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.
Coordinates in all four quadrants worksheet — GCSE Foundation
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- 1.Work out the coordinates of the point halfway between (−9, −2) and (−1, −2).
- 2.A line segment has one endpoint at (−6, 2) and its midpoint at (−1, 5). Work out the coordinates of the other endpoint.
- 3.Work out the distance of the point (−6, 4) from the x-axis.
- 4.Write down the coordinates of the point that is 4 units to the left of the origin and 7 units up.
- 5.Work out the coordinates of the reflection of (6, −3) in the y-axis.
- 6.ABCD is a parallelogram. A has coordinates (−3, 1), B has coordinates (2, 1) and C has coordinates (4, 4). Work out the coordinates of D.
- 7.Point P has coordinates (2, −5). P is reflected in the x-axis to point Q. Write down the coordinates of Q.
- 8.A park is drawn on a grid in which 1 unit represents 1 km. The car park is at the point (0, 0) and the lake is at the point (3, 5). Work out the direct distance, in km, from the car park to the lake, giving your answer to 1 decimal place.
- 9.A square has vertices at (−2, 3), (−2, −3) and (4, −3). Write down the coordinates of the fourth vertex.
- 10.The point A lies on the x-axis. Which statement about A must be true?
- 11.Three vertices of a rectangle are (−4, −1), (2, −1) and (2, 3). The sides of the rectangle are parallel to the axes. Write down the coordinates of the fourth vertex.
- 12.Point A has coordinates (4, −5). Point B has coordinates (4, 3). Work out the distance between A and B.
- 13.The point P has coordinates (0, 12). Write down the axis that P lies on and the y-coordinate of P.
- 14.The point B has coordinates (−2, −2). In which quadrant is B?
- 15.The point D has coordinates (3, −8). Write down the distance of D from the y-axis.
Answer key
- (b) (−5, −2) — Both points have the same y-coordinate, so the midpoint lies on the same horizontal line: y = −2. The x-coordinate is the average of −9 and −1: (−9 + (−1)) ÷ 2 = −10 ÷ 2 = −5, giving (−5, −2). (−10, −2) comes from adding the x-coordinates but forgetting to divide by 2. (−4, −2) comes from a sign error on the second x-coordinate, treating −1 as +1: (−9 + 1) ÷ 2 = −4. (5, −2) comes from dropping the negative sign on the x-coordinate.
- (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.
- (c) 4 — Method: the distance of a point from the x-axis is measured vertically, so it is the size of the y-coordinate taken without its sign. Working: the point (−6, 4) has y-coordinate 4, so moving straight down to the x-axis covers 4 units, and a distance is written as a positive number. Answer: 4. The distractors: 6 comes from using the x-coordinate, which measures the distance from the y-axis rather than from the x-axis; −6 comes from that same mistake with the minus sign left in place, although a distance is never negative; 10 comes from adding the two distances, 6 and 4, as though the question asked how far the point is from both axes together.
- (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.
- (a) (−6, −3) — Reflecting in the y-axis changes the sign of the x-coordinate and keeps the y-coordinate the same: (−6, −3). (6, 3) comes from reflecting in the x-axis instead, which changes the sign of the y-coordinate. (−6, 3) comes from reflecting in both axes. (6, −3) comes from not applying the reflection at all.
- (d) (−1, 4) — In parallelogram ABCD the side DC is parallel and equal to the side AB, so D = C − AB. The vector from A to B is (2 − (−3), 1 − 1) = (5, 0), so D = (4 − 5, 4 − 0) = (−1, 4). A candidate who adds this vector to C instead of subtracting it gets (4 + 5, 4 + 0) = (9, 4). A candidate who subtracts A's coordinates from C's rather than the vector AB, and drops the minus sign on −3 while doing so, works out (4 − 3, 4 − 1) and gets (1, 3). A candidate who makes only the y-part of that slip, working out 4 − 1 instead of 4 − 0, gets (−1, 3).
- (c) (2, 5) — Reflecting in the x-axis keeps the x-coordinate the same and changes the sign of the y-coordinate: Q = (2, 5). (−2, −5) comes from reflecting in the y-axis instead, which changes the sign of the x-coordinate. (−2, 5) comes from reflecting in both axes. (2, −5) comes from not applying the reflection at all.
- (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.
- (d) (4, 3) — The sides are parallel to the axes: the missing vertex must share the y-coordinate 3 with (−2, 3) and the x-coordinate 4 with (4, −3), giving (4, 3). (−4, 3) comes from a sign error on the x-coordinate. (4, −9) comes from continuing the pattern of the given points by subtracting 6 from the y-coordinate again instead of matching it to (−2, 3). (3, 4) comes from swapping the x- and y-coordinates.
- (a) y = 0 — Method: in the pair (x, y) the first coordinate measures how far left or right of the origin a point is and the second how far above or below the x-axis it is, so a point on an axis has one of those measurements equal to zero. Working: the x-axis is the horizontal line through the origin, so a point sitting on it is neither above nor below that line and its second coordinate is zero, while its first coordinate may be positive, negative or zero. Answer: y = 0. The distractors: x = 0 is the condition for lying on the y-axis, the other axis; x > 0 comes from assuming a point on the x-axis must be to the right of the origin, which is true only of part of that axis; x = y holds only at the origin, which is one point of the x-axis rather than a property shared by all of them.
- (b) (−4, 3) — Method: in a rectangle whose sides are parallel to the axes only two different x-coordinates and two different y-coordinates appear, and each of them is shared by a pair of vertices, so the missing vertex takes the x-coordinate and the y-coordinate that so far appear only once. Working: the x-coordinates given are −4, 2 and 2, so 2 is already used twice and −4 is used once; the y-coordinates given are −1, −1 and 3, so −1 is already used twice and 3 is used once; the fourth vertex therefore has x = −4 and y = 3. Answer: (−4, 3). The distractors: (3, −4) comes from picking the two unpaired coordinates correctly and then writing them in the wrong order; (−4, −5) comes from matching the 4-unit vertical side but measuring it downwards from (−4, −1) instead of upwards; (8, 3) comes from carrying on round the shape with the horizontal step used earlier, adding 6 to the x-coordinate of (2, 3) instead of closing the rectangle.
- (a) 8 — Both points share the x-coordinate, so the distance between them is the difference between the y-coordinates: 3 − (−5) = 8. A candidate who mistakenly uses the equal x-coordinates instead of the y-coordinates gets 4 − 4 = 0. A candidate who adds the y-coordinates instead of subtracting them gets 3 + (−5) = −2. A candidate who reads off only point B's y-coordinate as the distance gets 3.
- (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.
- (d) the third quadrant — Method: only the signs of the coordinates matter, so two coordinates equal in size still have to be read separately, one for each axis. Working: both coordinates of (−2, −2) are negative, so B lies to the left of the y-axis and below the x-axis; counting anticlockwise from the region where both coordinates are positive, left and below is the third region. Answer: the third quadrant. The distractors: 'the first quadrant' comes from ignoring both minus signs and treating the point as (2, 2); 'the second quadrant' comes from applying the minus sign to the x-coordinate only, as though the point were (−2, 2); 'the fourth quadrant' comes from applying it to the y-coordinate only, as though the point were (2, −2).
- (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.
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