Similarity — Paper 2 · IGCSE Mathematics 0580

18 Examiner's y Use 24 22 B D 20 18 C 16 14 12 G 10 H E A 8 6 4 F 2 x 0 2 4 6 8 10 12 14 16 18 20 22 Write down the letters of all the triangles which are (a) congruent to the shaded triangle, Answer(a) [2] (b) similar, but not congruent, to the shaded triangle. Answer(b) [2]

4 marks

Mark scheme: 18 (a) E, G 1, 1 (b) A, B 1, 1

This question in 0580/21 May/June 2010

9 For A Examiner's Use NOT TO 8 cm SCALE Q P 10 cm C 12 cm B APB and AQC are straight lines. PQ is parallel to BC. AP = 8 cm, PQ = 10 cm and BC = 12 cm. Calculate the length of AB. Answer AB = cm [2]

2 marks

Mark scheme: x 12 9 9.6 cao 2 M1 = oe 8 10 2

This question in 0580/23 Oct/Nov 2010

19 For Examiner′s Use D C A B Scale: 1 cm to 8 m The rectangle ABCD is a scale drawing of a rectangular football pitch. The scale used is 1 centimetre to represent 8 metres. (a) Construct the locus of points 40 m from A and inside the rectangle. [2] (b) Using a straight edge and compasses only, construct the perpendicular bisector of DB. [2] (c) Shade the region on the football pitch which is more than 40 m from A and nearer to D than to B. [1] _____________________________________________________________________________________

5 marks

Mark scheme: 19 (a) arc centre A radius 5 cm 2 B1 arc with centre A (b) ruled perpendicular bisector of DB with 2 2 B1 correct ruled line pairs of correct arcs B1 2 pairs of correct arcs (c) cao 1

This question in 0580/22 May/June 2013

18 NOT TO SCALE 10 cm 4 cm A solid cone has base radius 4 cm and height 10 cm. A mathematically similar cone is removed from the top as shown in the diagram. 1 The volume of the cone that is removed is of the volume of the original cone. 8 (a) Explain why the cone that is removed has radius 2 cm and height 5 cm. Answer(a) [2] (b) Calculate the volume of the remaining solid. 1 [The volume, V, of a cone with radius r and height h is V = πr 2h.] 3 Answer(b) … cm3 [4] __________________________________________________________________________________________ Question 19 is printed on the next page.

6 marks

Mark scheme: 10 4 1 1 = = 2 or 3 8 = 2 AND = 5 oe and18 (a) correct working 2 B2 for 3 8 2 2 2 oe or 1 1 1 = ( 3) or 3 8 or 8 = 23 or B1 for 3 8 8 2 IGCSE – May/June 2014 0580 21 7 1 2 (b) 147 or 146.5 to 146.6… 4 M3 for × × π × 4 × 10 8 3 or 1 2 M1 for × π × 4 × 10 3 and 1 2 M1 for × π × 2 × 5 3 and M1 for subtracting their volumes 1 i

This question in 0580/21 May/June 2014

20 (a) W V NOT TO 88° SCALE X Z 57° Y Two straight lines VZ and YW intersect at X. VW is parallel to YZ, angle XYZ = 57° and angle VXW = 88°. Find angle WVX. Answer(a) Angle WVX = … [2] (b) A NOT TO 7.2 cm SCALE P Q 8.4 cm B C 12.6 cm ABC is a triangle and PQ is parallel to BC. BC = 12.6 cm, PQ = 8.4 cm and AQ = 7.2 cm. Find AC. Answer(b) AC = … cm [2] __________________________________________________________________________________________

4 marks

Mark scheme: 20 (a) 35 2 M1 for [Z =] 180 − 88 − 57 or VWX = 57 or YZX = 35 AC 126. (b) 10.8 2 M1 for = oe 2.7 4.8

