Coordinates — Paper 2 · IGCSE Mathematics 0580

7 Find the co-ordinates of the mid-point of the line joining the points A(2, –5) and B(6, 9). Answer ( , ) [2]

2 marks

Mark scheme: 7 (4, 2) 2 2 + 6 − 5 + 9 M1 and oe 2 2 or a drawing used correctly IGCSE – May/June 2009 0580, 0581 21

This question in 0580/21 May/June 2009

8 Find the length of the line joining the points A(− 4, 8) and B(−1, 4). For Examiner's Use Answer AB = [2]

2 marks

Mark scheme: 8 5 www 2 M1 (–4 – –1)2 + (8 – 4)2 or better

This question in 0580/21 Oct/Nov 2009

15 Examiner's Use y NOT TO SCALE 4 A y = 4 B x 0 2x + y = 8 3x + y = 18 (a) The line y = 4 meets the line 2x + y = 8 at the point A. Find the co-ordinates of A. Answer(a) A ( , ) [1] (b) The line 3x + y = 18 meets the x axis at the point B. Find the co-ordinates of B. Answer(b) B ( , ) [1] (c) (i) Find the co-ordinates of the mid-point M of the line joining A to B. Answer(c)(i) M ( , ) [1] (ii) Find the equation of the line through M parallel to 3x + y = 18. Answer(c)(ii) [2]

5 marks

Mark scheme: 15 (a) (2, 4) 1 (b) (6, 0) 1 (c) (i) (4, 2) ft 1ft From (a) and (b) (ii) y = – 3x + 14 oe 2 M1 sub their (c)(i) into y = –3x + c oe 1

This question in 0580/22 May/June 2010

21 For y Examiner's A Use 8 7 6 5 4 3 2 1 0 x 1 2 3 4 5 6 7 –1 –2 –3 –4 B (a) Using a straight edge and compasses only, construct the perpendicular bisector of AB on the diagram above. [2] (b) Write down the co-ordinates of the midpoint of the line segment joining A(1, 8) to B(7, –4). Answer(b) ( , ) [1] (c) Find the equation of the line AB. Answer(c) [3] Question 22 is printed on the next page.

6 marks

Mark scheme: 21 (a) Bisector 2 B1 accurate line B1 two sets of correct arcs (b) (4, 2) 1 (c) y = –2x + 10 oe 3 B1 correct m B1 correct c M1 correct use of y = mx + c oe on answer line

This question in 0580/21 Oct/Nov 2010

13 For North Examiner's T Use B NOT TO SCALE 76° A C O P AOC is a diameter of the circle, centre O. AT is a straight line that cuts the circle at B. PT is the tangent to the circle at C. Angle COB = 76°. (a) Calculate angle ATC. Answer(a) Angle ATC = [2] (b) T is due north of C. Calculate the bearing of B from C. Answer(b) [2]

4 marks

Mark scheme: 13 (a) 52 2 M1 OAB or OBA = 38 or OCT = 90 (b) 322 2 M1 BCT = 38 or BCO = 52 IGCSE – October/November 2011 0580 22

This question in 0580/22 Oct/Nov 2011

15 For y Examiner's Use 6 5 B 4 3 2 A 1 x 0 1 2 3 4 5 6 The points A(1, 2) and B(5, 5) are shown on the diagram . (a) Work out the co-ordinates of the midpoint of AB. Answer(a) ( , ) [1] (b) Write down the column vector .     Answer(b) =   [1]     (c) Using a straight edge and compasses only, draw the locus of points which are equidistant from A and from B. [2]

4 marks

Mark scheme: 15 (a) (3, 3½) 1  4  (b)   1  3  (c) Correct perpendicular bisector with 2 B1 line through (3, 3½) perp to AB arcs B1 two sets of correct arcs

This question in 0580/22 Oct/Nov 2011

18 A (5, 23) and B (–2, 2) are two points. For Examiner′s Use (a) Find the co-ordinates of the midpoint of the line AB. Answer(a) ( … , … ) [2] (b) Find the equation of the line AB. Answer(b) … [3] (c) Show that the point (3, 17) lies on the line AB. Answer(c) [1] _____________________________________________________________________________________

