Coordinate geometryLength and midpoint
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10 Find the mid-point of AB where A = (w, r) and B = (3w, t). Give your answer in its simplest form in terms of w, r and t. (..................... , .....................) [2]
IGCSE Mathematics 0580 · Topic practice
Practise gradients, midpoints, lengths and equations of straight lines, including parallel and perpendicular lines. Sketch the points first so you can sanity-check the sign and size of your answer.
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You’re practising 2016–2018 papers. The 2019–2026 papers, including 2026, need a plan.
Unlock from $6/mo10 Find the mid-point of AB where A = (w, r) and B = (3w, t). Give your answer in its simplest form in terms of w, r and t. (..................... , .....................) [2]
17 NOT TO SCALE O C D y x The diagram shows the points C(–1, 2) and D(9, 7). Find the equation of the line perpendicular to CD that passes through the point (1, 3). Give your answer in the form y mx c = + . y = ............................................... [4]
24 (a) Point A has co-ordinates (1, 0) and point B has co-ordinates (2, 5). Calculate the angle between the line AB and the x-axis. .............................................. [3] (b) The line PQ has equation y = 3x - 8 and point P has co-ordinates (6, 10). Find the equation of the line that passes through P and is perpendicular to PQ. Give your answer in the form y = mx + c. y = .............................................. [3]
25 P is the point (16, 9) and Q is the point (22, 24). (a) Find the equation of the line perpendicular to PQ that passes through the point (5, 1). Give your answer in the form y mx c = + . y = ................................................ [4] (b) N is the point on PQ such that PN = 2NQ. Find the co-ordinates of N. ( .................... , .................... ) [2] 16
1 (a) 0 y x –5 –4 –7 –6 –3 –2 –1 1 2 3 4 5 6 7 –6 –5 –4 –7 –3 –2 –1 1 2 3 4 5 6 7 T P (i) Describe fully the single transformation that maps triangle T onto triangle P. ..................................................................................................................................................... ..................................................................................................................................................... [2] (ii) Translate triangle T by the vector 2 5 - - e o. [2] (iii) Rotate triangle T through 90° anticlockwise about (0, 0). [2] (iv) Enlarge triangle T by scale factor 2 1 - with centre (0, 0). [2] 3 (b) O x y A (3, 2) B (5, 6) NOT TO SCALE (i) Find the column vector AB. AB = f p [1] (ii) Find AB . AB = ................................................. [2] (iii) B is the mid-point of the line AC. Find the co-ordinates of C. ( ........................ , ....................... ) [2] (iv) Find the equation of the straight line that passes through A and B. .................................................. [3] (v) The straight line that passes through A and B cuts the y-axis at D. Write down the co-ordinates of D. ( ........................ , ....................... ) [1]
2 (a) (i) y 2x = Complete the table. x 0 1 2 3 4 y 2 4 8 [2] (ii) y x 14 2 = - Complete the table. x 0 1 2 3 4 y 13 10 5 [2] (b) On the grid, draw the graphs of y 2x = and y x 14 2 = - for x 0 4 G G . 0 2 −2 4 6 8 10 12 14 16 y x 1 2 3 4 [6] 5 (c) Use your graphs to solve the equations. (i) 2 12 x = x = ............................................... [1] (ii) x 2 14 x 2 = - x = ............................................... [1] (d) (i) On the grid, draw the line from the point (4, 2) that has a gradient of 4 - . [1] (ii) Complete the statement. This straight line is a .................................. to the graph of y x 14 2 = - at the point ( .......... , .......... ). [2]
6 Klaus buys x silver balloons and y gold balloons for a party. He buys • more gold balloons than silver balloons • at least 15 silver balloons • less than 50 gold balloons • a total of no more than 70 balloons. (a) Write down four inequalities, in terms of x and/or y, to show this information. ................................................. ................................................. ................................................. ................................................. [4] 9 (b) On the grid, show the information from part (a) by drawing four straight lines and shading the unwanted regions. 0 10 20 30 40 50 60 70 y x 10 20 30 40 50 60 70 [5] (c) Silver balloons cost $2 and gold balloons cost $3. Calculate the most that Klaus could spend. $ ............................................... [2]
7 The graph of y x 10 8 2 = - for . . x 1 5 1 5 G G - is drawn on the grid. y x – 2 2 4 6 8 10 12 – 4 – 6 – 8 – 1 – 1.5 – 0.5 0.5 1.5 1 0 11 (a) Write down the equation of the line of symmetry of the graph. ................................................ [1] (b) On the grid opposite, draw the tangent to the curve at the point where . x 0 5 = . Find the gradient of this tangent. ................................................ [3] (c) The table shows some values for y x x 3 4 3 = + + . x .1 5 - 1 - .0 5 - 0 0.5 1 1.5 y .3 9 - 5.6 8 11.9 (i) Complete the table. [3] (ii) On the grid opposite, draw the graph of y x x 3 4 3 = + + for . . x 1 5 1 5 G G - . [4] (d) Show that the values of x where the two curves intersect are the solutions to the equation x x x 8 3 6 0 3 2 + + - = . [1] (e) By drawing a suitable straight line, solve the equation x x 5 2 0 3 + + = for . . x 1 5 1 5 G G - . x = ............................................... [3]
