Similarity — Paper 3 · IGCSE Mathematics 0580

6 For y Examiner's Use B 6 4 2 A E C x –4 –2 0 2 4 6 8 10 12 –2 –4 –6 Triangle ABC is drawn on a 1cm2 grid. E is the point (0, 0). (a) Write down the gradient of the line AB. Answer(a) [2] (b) The gradient of BC is – 0.5 . Write down the equation of the line BC in the form y = mx + c. Answer(b) y = [2] (c) Write down the ratio AE : EC. For Give your answer in its simplest form. Examiner's Use Answer(c) : [2] (d) Measure angle ABE. Answer(d) Angle ABE = [1] (e) Triangle ABE is similar to triangle BCE. Explain what the word similar tells you about the triangles ABE and BCE. Answer(e) [2] (f) Calculate the area of triangle ABC. Answer(f) cm2 [3] (g) ABCD is a rectangle. (i) Mark point D on the grid. [1] (ii) Write down the co-ordinates of D. Answer(g)(ii) ( , ) [1]

14 marks

Mark scheme: (e) 6 × 10–3 4 M1 ‘50’ × ‘120’ figs seen in area calculation A1 for 6000 seen (implied by 0.006 later) M1 for dividing by 1000², 0.05 & 0.12 seen or ×10–6 oe somewhere B1 ft from ‘their 0.006’ provided SF power is –ve Or SC1 for 0.6 × 10–2 oe 9 (a) (i) 226 to 226.224 cm³ 3 M1 π × 3² × 8 B1 for units : cm³ (ii) 8 cao www 4 B1 1500 used M1ft 3 × their (a)(i) 4 their 1500 M1ft 3 × their (a)(i) 4 16 (b) 5.09 (5.092 to 5.10) 2 M1 π (c) 148 cm² 3 M2 for 2 × 4 × 5 + 2 × 4 × 6 + 2 × 5 × 6 SC1 for 2 × 4 × 5 oe or 4 × 5 + 4 × 6 + 5 × 6 implied by 40, 48, 60 or 74, or list of 20, 20, 24, 24, 30, 30 (d) (i) mv oe 1 (ii) msv oe 1ft Ft (d)(i) × s (iii) 1000 msv oe 1ft Ft (d)(ii) × 1000

This question in 0580/31 May/June 2012

4 (a) For Examiner's Use A B C D Complete the table for each of the different quadrilaterals A, B, C and D. Quadrilateral Mathematical name Number of lines of symmetry A B C D [8] (b) R C 9 cm NOT TO 14 cm SCALE 6 cm P 120° 120° B 10 cm A 21 cm Q The two triangles are similar. (i) Write down the angle in triangle PQR that corresponds to angle B in triangle ABC. Answer(b)(i) Angle [1] (ii) Work out PQ. Answer(b)(ii) PQ = cm [2]

11 marks

Mark scheme: 4 (a) Parallelogram 0 1,1 Kite 1 1,1 Rhombus 2 1,1 Trapezium 0 1,1 (b) (i) Q or RQP or PQR 1 (ii) 15 2 M1 for a complete correct method

This question in 0580/32 Oct/Nov 2012

1 For Examiner′s y Use 8 7 6 5 B 4 3 A 2 1 x –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 –1 –2 C –3 –4 –5 –6 –7 –8 Triangles A, B and C are shown on a 1 cm2 grid. (a) Write down the mathematical name for triangle A. Answer(a) … [1] (b) Complete the following statement. Triangles A, B and C are … triangles because they are the same shape and size. [1] (c) Describe fully the single transformation that maps For Examiner′s Use (i) triangle A onto triangle B, Answer(c)(i) … … [2] (ii) triangle A onto triangle C. Answer(c)(ii) … … [3] (d) Refl ect triangle A in the x-axis. Label the image P. [1] (e) Enlarge triangle A, scale factor 2, centre (0, 0). Label the image Q. [2] (f) Calculate the area of triangle Q. Answer(f) … cm2 [2] _____________________________________________________________________________________