6 marks

Mark scheme: 18 (a) (1.5, 12.5) oe 2 B1 for either coordinate (b) y = 3x + 8 oe 3 B2 for y = mx + 8 or y = 3x + c or 3x + 8 or B1 for gradient (or m) = 3 and B1 for c = 8 If 0 scored, SC1 for 23 = their m × 5 + c or for 2 = their m × –2 + c or for 12.5 = their m × 1.5 + c (c) Most common methods: 1 Correctly substituting P (3, 17) into y = 3x + 8 Showing the gradient of AP or BP = 3 Other methods possible. IGCSE – October/November 2013 0580 22

This question in 0580/22 Oct/Nov 2013

20 y 7 A 6 5 4 3 2 B 1 x 0 1 2 3 4 5 6 7 8 Point A has co-ordinates (3, 6). (a) Write down the co-ordinates of point B. ( … , … ) [1] (b) Find the gradient of the line AB. … [2] (c) Find the equation of the line that • is perpendicular to the line AB and • passes through the point (0, 2). … [3]

6 marks

Mark scheme: 20 (a) ( 7 , 1 ) 1 5 1 (b) ‒1.25 or − or −14 2 M1 for rise/run 4 4 4 −1 (c) y = x + 2 oe 3 B2 for x + 2 or y = x + 2 oe 5 5 their(b) 1 or M1 for −their ( b ) oe 4 or B1 for x seen or [ y = ] mx + 2 (m ≠ 0) 5

This question in 0580/22 Oct/Nov 2016

14 y 10 9 A 8 7 6 B 5 4 3 2 1 x 0 1 2 3 4 5 6 7 8 9 10 Points A and B are marked on the grid. - 4 BC = c 0m (a) On the grid, plot the point C. [1] (b) Write AC as a column vector. [1] f p (c) DE is a vector that is perpendicular to BC. The magnitude of DE is equal to the magnitude of BC. Write down a possible column vector for DE. [2] f p

4 marks

Mark scheme: 14 (a) Point at (3, 5) 1  1  JJJG (b)   1FT FT their AC  −3  0  0  (c)  or   2 M1 for a vector of magnitude 4 or of form 4  −4   0     ± k  20

This question in 0580/22 Feb/March 2017

6 y 5 4 A 3 2 1 x –5 –4 –3 –2 –1 0 1 2 3 4 5 –1 B –2 C –3 –4 –5 The diagram shows two sides of a rhombus ABCD. (a) Write down the co-ordinates of A. ( … , … ) [1] (b) Complete the rhombus ABCD on the grid. [1]

2 marks

Mark scheme: 6(a) (−2, 3) 1 6(b) Correct rhombus with 4th point at 1 (2,2)

This question in 0580/21 Oct/Nov 2017

23 A is the point (2, 3) and B is the point (7, -5). (a) Find the co-ordinates of the midpoint of AB. ( … , … ) [2] (b) Find the equation of the line through A that is perpendicular to AB. Give your answer in the form y = mx + c . y = … [4]

6 marks

Mark scheme: 23(a) (4.5, − 1) 2 B1 for each 23(b) 5 7 4 −−5 3 [ y = ]8 x + 4 M1 for 7 − 2 oe 8 M1 for –1/ their − 5 M1 for 3 = 2 × their gradient + c oe

This question in 0580/22 Feb/March 2019

15 A is the point (7, 12) and B is the point (2, - 1) . Find the length of AB. … [3]

3 marks

Mark scheme: 15 13.9 or 13.92 to 13.93 3 2 2 M2 for ( 7 − 2 ) + (12 −−1) oe or M1 for ( 7 − 2 ) 2 + (12 −−1) 2 oe

This question in 0580/23 May/June 2019

12 A is the point (2, 1) and B is the point (9, 4). Find the length of AB. … [3]

3 marks

Mark scheme: 12 7.62 or 7.615 to 7.616 3 2 2 M2 for ( 9 − 2 ) + ( 4 − 1) oe or M1 for ( 9 − 2 ) 2 + ( 4 − 1) 2 oe or 58

