Printable · GCSE Foundation · ages 14-16
Coordinates in all four quadrants worksheet — GCSE Foundation
Fifteen questions on "coordinates in all four quadrants" — DfE statement A8. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Answer key: Coordinates in all four quadrants worksheet — GCSE Foundation
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- (c) (3, 9) — Reflecting in the horizontal line y = 2 keeps the x-coordinate the same and maps y to 2 × 2 − y = 4 − (−5) = 9, so S = (3, 9). A candidate who uses k − y instead of 2k − y gets 2 − (−5) = 7, giving (3, 7). A candidate who reflects in the x-axis instead of the line y = 2, simply changing the sign of y, gets (3, 5). A candidate who also changes the sign of the x-coordinate, as if reflecting in both axes, gets (−3, 9).
- (b) 3 — −6 + 9 = 3. A candidate who ignores the negative sign on the x-coordinate and adds the two positive values gets 6 + 9 = 15. A candidate who treats the y-coordinate as negative too gets −6 + (−9) = −15. A candidate who works out 9 − 6 correctly as 3 but then writes the answer with the wrong sign gets −3.
- (b) 3 — Distance from the y-axis is given by the size of the x-coordinate, which is 3. 8 comes from using the size of the y-coordinate instead of the x-coordinate. −8 comes from using the y-coordinate and keeping its negative sign, but a distance is never negative. 11 comes from adding the sizes of both coordinates, 3 + 8, instead of using the x-coordinate alone.
- (b) (−5, −2) — A rotation of 180° about the origin reverses the sign of both coordinates, so Q = (−5, −2). A candidate who reverses the sign of only the y-coordinate, as if reflecting in the x-axis, gets (5, −2). A candidate who reverses the sign of only the x-coordinate, as if reflecting in the y-axis, gets (−5, 2). A candidate who swaps the coordinates instead of reversing their signs gets (2, 5).
- (b) x = 0 — Method: the first coordinate of a point measures how far to the left or right of the origin it is, so a point that is neither left nor right of the origin has a first coordinate of zero. Working: the y-axis is the vertical line through the origin; every point on it lies directly above or directly below the origin with no sideways movement at all, so its first coordinate is zero while its second coordinate may take any value. Answer: x = 0. The distractors: y = 0 is the condition for lying on the x-axis, the other axis; x = 1 comes from confusing the y-axis with the vertical line one unit to the right of it; x = y is the condition for the diagonal through the origin, which meets the y-axis at the origin only.
- (d) (4, 8) — The other endpoint is found from 2 × midpoint − known endpoint: x = 2 × (−1) − (−6) = −2 + 6 = 4, y = 2 × 5 − 2 = 10 − 2 = 8, giving (4, 8). (−3.5, 3.5) comes from averaging the given endpoint and the midpoint as if they were the two endpoints of a segment, instead of working backwards from the midpoint. (5, 3) comes from working out (−1 − (−6), 5 − 2) instead of doubling the midpoint before subtracting. (4, 5) comes from correctly finding the x-coordinate but copying the midpoint's y-coordinate of 5 instead of doubling it.
- (a) the second quadrant — Method: the quadrant is settled by the signs of the two coordinates, not by their size; the quadrants are numbered anticlockwise, starting from the region where both coordinates are positive. Working: the x-coordinate −3 is negative, so the point lies to the left of the y-axis; the y-coordinate 0.5 is positive, so it lies above the x-axis; the region that is both left of the y-axis and above the x-axis is the second. Answer: the second quadrant. The distractors: 'the first quadrant' comes from ignoring the minus sign on −3; 'the third quadrant' comes from treating 0.5 as a negative value because it is smaller than 1, when in fact any number above zero is positive; 'the fourth quadrant' comes from reading the pair the wrong way round, as though the point were (0.5, −3).