12 marks

Mark scheme: 1 (a) Scalene [triangle] 1 1 (b) Congruent (c) (i) translation 1 − 6    1 Accept 6 left and 2 up. 2   (ii) rotation 1 SC1, 1, 1 for 180° 1 Enlargement, [SF=] –1,(0,0) [Centre] ( 0,0 ) 1 (d) Image (1, –2), (4, –2), (2, –3) 1 (e) Image (2, 4), (8, 4), (4, 6) 2 B1 for 2 times enlargement, incorrect centre (f) 6 2FT M1 for 0.5 × their base × their height IGCSE – October/November 2013 0580 32 5 80

This question in 0580/32 Oct/Nov 2013

4 (a) The diagram shows the plan of part of Rachel’s garden. 14.6 m 3.2 m NOT TO SCALE 10.9 m 8.5 m Calculate the area. … m2 [3] (b) Rachel has a pond in her garden in the shape of a circle. The circumference of the pond is 4.25 m. Calculate the diameter of the pond. Give your answer in centimetres. … cm [3] (c) A plant pot is a cylinder with radius 15 cm and height 24 cm. Calculate the volume of the pot. … cm3 [2] (d) The diagram shows two mathematically similar plant pots. h NOT TO 21.6 cm SCALE 27 cm 33 cm The smaller pot has height 21.6 cm and diameter 27 cm. The larger pot has diameter 33 cm. Find the height, h, of the larger pot. h = … cm [2] (e) A shop sells bags of compost in three different sizes. Small Medium Large 30 litres 50 litres 75 litres $5.82 $9.45 $14.50 Work out which size of bag gives the best value. Show how you decide. … [3]

13 marks

Mark scheme: 4(a) 112.17 3 M2 for 14.6 × 10.9 – (14.6 – 8.5) × (10.9 – 3.2) oe or M1 for a correct method to find one of the six areas or B1 for 7.7 or 6.1 soi 4(b) 135 or 135.26 to 135.3 3 4.25 M1 for oe π B1 for their answer in metres seen correctly converted to cm or 425 [cm] seen 4(c) 17 000 or 16 960 to 16 970 2 M1 for π × 152 × 24 oe 4(d) 26.4 2 21.6 27 M1 for = oe or better h 33 4(e) Medium 3 M2 for 3 correct consistent divisions shown With correct comparisons made of the but either not evaluated to enough accuracy 3 bags with suitable accuracy shown or wrong bag selected or M1 for 2 correct consistent divisions shown for 2 bags

This question in 0580/33 Oct/Nov 2020

6 The diagram shows three triangles, A, B and C, on a 1 cm2 grid. y 44 33 C 22 11 x –– 99 –– 88 –– 77 –– 66 –– 55 –– 44 –– 33 –– 22 –– 11 0 11 22 33 44 55 –– 11 –– 22 B –– 33 –– 44 A b –– 55 –– 66 –– 77 (a) Describe fully the single transformation that maps (i) triangle A onto triangle B, … … [3] (ii) triangle A onto triangle C. … … [3] (b) On the grid, draw the image of - 5 (i) triangle A after a translation by the vector [2] e 4o, (ii) triangle A after a reflection in the line x =- 4.5 . [2] (c) The diagram also shows an angle b in triangle B. Use trigonometry to show that angle b is 63.4°, correct to 1 decimal place. [2] (d) D B E 63.4° 63.4° Two new triangles, D and E, are made from triangle B, as shown in the diagram. Are all three triangles similar? Give a reason for your answer. … because … … [2]

14 marks

Mark scheme: 6(a)(i) Enlargement 3 B1 for each [centre] ( − 4, − 5) [scale factor] 4 6(a)(ii) Rotation 3 B1 for each [centre] (0, 0) oe 90° clockwise oe 6(b)(i) Correct translation 2 k  −5  ( −7,1) ,( −−7, 1) , ( −−8, 1) B1 for translation  or   4  k  6(b)(ii) Correct reflection 2 B1 for reflection in x = k , k ≠−4.5 = − 4.5 ( −−6, 5) ,( −−7, 3) , ( −−7, 5) or in y 6(c) 8 M1 tan b = oe 4 63.43… A1 6(d) Yes correct reason 2 B1 for evidence that all three triangles each have angles 63.4°, 26.6° and 90° B1 for yes and statement that triangles are similar because they have the same 3 angles oe

