60 7(c) 2890 to 2895 1 FT 12 × their (b)

This question in 0607/43 May/June 2020

5 P NOT TO SCALE 10 m 20 m C B 35° A A, B and C are points on horizontal ground. BP is a vertical pole. BC = 20 m and BP = 10 m. Angle PAB = 35°. (a) Show that PC = 22.36 m correct to 2 decimal places. [2] (b) Show that AB = 14.28 m correct to 2 decimal places. [2] (c) Calculate AP. AP = … m [2] (d) Angle ABC = 125°. Calculate AC. AC = … m [3] (e) Calculate angle APC. Angle APC = … [3]

12 marks

Mark scheme: 5(a) 202 + 102 M1 22.360 to 22.361 A1 5(b) 10 M1 sin35 sin55 tan35 = oe = , i.e correct implicit AB 10 AB 14.281... A1 5(c) 17.4 or 17.43... 2 10 M1 for sin35 = oe AP or 14.282 + 102 5(d) 30.5 or 30.52... 3 M1 for 20 2 + 14.282 −×2 20 × 14.28 × cos125 A1 for 931.5 to 931.6... 5(e) 99.2 to 99.5 3 M2 for [cos = ] their 30.5 22.36 2 + ( their 17.4 ) 2 − ( 2 ) 2 × 22.36 × ( their 17.4 ) or M1 for ( their 30.5 ) 2 = 22.36 2 + ( their17.4 ) 2 −×2 22.36 × ( their17.4 ) × cos APB

This question in 0607/42 May/June 2021

7 North B NOT TO SCALE 535 m 420 m C A 28° 750 m D The diagram shows four points A, B, C and D. B is due north of C and C is due east of A. AC = 420 m, AD = 750 m, BC = 535 m and angle CAD = 28°. (a) Find the bearing of (i) D from A, … [1] (ii) A from D. … [1] (b) Calculate AB. AB = … m [2] (c) Calculate CD. CD = … m [3] (d) Calculate the area of quadrilateral ABCD. … m2 [3] (e) Angle ACD is obtuse. Find the bearing of D from C. … [4]

14 marks

Mark scheme: 7(a)(i) 118 1 7(a)(ii) 298 cao 1 7(b) 680 or 680.1 to 680.2 2 M1 for 4202 + 5352 7(c) 427 or 427.3 to 427.4 3 M2 for  CD   420 2  750 2  2  420  750  cos28 or M1 for  CD 2  420 2  750 2 2 420  750  cos28   7(d) 186000 or 186200 to 186300 3 M1 for area ABC = 0.5  420  535 M1 for area ACD = 0.5  420  750 x sin28 7(e) 145 or 145 to 146 nfww 4 750  sin 28 M2 for sin ACD  theirCD 750 theirCD or M1 for  sin ACD sin28 And M1 for 360 – 90 – (180 – their acute C) OR 420 2  427.367 2  750 2 M2 for cos ACD = 2  420  427.367 or M1 for 7502 = 4202 + 427.372 – 2  420  427.37 cos C And M1 for 360 – 90 – their obtuse C

This question in 0607/42 May/June 2022

11 F X E NOT TO SCALE B A C D The diagram shows a triangular prism ABCDEF. X is a point on FE. AB = 8 m , AD = 15 m , AF = 10 m , EC = 6 m and FX = 5 m . Angle ABF = 90° and angle DCE = 90° . (a) Calculate angle CDE. Angle CDE = … [2] (b) Calculate AC. AC = … m [2] (c) Calculate angle CXA. Angle CXA = … [5] (d) Calculate the area of triangle CXA. … m2 [2] Question 12 is printed on the next page.

11 marks

Mark scheme: 11(a) 36.9 or 36.86 to 36.87 2 6 6 M1 for tan[CDE ] = or sin[CDE ] = 8 10 8 or cos[CDE ] = 10 11(b) 17 2 M1 for [ AC 2 ] = 15 2 + 8 2 11(c) 96.2 or 96.16… 5 M1 for AX 2 = 52 + 10 2 M1 for CX 2 = 10 2 + 6 2 M2 dep for their AX 2 + theirCX 2 − their AC 2 [cos CDX =] 2  their AX  theirCX their AC 2 = their AX 2 + theirCX 2 or M1dep for −2 their AX  theirCX  cos CDX both dependent on Pythagoras or trigonometry used for AX and CX 11(d) 64.8 or 64.81 to 64.82 2 M1 dep for area CXA = 0.5  theirCX  their AX  sin(theirCXA) dependent on Pythagoras or trigonometry used for AX and CX