This question in 0580/22 Feb/March 2015

21 (a) NOT TO SCALE 25 cm 15 cm 7.2 cm x cm The diagram shows two jugs that are mathematically similar. Find the value of x. Answer(a) x = … [2] (b) NOT TO 16 cm SCALE y cm The diagram shows two glasses that are mathematically similar. The height of the larger glass is 16 cm and its volume is 375 cm3. The height of the smaller glass is y cm and its volume is 192 cm3. Find the value of y. Answer(b) y = … [3]

5 marks

Mark scheme: 2.7 15 25 21 (a) 12 2 M1 for = oe or better eg 2.7 × x 25 15 192 3 oe (b) 12.8 3 M2 for 16× 375 or 3 192 375  16  375 M1 for 3 or 3 oe or oe =  375 192   y  192

This question in 0580/22 Oct/Nov 2015

14 Two containers are mathematically similar. Their volumes are 54 cm3 and 128 cm3. The height of the smaller container is 4.5 cm. Calculate the height of the larger container. Answer … cm [3] __________________________________________________________________________________________

3 marks

Mark scheme: 128 14 6 3 M2 for 5.4 × 3 oe or better 54 3 128 54 54  5.4  M1 for 3 or 3 oe or =   oe 54 128 128  x  8 2 3

This question in 0580/23 Oct/Nov 2015

5 Triangle ABC is similar to triangle PQR. Q B NOT TO SCALE 5.2 cm A C P R 12.4 cm 21.7 cm Find PQ. PQ = … cm [2]

2 marks

Mark scheme: 5.2 12.4 5 9.1 oe 2 M1 for = oe PQ 21.7 4

This question in 0580/22 Feb/March 2016

21 (a) B 4.8 cm NOT TO E 6 cm SCALE 9 cm k cm C D A Triangles CBA and CED are similar. AB is parallel to DE. AB = 9 cm, BE = 4.8 cm, EC = 6 cm and ED = k cm. Work out the value of k. k = … [2] (b) h NOT TO 20 cm SCALE Vase A Vase B The diagram shows two mathematically similar vases. Vase A has height 20 cm and volume 1500 cm3. Vase B has volume 2592 cm3. Calculate h, the height of vase B. h = … cm [3]

5 marks

Mark scheme: 9 6 + 4.8 21 (a) 5 2 M1 for = oe k 6 2592 (b) 24 3 M2 for 3 × 20 oe 1500 2592 1500 or M1 for 3 or 3 1500 2592

This question in 0580/23 May/June 2016

10 The length of a backpack of capacity 30 litres is 53 cm. Calculate the length of a mathematically similar backpack of capacity 20 litres. … cm [3]

3 marks

Mark scheme: 20 10 46.3 or 46.29 to 46.30 3 M2 for 53 × 3 oe 30 20 30  53 3 30 or or M1 for 3 or 3 or  = 30 20  x  20 better

This question in 0580/22 Oct/Nov 2016

11 The two barrels in the diagram are mathematically similar. NOT TO SCALE 90 cm h cm The smaller barrel has a height of h cm and a capacity of 100 litres. The larger barrel has a height of 90 cm and a capacity of 160 litres. Work out the value of h. h = … [3]

3 marks

Mark scheme: 11 76.9 or 76.94 to 76.95 3 160 100 M2 forr 90 ÷ 3 or 90 × 3 100 160 160 100 or M1 for 3 sooi or 3 soi or 100 160  h  3 100   = oe  90  160

This question in 0580/21 May/June 2017

20 (a) 20 cm NOT TO SCALE A cylinder has height 20 cm. The area of the circular cross section is 74 cm2. Work out the volume of this cylinder. … cm3 [1] (b) Cylinder A is mathematically similar to cylinder B. NOT TO 10 cm A B SCALE The height of cylinder A is 10 cm and its surface area is 440 cm2. The surface area of cylinder B is 3960 cm2. Calculate the height of cylinder B. … cm [3]