This question in 0580/31 May/June 2021

8 (a) D NOT TO SCALE A 74.9 cm 21.4 cm B 14.8 cm C E F Right-angled triangles ABC and DEF are similar. (i) Calculate EF. EF = … cm [2] (ii) Calculate angle BCA. Angle BCA = … [2] (b) The diagram shows two congruent rectangular tiles placed together. H NOT TO 32.5 cm SCALE G The width of each tile is 32.5 cm and GH = 84.5 cm . Find the length of each tile. … cm [4] (c) Town B is 72 km from town A on a bearing of 058°. Town C is 60 km due east of town B. (i) Using a scale of 1 cm to represent 12 km, complete the scale drawing to show the positions of town B and town C. North A Scale: 1 cm to 12 km [3] (ii) Measure the bearing of town C from town A. … [1]

12 marks

Mark scheme: 8(a)(i) 51.8 2 EF 14.8 M1 for = oe or better 74.9 21.4 8(a)(ii) 46.2 or 46.24… 2 14.8 M1 for cos [ =] oe 21.4 8(b) 39 4 B3 for 78 1 2 2 or M3 for × 84.5 − 32.5 or better 2 or M2 for 84.5 2 − 32.5 2 or better or M1 for [ ]2 + 32.52 = 84.52 or better 8(c)(i) Accurate scale drawing 3 B1 for accurate bearing at A of 058° B1 for AB of length of 6 cm B1 for BC of length of 5 cm in direction east 8(c)(ii) Correct bearing 1 Strict FT on bearing of C from A

This question in 0580/32 Oct/Nov 2021

10 (a) U P 21.4 cm 14.4 cm NOT TO 9.6 cm SCALE Q R V W 12.8 cm Triangle PQR is mathematically similar to triangle UVW. Calculate VW. VW = … cm [2] (b) ABC is a right-angled triangle. A NOT TO 9.1 cm SCALE 3.5 cm C B Calculate BC. BC = … cm [3] (c) DEF is a right-angled triangle. D 8.4 cm NOT TO SCALE 35° F E Calculate EF. EF = … cm [2] (d) JKL is a right-angled triangle. 10 cm J L NOT TO 8 cm SCALE K Calculate angle JKL. Angle JKL = … [2]

9 marks

Mark scheme: 10(a) 19.2 2 14.4 VW M1 for = oe or better 9.6 12.8 10(b) 8.4 3 M2 for 9.12 – 3.52 oe or better or M1 for […]2 + 3.52 = 9.12 oe or better 10(c) 6.88 to 6.881 2 [...] M1 for cos35 = oe or better 8.4  or sin 55 = oe or better 8.4 10(d) 51.3[4…] 2 10 M1 for tan[… =] oe or better 8

This question in 0580/31 Oct/Nov 2023

3 (a) In triangle DEF, DE = 6 cm and DF = 4. 8 cm . Using a ruler and compasses only, construct triangle DEF. Leave in your construction arcs. The line EF has been drawn for you. E F [2] (b) B A C T D F G E (i) Write down the letter of the triangle that is congruent to triangle T. … [1] (ii) Write down the letter of the triangle that is similar but not congruent to triangle T. … [1] (c) NOT TO SCALE h 62° 62° 7 cm The diagram shows an isosceles triangle. (i) Show that the perpendicular height, h, is 6.58 cm, correct to 3 significant figures. [3] (ii) Calculate the area of the triangle. Give the units of your answer. … … [3] (iii) Kalpit tries to arrange some of these triangles to make a regular polygon with centre O. O NOT TO SCALE 7 cm 7 cm 62° 62° 7 cm Show that Kalpit cannot make a regular polygon. [3]