- (c) the fourth quadrant — Method: read the sign of each coordinate in turn, then count round the regions, which are numbered anticlockwise from the one where both coordinates are positive. Working: the x-coordinate 5 is positive, so A is to the right of the y-axis; the y-coordinate −3 is negative, so A is below the x-axis; right of the y-axis and below the x-axis is the fourth of the four regions. Answer: the fourth quadrant. The distractors: 'the first quadrant' comes from ignoring the minus sign on the y-coordinate and treating the point as (5, 3); 'the second quadrant' comes from writing the pair the wrong way round and locating (−3, 5) instead; 'the third quadrant' comes from assuming that any point with a negative coordinate belongs to the region where the minus signs are, without checking the other coordinate.
- (d) 10 — The right angle is at (1, 1). The vertical side has length 5 − 1 = 4 and the horizontal side has length 6 − 1 = 5, so the area is (4 × 5) ÷ 2 = 20 ÷ 2 = 10. A candidate who forgets to halve the product of the two sides gets 4 × 5 = 20. A candidate who forgets to subtract the shared vertex's coordinate and uses the raw coordinates 6 and 5 as the side lengths gets (6 × 5) ÷ 2 = 30 ÷ 2 = 15. A candidate who uses only one side length as the area gets 5.
- (d) the y-axis, and 12 — Method: coordinates are written (x, y), so the first number is the distance across and the second the distance up; a point whose first coordinate is 0 has not moved across from the origin and therefore lies on the vertical axis. Working: in (0, 12) the first number is 0, so P is on the y-axis, and the second number, 12, is the y-coordinate of P. Answer: the y-axis, and 12. The distractors: 'the x-axis, and 12' comes from mixing up which axis the condition 'the first coordinate is 0' describes; 'the y-axis, and 0' comes from placing P correctly but reading the pair the wrong way round, so that the first number is quoted as the y-coordinate; 'the x-axis, and 0' comes from making both of those mistakes at once.
- (a) (2, 7) — Midpoint = ((x1+x2)/2, (y1+y2)/2) = ((−3+7)/2, (5+9)/2) = (4/2, 14/2) = (2, 7). (4, 14) comes from adding the coordinates correctly but forgetting to divide by 2. (2, 9) comes from correctly averaging the x-coordinates but simply copying the y-coordinate of the second point instead of averaging the y-coordinates. (5, 2) comes from subtracting the coordinates instead of adding them before halving: ((7−(−3))/2, (9−5)/2) = (5, 2).
- (b) (−1, 6) — '6 units above the x-axis' gives a y-coordinate of 6, and the x-coordinate is given as −1, so F = (−1, 6). (6, −1) comes from swapping the x- and y-coordinates. (−1, −6) comes from treating 'above' as a negative direction instead of positive. (1, 6) comes from dropping the negative sign on the given x-coordinate.
- (a) 5.8 — The horizontal distance is 3 and the vertical distance is 5, so using Pythagoras' theorem the distance is √(3² + 5²) = √34 = 5.8 (1 d.p.). A candidate who adds the two differences instead of using Pythagoras gets 3 + 5 = 8.0. A candidate who works out 3² + 5² = 34 but forgets to take the square root gets 34.0. A candidate who subtracts the squares instead of adding them gets √(5² − 3²) = √16 = 4.0.
- (a) 9 km — Ravi's house and the library have the same y-coordinate, so the distance between them is the difference between their x-coordinates: 6 − (−3) = 9 km. 3 km comes from dropping the negative sign on Ravi's x-coordinate and working out 6 − 3 instead. 0 km comes from using the y-coordinates, which are equal, instead of the x-coordinates. −9 km comes from working out −3 − 6 = −9 and not converting it to a positive distance.
- (c) (−4, 7) — Left of the origin means negative x, and up means positive y, so the point is (−4, 7). (4, 7) comes from forgetting that 'left' means the x-coordinate is negative. (−4, −7) comes from treating 'up' as a negative direction instead of positive. (7, −4) comes from swapping the x- and y-coordinates.
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