Pythagoras’ theorem — Paper 1 · IGCSE Mathematics 0580

8 Town E is 13 kilometres due east of D. Town F is due south of E, and DF = 16 kilometres. Calculate the distance from E to F. 13 km D E North NOT TO SCALE 16 km F Answer km [2]

2 marks

Mark scheme: 8 9.33 or 9.327(…) 2 M1 for 162 − 132 as chosen method. Alt. Trig must be complete correct method for M1.

This question in 0580/11 May/June 2009

21 Examiner's B Use C 16 cm 9 cm NOT TO SCALE 24° A D The diagram shows a quadrilateral ABCD. AB = 16 cm, BD = 9 cm and angle BDC = 24°. Angle ADB = 90° = angle BCD. Calculate the length of (a) AD, Answer(a) AD = cm [3] (b) CD. Answer(b) CD = cm [2]

5 marks

Mark scheme: 21 (a) 13.2 3 M2 for √(162 – 92) or √175 or 13.22 to 13.23 or M1 for 162 = x2 + 92 or better CD (b) 8.22 to 8.23 2 M1 for cos 24 = or better 9

This question in 0580/13 May/June 2010

18 For Examiner's D C Use NOT TO SCALE 18 cm A B The diagram shows a square ABCD. The length of the diagonal AC is 18 cm. (a) Calculate (i) the length of the side of the square, Answer(a)(i) cm [2] (ii) the area of the square. Answer(a)(ii) cm2 [2] (b) A, B, C and D lie on a circle with diameter AC. Calculate the area of this circle. Answer(b) cm2 [2]

6 marks

Mark scheme: x x 18 (a) (i) 12.7 to 12.73 2 M1 for = sin45 or = cos45 or better 18 18 (ii) 161 to 162.1 2ft M1 for method for squaring their (a)(i). (b) 254 to 255 2 M1 for π × 92

This question in 0580/12 Oct/Nov 2010

10 C NOT TO 9 cm SCALE B A 17 cm In the triangle ABC, AB = 17 cm, BC = 9 cm and angle ACB = 90°. Calculate AC. Answer AC = cm [3]

3 marks

Mark scheme: 10 14.4(……) 3 M2 for (17 2 − 9 2 ) or M1 for 172 = x2 + 92 or better seen

This question in 0580/12 May/June 2011

11 D C 12 cm NOT TO SCALE A 9 cm B In the rectangle ABCD, AB = 9 cm and BD = 12 cm. Calculate the length of the side BC. Answer BC = cm [3]

3 marks

Mark scheme: 11 7.94 or 7.937(…..) www 3 M2 for (12 2 − 9 2 ) or M1 for 122 = x2 + 92 oe or better IGCSE – May/June 2011 0580 13 7

This question in 0580/13 May/June 2011

17 For Examiner's Use NOT TO SCALE 8 m y° 3 m The diagram shows a ladder, of length 8 m, leaning against a vertical wall. The bottom of the ladder stands on horizontal ground, 3 m from the wall. (a) Find the height of the top of the ladder above the ground. Answer(a) m [3] (b) Use trigonometry to calculate the value of y. Answer(b) y = [2]

5 marks

Mark scheme: 17 (a) 7.42 or 7.416... cao 3 M2 for (8 2 − 3 2 ) or complete alternate method or M1 for x2 + 32 = 82 or better 3 (b) 67.97 to 68(.0) cao 2 M1 for cos (y) = oe 8 500 × 5 × 3

This question in 0580/11 Oct/Nov 2011

24 For C Examiner's Use NOT TO SCALE 13 cm 12 cm 7 cm A B D In triangle ABC, D is on AB so that angle ADC = angle BDC = 90°. AC = 13 cm, BC = 12 cm and CD = 7 cm. (a) Calculate the length of DB. Answer(a) DB = cm [3] (b) Use trigonometry to calculate angle CAD. Answer(b) Angle CAD = [2]

5 marks

Mark scheme: 24 (a) (DB =) 9.75 or 9.746 to 9.747 3 M2 for (12 2 − 7 2 ) or M1 for 122 = 72+ x2 or better 7 (b) (Angle CAD =) 32.6 or 32.57 to 32.58 2 M1 for sin 13

This question in 0580/13 Oct/Nov 2011

19 For B Examiner's Use NOT TO SCALE 7.4 cm 9.3 cm 6.5 cm A C D (a) Calculate AD. Answer(a) AD = cm [3] (b) Use trigonometry to calculate angle BCD. Answer(b) Angle BCD = [2] Question 20 is printed on the next page.