13 marks

Mark scheme: 3(a) Correct triangle drawn with construction 2 B1 for correct triangle drawn with arcs. incorrect or no arcs or for 2 correct arcs drawn If 0 scored, SC1 for reversed triangle with arcs. 3(b)(i) G 1 3(b)(ii) F 1 3(c)(i) 7 M2 h [h=] tan 62 oe M1 for tan62 = oe 2 7 2 6.582 to 6.583 A1 3(c)(ii) 23[.0] or 23.03 to 23.04… 2 1 M1 for 7 6.58 oe 2 cm2 1 3(c)(iii) 360 M2 M1 for 180 −2 62 oe 180 −2 62 6.4… which is not integer oe A1

This question in 0580/32 Feb/March 2024

7 (a) The scale drawing shows the position of a castle, C. The scale is 1 centimetre represents 2 kilometres. North C Scale: 1 cm to 2 km Micah walks at 4.8 km/h for 3 hours on a bearing of 236° from C, to his house, H. Mark the position of H on the scale drawing. [3] (b) A restaurant, R, is on a bearing of 083° from C. (i) Work out the bearing of C from R. … [2] (ii) The distance, d km, from R to C is 5 km, correct to the nearest kilometre. Complete this statement about the value of d. … G d 1 … [2] (c) A model of the castle is made. On the model, the length of one of the castle walls is 5 centimetres. The actual length of the wall is 90 metres. Find the scale of the model in the form 1| n . 1| … [2] (d) The castle has two kitchens with rectangular floors. F G B C NOT TO SCALE 5.6 m 9.2 m A 8.4 m D E x m H Rectangle ABCD is mathematically similar to rectangle EFGH. Calculate the value of x. x = … [2] (e) The castle has a cylindrical tower. The tower has a radius of 2.4 m and a height of 25 m. Calculate the curved surface area of the tower. Give the units of your answer. … … [3] Question 8 is printed on the next page.

14 marks

Mark scheme: 7(a) H correctly marked 3 B2 for H 7.2 cm from C 7.2 cm from C on bearing 236° or M1 for 4.8×3 B1 for H on bearing 236 from C 7(b)(i) 263 2 M1 for 83  180 oe or indicates correct angle on diagram 7(b)(ii) 4.5 5.5 2 B1 for each If 0 scored SC1 for both correct but reversed 7(c) [1 : ] 1800 2 M1 for 5 : 9000 or 0.05:90 9000 90 or or 5 0.05 or B1 for answer figs 18 7(d) 13.8 2 9.2 5.6 M1 for  oe or better x 8.4 7(e) 377 or 376.9 to 377.04 2 M1 for 2 2.4  25 m2 1

This question in 0580/31 May/June 2024

24 V B NOT TO SCALE 14 cm A C 12 cm U W 21 cm Triangle ABC is mathematically similar to triangle UVW. Calculate AB. AB = … cm [2]

2 marks

Mark scheme: 24 8 2 12 21 M1 for = oe AB 14

This question in 0580/31 May/June 2025

24 Triangles ABC and PQR are mathematically similar. Q B NOT TO 42 cm SCALE A P 12 cm C 28 cm R Calculate AB. AB = … cm [2]

2 marks

Mark scheme: 24 18 2 42 28 M1 for = oe or better x 12

This question in 0580/32 Oct/Nov 2025

29 (a) NOT TO SCALE 20.8 cm A x cm B 9.1 cm 1.4 cm The diagram shows two similar triangles, A and B. Calculate the value of x. x = … [2] (b) 5.4 cm 38° y cm The diagram shows a right-angled triangle. Calculate the value of y. y = … [2]

4 marks

Mark scheme: 29(a) 3.2 2 20.8 9.1 M1 for = oe x 1.4 29(b) 4.26 or 4.255.. 2 y M1 for cos38 = or better 5.4 y or sin 52 = or better 5.4

This question in 0580/33 Oct/Nov 2025