This question in 0580/23 Oct/Nov 2019

9 A is the point (5, - 5 ) and B is the point (9, 3). (a) Find the coordinates of the midpoint of AB. ( … , … ) [2] (b) Find the length of AB. … [3]

5 marks

Mark scheme: 9(a) (7, − 1) 2 B1 for each 9(b) 8.94 or 8.944… 3 2 2 M2 for ( 9 − 5 ) + ( 3 −−5 ) oe 2 2 or M1 for ( 9 − 5 ) + ( 3 −−5 ) oe

This question in 0580/21 May/June 2021

15 C is the point ( 5, - 1) and D is the point (13, 15). (a) Find the midpoint of CD. ( … , … ) [2] (b) Find the gradient of CD. … [2] (c) Find the equation of the perpendicular bisector of CD. Give your answer in the form y = mx + c . y = … [3]

7 marks

Mark scheme: 15(a) (9, 7) 2 B1 for each 15(b) 2 2 15 – –1 M1 for oe 13 – 5 15(c) 1 23 3 [y =] – x + oe 2 2 final answer 1 M1 for gradient = their (b) oe M1 for correct substitution of their (a) into y = (their m)x + c oe

This question in 0580/22 May/June 2023

1 y 8 7 6 A 5 4 3 C 2 1 B x – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 – 1 – 2 The diagram shows two sides of a parallelogram ABCD. Find the coordinates of point D. ( … , … ) [2]

2 marks

Mark scheme: Question Answer Marks Partial Marks 1 (–3, 7) 2 B1 for correct diagram or correct coordinates for their point D or for   3, k  or  k , 7 

This question in 0580/21 May/June 2024

24 (a) A is the point (a, 12) and B is the point (b, 27). (i) Find the y-coordinate of the midpoint of AB. … [1] (ii) The line AB has gradient 3. Find an expression for a in terms of b. a = … [3] (b) D is the point (22, 34) and E is the point (23, 39). D is the point on CE such that 2CE = 5DE. Find the coordinates of C. ( … , … ) [3]

7 marks

Mark scheme: 24(a)(i) 19.5 1 24(a)(ii) [a =] b – 5 oe 3 27 − 1 2 M1 for 3 = oe b − a M1 for 3(b – a) = 27 – 12 or better 27 − 1 2 e.g. [a =] b – 3 24(b) 41 53 3 SC2 for answer (25.5, 51.5) or (20.5, 26.5) or ( , ) 2 2 51 103 ( , ) 2 2 OR 5 M2 for 23 − ( 23 − 22 ) 2 5 or 39 − ( 39 − 34 ) oe 2 or sketch 12.5 2.5  2.5   −2.5  or vector e.g.   or    12.5   −12.5  or M1 for 5 1 5 5 ( 23 − 22 ) or ( 39 − 34 ) 2 2 1  −1  or vector e.g.  or   5  −5 

This question in 0580/23 May/June 2025

17 B is the point (-3, 1) and D is the point (-5, 9). BD is a diagonal of the kite ABCD. (a) The ratio of the lengths of the diagonals BD : AC = 2 : 3. Work out the length of AC. Give your answer as a surd in its simplest form. … [5] (b) Find the coordinates of the midpoint of BD. ( … , … ) [2] (c) The diagonal AC of the kite passes through the midpoint of BD. Find an equation of AC. Give your answer in the form y = mx + c . y = … [4]

11 marks

Mark scheme: 17(a) 3 17 cao 5 B4 for answer equivalent to 3 17 but 3 68 not in the correct form e.g. , 2 6 17 , 153 2 OR B3 for 68 or 2 17 or M2 for (9 – 1)2 + (–5 – – 3)2 oe or M1 for (9 – 1) or (–5 – – 3) oe and 3 M1 for their 68  oe 2 17(b) (– 4, 5) 2 B1 for each coordinate 17(c) 1 4 − 1 y = x + 6 final answer M1 for  grad BD = 9 oe 4 −−−5 3 −1 M1 for  grad AC =  their grad BD M1 for their (– 4, 5) substituted into y = their mx + c oe

This question in 0580/22 Oct/Nov 2025