5 marks

Mark scheme: 19 (a) 3.54 3 M2 for √(7.42 – 6.52) or M1 for 7.42 = AD2 + 6.52 or better 6 . 5 (b) 44.3 2 M1 for sin [BCD] = or better 9 . 3

This question in 0580/11 Oct/Nov 2012

19 For D Examiner's Use NOT TO SCALE 7.8 m 5.6 m A B C 2.9 m (a) Calculate BD. Answer(a) BD = m [3] (b) DC = 7.8 m . Use trigonometry to calculate angle BCD. Answer(b) Angle BCD = [2]

5 marks

Mark scheme: 19 (a) 4.79[1] or 4.79[06…] 3 M2 for √(5.6 2 − 2.9 2) or better, or 2 2 2 M1 for 2.9 + BD = 5.6 or better. (b) 37.879 or 37.9[0] 2 ft M1 for sin [BCD =] their (a) / 7.8 or better

This question in 0580/12 Oct/Nov 2012

10 B NOT TO SCALE 4.6 cm A C 7.2 cm Calculate AB. Answer … cm [2] _____________________________________________________________________________________

2 marks

Mark scheme: 10 8.54[4 … ] 2 M1 for 7.22 + 4.62 or better

This question in 0580/12 Oct/Nov 2013

17 Q NOT TO 17 cm SCALE 9 cm P R O The diagram shows a circle, centre O. P, Q and R are points on the circumference. PQ = 17 cm and QR = 9 cm. (a) Explain why angle PQR is 90°. Answer(a) … … [1] (b) Calculate the length PR. Answer(b) PR = … cm [2] __________________________________________________________________________________________

3 marks

Mark scheme: 17 (a) Angle [in a] semi-circle 1 (b) 19.2 or 19.23 to 19.24 2 M1 for 172 + 92 16 21 8 25

This question in 0580/11 May/June 2014

22 16 km P North NOT TO SCALE 9 km Q The diagram shows the route of a ship that leaves a port, P. It travels due west for 16 km and then changes course to due south for 9 km. (a) Calculate the straight line distance PQ. Answer(a) PQ = … km [2] (b) Use trigonometry to calculate the bearing of P from Q. Answer(b) … [2]

4 marks

Mark scheme: 22 (a) 18.4 2 M1 for [PQ2 =] 162 + 9 2 or better 16 (b) [0]60.4 to [0]60.73 2 M1 for tan [... = ] or better 9 16 or sin [... = ] or better their (a) 9 or cos[... = ] or better their (a) If zero scored, SC1 for answer [0]29.3 to [0]29.4

This question in 0580/11 Oct/Nov 2014

21 C 5 cm B NOT TO SCALE 8 cm 32° A A, B and C lie on a circle with diameter AB. Angle CAB = 32°, AC = 8 cm and BC = 5 cm. (a) Work out the size of angle CBA. Answer(a) Angle CBA = … [2] (b) Work out the length of AB. Answer(b) AB = … cm [2] __________________________________________________________________________________________

4 marks

Mark scheme: 21 (a) 58 2 B1 for ACB = 90o soi as angle at C or 8 M1 for tan 5 2 + 5 2 (b) 9.43 to 9.44 2 M1 for [AB 2 = ] 8 5 8 or sin 32 = or cos 32 = oe AB AB

This question in 0580/13 Oct/Nov 2014

16 P 11 cm NOT TO SCALE 37° A O In the diagram, AP is a tangent to the circle at P. O is the centre of the circle, angle PAO = 37° and AP = 11 cm. (a) Write down the size of angle OPA. Answer(a) Angle OPA = … [1] (b) Work out the radius of the circle. Answer(b) … cm [2] __________________________________________________________________________________________

3 marks

Mark scheme: 16 (a) 90 1 OP (b) 8.29 or 8.289 to 8.29 2 M1 for = tan 37° oe 11

This question in 0580/11 Oct/Nov 2015

19 NOT TO 8 cm SCALE 5 cm x cm Calculate the value of x. Answer x = … [3]

3 marks

Mark scheme: 19 6.24 or 6.244 to 6.245 3 M2 for 8 2 − 5 2 or M1 for 82 = 52 + x2 or better 3 15 27 9× 3