This question in 0607/41 Oct/Nov 2022

11 North North B NOT TO SCALE 72 km North 85 km A 102 km C A, B, and C are three ports. The bearing of B from A is 040c. (a) Show that angle ABC = 80.6c , correct to 1 decimal place. [3] (b) Find the bearing of B from C. … [2] (c) A ship leaves port A at 13 00. It sails directly towards C at a speed of 32 km/h. At point P the ship is at its shortest distance from B. Find the time when the ship reaches point P. Give your answer correct to the nearest minute. … [6]

11 marks

Mark scheme: 11(a) 72 2  85 2  102 2 M2 M1 for 1022 = 722 + 852 – 2 × 72 × 85 × cos[B] [cosB] = 2  72  85 80.57... A1 11(b) 319.4 2 B1 for 40.6 or 139.4 11(c) 14 17 6 85sin80.6 M2 for sin [A] = 102 72 2  102 2  85 2 or cos[A] = 2  72  102 85 102 or M1 for  sin A sin80.6 or 85 2  72 2  102 2  2  72  102cos A M1 for [AP =] 72 cos their A oe M1 for [time =] their AP ÷ 32 M1 adding their time to 13 00 If 0 scored, SC1 for showing BP on diagram with right angle correctly placed on AC

This question in 0607/43 May/June 2024

5 B NOT TO 26.3 cm SCALE 115° A C The area of triangle ABC is 262 cm 2. (a) Show that AC = 22.0 cm , correct to 1 decimal place. [2] (b) Find BC. BC = … cm [3] (c) Use the sine rule to find angle ABC. Angle ABC = … [3] (d) Find the length of the perpendicular line from A to the line BC. … cm [2]

10 marks

Mark scheme: 5(a) 1 M1  26.3  AC  sin115 = 262 2  262  2  A1 AC = = 21.98  = 22.0     26.3  sin115  5(b) 40.8 or 40.78 to 40.80… 3 M2 for  BC = 26.32 + 222 −2 26.3  22  cos115 or M1 for  BC 2 =  26.32 + 22 2 −2 26.3  22  cos115   5(c) 22  sin115 M2 22 their 40.8 sin ABC = oe M1 for = oe their 40.8 sin ABC sin115 29.2 or 29.3 or 29.23 to 29.26 B1 5(d) 12.8 to 12.9 2 M1 for 262 = 0.5 x their 40.8 Or for x = 26.3  sin(their 29.2)

This question in 0607/41 Oct/Nov 2024

12 B 68° NOT TO 140 m SCALE 260 m A C The diagram shows a triangular field. AB = 140 m and BC = 260 m. Angle ABC = 68°. (a) Show that AC = 244.8 m correct to 1 decimal place. [3] (b) Giselle walks directly from A to C. Calculate the distance Giselle is from A when she is closest to B. … m [5]

8 marks

Mark scheme: 12(a) 2 2 M2 M1 for 140 2 + 260 2 −2 140  260  cos68 140 + 260 −2 140  260  cos68 244.80… A1 12(b) 24.3 to 24.4 5 M2 for complete explicit method for angle A or angle C or BN 260sin68 sin A = oe 244.8 140 2 + 244.82 − 260 2 or cos A = oe 2  140  244.8 140sin68 or sin C = oe 244.8 260 2 + 244.82 − 140 2 or cos C = oe 2  260  244.8 1  140  260  sin68 2 or BN = oe 1  244.8 2 or M1 for implicit method for angle A or angle C or BN sin A sin68 e.g. = oe 260 244.8 or 2602 = 1402 + 244.82 −2 140  244.8  cos A oe sin C sin68 or = oe 140 244.8 or 1402 = 2602 + 244.82 −2 260  244.8  cosC oe 1 1 or  BN  244.8 = 140  260  sin68 oe 2 2 AND M2 dep for 140 cos their A oe or 244.8 – 260 cos their C oe or 140 2 – theirBN 2 oe dependent on at least M1 above or M1 for clear indication of perpendicular from B to AC

This question in 0607/41 May/June 2025

6 NOT TO 6 cm SCALE w° 11 cm Calculate the value of w. w = … [2]

2 marks

Mark scheme: 6 28.6 or 28.61... 2 6 M1 for tan = oe 11

This question in 0607/43 May/June 2025

7 A 23 cm B NOT TO SCALE D C The diagram shows a trapezium, ABCD. AD = DC = CB and AB = 23 cm. The perimeter of the trapezium is 62 cm. (a) Calculate the area of the trapezium. … cm2 [5] (b) Calculate angle BAD. Angle BAD = … [2]

7 marks

Mark scheme: 7(a) 216 5 B1 for 13 23 −their13 M1 for 2 M1 for (their13)2 = (their5)2 + h2 1 M1 for (23 + their13)(theirh ) oe h < 13 2 7(b) 67.4 or 67.38… 2 their 5 M1 for cos = oe, h < 13 if used in their13 trig

This question in 0607/41 Oct/Nov 2025