4 marks

Mark scheme: 20(a) 1480 1 20(b) 30 3 3960 440 M2 for 10 × or 10 ÷ 440 3960 3960 440 or M1 for or or 440 3960  h 2 3960 oe  =  10  440

This question in 0580/21 Oct/Nov 2017

11 T 3 cm U NOT TO 12 cm SCALE The diagram shows two mathematically similar triangles, T and U. Two corresponding side lengths are 3 cm and 12 cm. The area of triangle T is 5 cm2. Find the area of triangle U. … cm2 [2]

2 marks

Mark scheme: 11 80 2 2 2 2 2  12   3  3 12 M1 for or oe or = oe      3   12  5 A

This question in 0580/22 May/June 2018

25 (a) A NOT TO SCALE P Q B C In the diagram, PQ is parallel to BC. APB and AQC are straight lines. PQ = 8 cm, BC = 10 cm and AB = 9 cm. Calculate PB. PB = … cm [2] (b) 13 cm NOT TO SCALE The diagram shows two glasses which are mathematically similar. The larger glass has a capacity of 0.5 litres and the smaller glass has a capacity of 0.25 litres. The height of the larger glass is 13 cm. Calculate the height of the smaller glass. … cm [3]

5 marks

Mark scheme: 25(a) 1.8 2 10 9 M1 for = oe 8 AP 25(b) 10.3 or 10.31 to 10.32 3 0.25 M2 for 13 × 3 oe 0.5 0.5 0.25 0.5  13  3 or M1 for 3 oe or 3 oe or =   oe 0.25 0.5 0.25  h 

This question in 0580/23 May/June 2018

2 A NOT TO SCALE B C N In the diagram, AB = AC and BN = NC. Complete the statement using a mathematical term. Triangle ABN is … to triangle ACN. [1]

1 marks

Mark scheme: 2 Congruent 1

This question in 0580/21 Oct/Nov 2018

16 A paperweight has height 4 cm and volume 38.4 cm3. A mathematically similar paperweight has height 7 cm. Calculate the volume of this paperweight. … cm3 [3]

3 marks

Mark scheme: 16 205.8 3 3  7  M2 for 38.4 ×   oe  4   7  3  4  3 or M1 for   or   oe or  4   7  7 v = 3 oe 4 38.4

This question in 0580/23 Oct/Nov 2020

20 (a) P NOT TO A SCALE x cm 1.61 cm R 3.2 cm Q C 2.8 cm B Triangle ABC is mathematically similar to triangle PQR. Find the value of x. x = … [2] (b) NOT TO SCALE The diagram shows two mathematically similar bowls. The larger bowl has capacity 7.8 litres and height 11.5 cm. The smaller bowl has capacity 4 litres. Calculate the height of the smaller bowl. … cm [3]

5 marks

Mark scheme: 20(a) 1.84 2 1.61 2.8 M1 for = oe x 3.2 20(b) 9.20 or 9.204 to 9.205 3 4 M2 for 11.5 × 3 oe 7.8 4 7.8 or M1 for 3 or 3 oe seen 7.8 4 11.53 7.8 or for = oe x 3 4

This question in 0580/22 Feb/March 2021

15 6 cm A B NOT TO SCALE X 8 cm 7 cm C D 12 cm In the diagram, AB is parallel to CD. AD and BC intersect at X. AB = 6 cm , CD = 12 cm , CX = 8 cm and DX = 7 cm . (a) Complete the statement. Triangle ABX is … to triangle DCX. [1] (b) Work out the length of BX. BX = … cm [2] (c) The area of triangle DCX is 26. 906 cm 2. Use this value to find the area of (i) triangle ABX, … cm2 [2] (ii) triangle ACX. … cm2 [1]

6 marks

Mark scheme: 15(a) Similar 1 15(b) 4 2 12 8 M1 for = oe or better 6 BX If 0 scored SC1 for answer 3.5 15(c)(i) 6.7265 or 6.73 or 6.726 to 6.727 2 2  1  M1 for scale factor 22 or   oe soi  2  15(c)(ii) 13.453 or 13.5 or 13.45 to 13.46 1 FT their (c)(i) × 2