This question in 0580/12 Oct/Nov 2015

13 NOT TO SCALE x cm 18 cm 26 cm Calculate the value of x. Answer x = … [2] __________________________________________________________________________________________

2 marks

Mark scheme: 13 31.6 [2 … ] 2 M1 for 18 2 + 26 2

This question in 0580/13 Oct/Nov 2015

19 A D 93.5 m NOT TO SCALE B C 82.5 m The diagram shows a rectangular field, ABCD, with a straight path, BD. (a) Calculate the distance from C to D. … m [3] (b) Yan walks along the edge of the field from B to C and then from C to D. Lee walks along the straight path BD. Work out how much further Yan walks than Lee. … m [1]

4 marks

Mark scheme: 19 (a) 44 3 M2 for 93.52 − 82.5 2 or M1 for CD2 + 82.52 = 93.52 (b) 33 1FT FT 93.5 – (82.5 + their (a))

This question in 0580/13 May/June 2016

20 (a) A NOT TO 6.3 m 6.3 m SCALE B C Calculate the length BC. BC = … m [2] (b) D NOT TO 223 cm SCALE 52 cm F E Calculate angle DFE. Angle DFE = … [2]

4 marks

Mark scheme: 20(a) 8.91 2 M1 for [ BC2 =] 6.32 + 6.32 or 6.3 ÷ sin 45 or 6.3 ÷ cos 45 20(b) 13.5 or 13.48… 2 52 M1 for sin [=] 223

This question in 0580/13 May/June 2017

16 A NOT TO SCALE 18 cm 8 cm B C Calculate the length of BC. BC = … cm [3]

3 marks

Mark scheme: 16 16.1 or 16.12 to 16.13 3 M2 for √(182 – 82) or better or M1 for 182 = […]2 + 82 or better

This question in 0580/11 Oct/Nov 2017

18 D C NOT TO 35.1 m SCALE A B 23.4 m The diagram shows a rectangular playground ABCD. AB = 23.4 m and AC = 35.1 m. Calculate BC. BC = … m [3]

3 marks

Mark scheme: 18 26.2 or 26.16( … ) 3 2 2 M2 for 35.1 − 23.4 or better or M1 for 35.12 = 23.42 + BC2 or better

This question in 0580/12 May/June 2018

26 C y cm D 8.2 cm NOT TO 5.3 cm SCALE 4.4 cm x° A B Triangles ABC and BCD are both right-angled triangles. (a) Calculate the value of y. y = … [3] (b) Calculate the value of x. x = … [2]

5 marks

Mark scheme: 26(a) 2.95 or 2.954 to 2.955 3 2 2 M2 for 5.3 − 4.4 or better or M1 for 5.32 = 4.4 2 + y 2 or better 26(b) 40.3 or 40.26 to 40.27… 2 M1 for sin [ x = ]5.3 8.2

This question in 0580/12 Feb/March 2019

24 D NOT TO SCALE 15 m 38° A B 10.7 m C A vertical flagpole, BD, stands on horizontal ground and is held by two ropes, AD and CD. AD = 15m , BC = 10.7m and angle DAB = 38 ° . (a) Using trigonometry, calculate BD. BD = … m [2] (b) Calculate CD. CD = … m [2]

4 marks

Mark scheme: 24(a) 9.23 or 9.234 to 9.235 2 BD M1 for sin38 = or better 15 24(b) 14.1 or 14.13 … 2 FT their (a) M1 ( their (a))2 + 10.72

This question in 0580/13 May/June 2019

19 NOT TO SCALE 8.5 cm 3.9 cm x cm The diagram shows a right-angled triangle. Calculate the value of x. x = … [3]

3 marks

Mark scheme: 19 7.55 or 7.552… 3 M2 for 8.52 − 3.92 oe or M1 for 8.52 = 3.92 + x2 oe

This question in 0580/12 Oct/Nov 2019

14 The diagram shows a right-angled triangle. NOT TO SCALE x cm 28 cm 31 cm Show that the value of x is 41.8, correct to 3 significant figures. [2]