8 –3 –2 –1 0 1 2 3 4 5 6 7 8 y x –5 –4 –3 –2 –1 1 2 3 4 5 6 A l B (a) Write down the co-ordinates of A. ( ..................... , ......................) [1] (b) Find the equation of line l in the form y mx c = + . y = ............................................... [3] (c) Write down the equation of the line parallel to line l that passes through the point B. ................................................ [2] (d) C is the point (8, 14). (i) Write down the equation of the line perpendicular to line l that passes through the point C. ................................................ [3] (ii) Calculate the length of AC. ................................................ [3] (iii) Find the co-ordinates of the mid-point of BC. ( ..................... , ......................) [2]
9 (a) Find the equation of the straight line that is perpendicular to the line y x 2 1 1 = + and passes through the point (1, 3). ................................................ [3] (b) 0 1 2 3 4 5 6 7 1 2 3 R 4 5 6 7 8 9 10 11 12 8 y x (i) Find the three inequalities that define the region R. ................................................ ................................................ ................................................ [4] (ii) Find the point (x, y), with integer co-ordinates, inside the region R such that x y 3 5 35 + = . ( .................... , ....................) [2]
10 (a) NOT TO SCALE x B (6, –2) A (–3, 4) 0 y Calculate the length of AB. ................................................ [3] (b) The point P has co-ordinates , 10 12 ^ h and the point Q has co-ordinates ,2 4 - ^ h. Find (i) the co-ordinates of the mid-point of the line PQ, ( ....................... , .......................) [2] (ii) the gradient of the line PQ, ................................................ [2] (iii) the equation of a line perpendicular to PQ that passes through the point ,2 3 ^ h. ................................................ [3]
6 –1 0 –2 –3 –4 –5 –5 5 4 3 2 1 –4 –3 –2 –1 1 2 3 4 5 x y A B C 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]
12 A line has gradient 5. M and N are two points on this line. M is the point (x, 8) and N is the point (k, 23). Find an expression for x in terms of k. x = ....................................... [3]
14 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 x y A B Points A and B are marked on the grid. BC 4 0 = - c m (a) On the grid, plot the point C. [1] (b) Write AC as a column vector. f p [1] (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. f p [2]
14 (a) D is the point (2, ‒5) and DE 7 1 = c m. Find the co-ordinates of the point E. ( ..................... , ..................... ) [1] (b) t v 12 = c m and v 13 = . Work out the value of t, where t is negative. t = .................................................. [2]
20 y x –3 –2 –1 1 2 3 4 5 5 4 3 2 1 0 –1 –2 –3 l (a) Find the equation of the line l. Give your answer in the form y = mx + c. y = .................................................. [3] (b) A line perpendicular to the line l passes through the point (3, −1). Find the equation of this line. ................................................... [3] Question 21 is printed on the next page.
27 O x y A B NOT TO SCALE A is the point (-2, 0) and B is the point (0, 4). (a) Find the equation of the straight line joining A and B. ................................................. [3] (b) Find the equation of the perpendicular bisector of AB. ................................................. [4]
3 The table shows some values for 2 4 y x x 3 2 = + . x –2.2 –2 –1.5 –1 –0.5 0 0.5 0.8 y –1.94 0.75 0 3.58 (a) Complete the table. [4] (b) Draw the graph of 2 4 y x x 3 2 = + for 2.2 0.8 x G G - . x y –2.5 –2 –2 1 2 3 4 –1 –1.5 –1 –0.5 0.5 1 0 [4] (c) Find the number of solutions to the equation 2 4 3 x x 3 2 + = . ................................................... [1] 7 (d) (i) The equation 2 4 1 x x x 3 2 + - = can be solved by drawing a straight line on the grid. Write down the equation of this straight line. y = .................................................. [1] (ii) Use your graph to solve the equation 2 4 1 x x x 3 2 + - = . x = ............................ or x = ............................ or x = ............................[3] (e) The tangent to the graph of 2 4 y x x 3 2 = + has a negative gradient when x k = . Complete the inequality for k. ...................... 1 k 1 ......................[2]
4 The diagram shows the graph of ( ) f y x = for . x 2 5 2 G G - . –8 –9 –10 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 –1 1 2 –2 x y 9 (a) Find f(1). ................................................. [1] (b) Solve ( ) f x 3 = . x = ................................................ [1] (c) The equation ( ) f x k = has only one solution for . x 2 5 2 G G - . Write down the range of values of k for which this is possible. ................................................. [2] (d) By drawing a suitable straight line, solve the equation f(x) = x – 5. x = ..................... or x = ..................... or x = ..................... [3] (e) Draw a tangent to the graph of ( ) f y x = at the point where x = 1. Use your tangent to estimate the gradient of ( ) f y x = when x = 1. ................................................. [3]
7 A line joins the points ( , ) A 3 8 - and ( , ) B 2 2 - . (a) Find the co-ordinates of the midpoint of AB. (....................... , .......................) [2] (b) Find the equation of the line through A and B. Give your answer in the form y mx c = + . y = ....................................... [3] (c) Another line is parallel to AB and passes through the point (0, 7). Write down the equation of this line. ................................................. [2] (d) Find the equation of the line perpendicular to AB which passes through the point (1, 5). Give your answer in the form ax by c 0 + + = where a, b and c are integers. ................................................. [4]