This question in 0580/23 May/June 2021

11 C R NOT TO 6 cm SCALE A B P Q 8 cm 6 cm Triangle ABC is mathematically similar to triangle PQR. (a) Calculate QR. QR = … cm [2] (b) The two triangles are the cross-sections of two mathematically similar prisms. The volume of the larger prism is 320 cm 3 . Calculate the volume of the smaller prism. … cm3 [2]

4 marks

Mark scheme: 11(a) 4.5 oe 2 8 6 M1 for oe or better 6 QR 11(b) 135 2 3 3 3  6   8   their 4.5  M1 for   or   or   oe  8   6   6 

This question in 0580/21 May/June 2022

12 C R NOT TO SCALE P Q A 9 cm B PR 2 Triangle PQR is similar to triangle ABC with = . AC 3 AB = 9 cm and the area of triangle ABC is 18 cm2. (a) Find the length of PQ. … cm [1] (b) Find the area of triangle PQR. … cm2 [2]

3 marks

Mark scheme: 12(a) 6 1 12(b) 8 2 2 2  2   3  M1 for   or   oe seen  3   2 

This question in 0580/21 Oct/Nov 2022

9 The diagram shows three triangles A, B and C. 60° 60° 6 cm NOT TO 60° 25° SCALE 6 cm 25° 6 cm 25° A B C (a) Which two of the triangles A, B and C are congruent with each other? … [1] (b) Draw a ring around the congruence criterion that can be used to support your answer to part (a). SSS ASA SAS RHS [1]

2 marks

Mark scheme: 9(a) A and C 1 9(b) ASA 1

This question in 0580/22 Oct/Nov 2022

22 The volumes of two mathematically similar objects are 56 cm 3 and 875 cm 3. The height of the smaller object is 18 cm. Find the height of the larger object. … cm [3]

3 marks

Mark scheme: 22 45 3 875 M2 for 3  18 oe 56 875 56 or M1 for 3 or 3 oe or 56 875 183 56 = oe h 3 875

This question in 0580/22 Oct/Nov 2022

10 3.0 cm A B 2.0 cm 2.7 cm X NOT TO SCALE C D 7.5 cm In the diagram, AB and CD are parallel. The lines CB and AD intersect at X. AB = 3. 0 cm, AX = 2. 0 cm, BX = 2. 7 cm and CD = 7. 5 cm . Find the length of BC. BC = … cm [3]

3 marks

Mark scheme: 10 9.45 3 2.7  7.5 M2 for + 2.7 oe 3 OR B2 for 6.75 oe 3 2.7 or M1 for = oe 7.5 XC If 0 scored SC1 for answer 7.7

This question in 0580/22 Feb/March 2023

9 D A NOT TO h cm 5.6 cm SCALE B C E F 7.2 cm 8.1 cm Triangle ABC is similar to triangle DEF. Calculate the value of h. h = … [2]

2 marks

Mark scheme: 9 6.3 2 5.6 7.2 M1 for  oe or better h 8.1

This question in 0580/21 May/June 2023

12 (a) A NOT TO SCALE O C B AO, OB and OC are all radii of the circle. AB = BC. Therefore triangle AOB is congruent to triangle COB. Draw a ring around the correct criterion for this statement. SAS RHS SSS ASA [1] (b) S P O NOT TO SCALE R 42°42° T Q U P, Q, R and S are points on the circle and TQU is a tangent to the circle at Q. PR and SQ intersect at the centre of the circle, O, and PQ is parallel to SR. Angle RQU = 42°. Calculate (i) angle QSR Angle QSR = … [1] (ii) angle PQS Angle PQS = … [1] (iii) angle POS. Angle POS = … [1]