2 marks

Mark scheme: 14 2 2 M1 31 + 28 oe 41.77… A1

This question in 0580/13 Oct/Nov 2019

21 (a) NOT TO SCALE 10 cm x° 18 cm Calculate the value of x. x = … [2] (b) NOT TO k cm SCALE 8 cm 15 cm Calculate the value of k. k = … [2]

4 marks

Mark scheme: 21(a) 29.1 or 29.05… 2 10 M1 for tan [x =] 18 21(b) 17 2 M1 for 82 + 152

This question in 0580/11 Oct/Nov 2020

19 The length of one side of a rectangle is 12 cm. The length of the diagonal of the rectangle is 13 cm. Calculate the area of the rectangle. … cm2 [3]

3 marks

Mark scheme: 19 60 3 2 2 M2 for 12 × 13 − 12 or M1 for 132 − 122 or for 12 × their 5 from Pythagoras or trig

This question in 0580/12 Oct/Nov 2020

22 P Q NOT TO SCALE 7.2 cm 11.5 cm R Calculate PQ. PQ = … cm [3]

3 marks

Mark scheme: 22 8.97 or 8.967… 3 M2 for 11.52 – 7.22 or better or M1 for PQ2 + 7.22 = 11.52 or better

This question in 0580/13 Oct/Nov 2020

24 (a) 12.9 cm NOT TO 8.6 cm SCALE x cm Calculate the value of x. x = … [2] (b) 50° NOT TO SCALE y cm 3.8 cm Show that the value of y is 5.9, correct to 2 significant figures. [3]

5 marks

Mark scheme: 24(a) 15.5 or 15.50... 2 M1 for 12.92 + 8.62 or better 24(b) 3.8 M2 3.8 y = M1 for cos 50 [=] or better cos50 y 5.91... A1

This question in 0580/13 Oct/Nov 2021

14 NOT TO SCALE 5 cm 5 cm h 4 cm (a) Calculate the height, h, of the triangle. h = … cm [3] (b) The triangle is one face of a square-based pyramid. On the 1 cm2 grid, draw a net of this pyramid. [3]

6 marks

Mark scheme: 14(a) 4.58 or 4.582 to 4.583 3 M2 for 52 − 22 or better or M1 for 52 = 22 + h2 or better 14(b) Correct ruled net of the pyramid 3 B2 for a 4 by 4 square and 4 isosceles triangles on the sides of the square or B1 for a 4 by 4 square or at least one isosceles triangle drawn on a side of a square

This question in 0580/13 May/June 2022

18 NOT TO 14 cm SCALE h cm 10 cm The diagram shows a right-angled triangle. (a) Calculate the value of h. h = … [3] (b) Find the perimeter of this triangle. … cm [1]

4 marks

Mark scheme: 18(a) 9.8[0] or 9.797 to 9.798 3 M2 for 142 –102 oe or better or M1 for 102 + h2 = 142 oe or better 18(b) 33.8 or 33.79 to 33.80 1 FT 24 + their (a)

This question in 0580/13 Oct/Nov 2022

25 A x cm NOT TO D SCALE 39 cm 25 cm B C 52 cm The diagram shows two right-angled triangles, ABC and ACD. Work out the value of x. x = … [4]

4 marks

Mark scheme: 25 60 cao 4 M3 for 39 2 + 52 2 − 252  = x 2  oe   or M2 for 39 2 + 52 2 = 252 + x 2 oe or M1 for 39 2 + 52 2 or 252 + x 2 [= their AC2]

This question in 0580/12 Feb/March 2023

14 C NOT TO SCALE 4 cm D 6 cm A 6 cm B The diagram shows a triangular prism. ABC is an isosceles triangle with AC = BC. The perpendicular height of triangle ABC is 4 cm. AB = 6 cm and BD = 6 cm. (a) Complete this statement. The prism has … faces and … edges. [2] (b) Show that the length of BC is 5 cm. [2] (c) Complete the net of the prism on the 1 cm 2grid. The base has been drawn for you. [3]

7 marks

Mark scheme: 14(a) 5 9 2 B1 for each 14(b) 2 2 2 2 2 3 + 4 [=5] M1 for 3 + 4 14(c) correct net drawn 3 B2 for 3 correct extra faces drawn in correct places or B1 for 1 correct extra face drawn in correct place

This question in 0580/12 Feb/March 2025