4 marks

Mark scheme: 12(a) SSS 1 12(b)(i) 42 1 12(b)(ii) 42 1 FT their part (i) 12(b)(iii) 84 1 FT 2  their part (ii)

This question in 0580/21 May/June 2023

11 NOT TO 6.3 cm SCALE h cm 5 cm 9 cm The two shapes are mathematically similar. (a) Find the value of h. h = … [2] (b) The area of the smaller shape is 16 cm 2. Calculate the area of the larger shape. … cm2 [2]

4 marks

Mark scheme: 11(a) 3.5 2 9 6.3 M1 for = oe 5 h 11(b) 51.84 2 2 2  9   5  M1 for   or   oe  5   9   6.3  2  their ( a )  2 or   or   oe   6.3   their ( a )  

This question in 0580/22 Oct/Nov 2023

20 Two parcels are mathematically similar. The larger parcel has volume 80 cm 3 and height 5.2 cm. The smaller parcel has volume 33.75 cm 3. Calculate the height of the smaller parcel. … cm [3]

3 marks

Mark scheme: 20 3.9 3 33.75 M2 for 5.2  3 oe 80 3 33.75 3 80 or M1 for oe or oe 3 80 3 33.75 h 3 33.75 or  oe 5.2 3 80

This question in 0580/22 May/June 2024

23 NOT TO SCALE x cm 3.5 cm Small box Large box The small box is mathematically similar to the large box. The volume of the large box is 72.8% greater than the volume of the small box. The small box has length 3.5 cm and the large box has length x cm. Calculate the value of x. x = … [3]

3 marks

Mark scheme: 23 4.2 oe 3 72.8 M2 for 3 1 +  3.5 oe 100 3 172.8 or M1 for oe 3 100 3 100 or oe 3 172.8 x 3 172.8 or = oe 3.5 3 100

This question in 0580/22 Oct/Nov 2024

14 C 55° NOT TO B SCALE X 88° A D E A, B, C, D and E lie on the circle. AC and BD intersect at X. Angle ACD = 55° and angle CXD = 88°. (a) Complete the statements, giving a geometrical reason in each part. Angle CDB = … because … … Angle ABD = … because … … Angle AED = … because … … [6] (b) Triangle CXD is mathematically similar to triangle BXA. DX = 8.0 cm, BX = 2.7 cm and AX = 4.0 cm. (i) Work out the length of CX. CX = … cm [2] (ii) Complete the statement. Area of triangle CXD : area of triangle BXA = … : … [1]

9 marks

Mark scheme: 14(a) 37 2 B1 for each Angle sum of triangle = 180 55 2 B1 for each Angles in the same segment are equal 125 2 B1 for each Opposite angles of cyclic quadrilateral sum to 180 14(b)(i) 5.4 2 4 2.7 M1 for = oe 8 CX 14(b)(ii) 4 : 1 oe 1

This question in 0580/21 Oct/Nov 2025

11 P NOT TO S T SCALE Q R In the diagram, S lies on PQ and T lies on PR. ST is parallel to QR. (a) Explain why triangle PQR is mathematically similar to triangle PST. Give a geometrical reason for each statement you make. … … … … [3] (b) ST = 3 cm, QR = 9 cm and PS = 5 cm. Work out PQ. PQ = … cm [2] (c) The area of triangle PST is 2k cm2. Find, in terms of k, the area of quadrilateral QRTS. … cm2 [2]

7 marks

Mark scheme: 11(a) Three correct pairs with correct reasons 3 B2 for two correct pairs with correct reasons PQR = PST and corresponding angles or B1 for one correct pair with correct PRQ = PTS and corresponding angles reason QPR = TPS and common angle OR Two correct pairs with correct reasons and correct similarity condition e.g. AAA 11(b) 15 2 3 5 M1 for = oe or better 9 PQ 11(c) 16k 2 2 1 M1 for 32 or  oe 3

This question in 0580/22 Oct/Nov 2025