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This page sets out the method for coordinates in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Coordinates topic page.
When you see coordinates like (3, −2), read the x first and the y second.
When you see three vertices given and one missing, use the fact that opposite sides are equal and parallel.
The first number is across. The second number is up.
(2, 5) is not the same point as (5, 2).
The origin is (0, 0), where the axes cross.
Positive x is to the right of the origin. Positive y is up.
Go across to the x-number, then up to the y-number.
Count the grid lines, not the squares in between, when the scale is 1.
Moving right adds to x. Moving up adds to y.
Moving left subtracts from x. Moving down subtracts from y.
The horizontal axis is the x-axis. The vertical axis is the y-axis.
A point on the x-axis has y = 0.
Points with the same x sit on a vertical line.
Points with the same y sit on a horizontal line.
Start at the origin every time. Move along the x-axis first. A positive x-coordinate goes right; a negative x-coordinate goes left. Then move parallel to the y-axis. Positive goes up; negative goes down. The point (-4, 3) is 4 squares left and 3 squares up. The point (4, -3) is a different place: 4 right and 3 down. Swapping the numbers swaps the point.
Before you plot, read the scale on both axes. A grid is not always one square per unit. If the labels go 0, 2, 4, 6, each square is worth 2. A point halfway between 4 and 6 has x-coordinate 5, even though you have not landed on a grid line. Mark a point with a small cross, and write its letter beside the cross so the point can be checked.
Method: 1. Start every plot at the origin, then read the coordinates in order. 2. Move along the horizontal axis using the first number. 3. Travel right for a positive first number and left for a negative one, then move up or down for the second number.
If two points share a y-coordinate, the line between them is horizontal. The length is the difference of the x-coordinates. If they share an x-coordinate, the line is vertical and the length is the difference of the y-coordinates. Difference means the positive distance, so subtract the smaller number from the larger one, or count the jumps.
The count has to include the jump across zero. From x=-3 to x=5 you travel 3 units to reach 0 and then 5 more, which is 8. Subtracting the digits 5-3=2 ignores the gap between the negatives and the positives. The same rule works for y. From y=2 down to y=-6 is 2+6=8 units.
Method: 1. If the y-coordinates match, measure a horizontal distance. 2. Take the positive difference of the two x-coordinates. 3. If the x-coordinates match, measure a vertical distance from the y-coordinates instead.
A rectangle drawn on the grid has sides parallel to the axes, unless the question says otherwise. Opposite corners decide the other two. If A is (1, 2) and the opposite corner C is (6, -3), then one remaining corner keeps the x-coordinate of A and the y-coordinate of C, which is (1, -3). The other keeps the x-coordinate of C and the y-coordinate of A, which is (6, 2). Letters written in order, such as ABCD, go around the outside. Do not jump diagonally from B to D.
After the four corners are written, check the sides. Horizontal sides share a y-coordinate and should have the same length. Vertical sides share an x-coordinate and should have the same length. If a length does not match its opposite side, a corner has taken the wrong coordinate from the wrong end. Go around the outside in letter order. The diagonal joins opposite corners and is not a side.
Method: 1. Unless the question says otherwise, draw the rectangle with sides parallel to the axes. 2. Use the pair of opposite corners to fix the other two corners. 3. One new corner keeps the first point’s x-coordinate and the opposite point’s y-coordinate.
The two axes split the grid into four quadrants. The signs of the coordinates tell you which one a point is in. Top right has (+, +), top left has (-, +), bottom left has (-, -) and bottom right has (+, -). A point such as (-6, 2) is always left of the y-axis and above the x-axis, so it cannot be in any other quadrant.
Exam questions use this in two ways. Either you are given the signs and have to place the point, or you are given a point and asked to reflect it. Reflecting in the y-axis changes the sign of the x-coordinate only. Reflecting in the x-axis changes the sign of the y-coordinate only. A point on an axis has a zero: (5, 0) is on the x-axis and (0, -3) is on the y-axis.
Method: 1. Read the x-coordinate first: left is negative, right is positive. 2. Read the y-coordinate second: down is negative, up is positive. 3. For a reflection in the y-axis, keep y and change the sign of x. 4. For a reflection in the x-axis, keep x and change the sign of y.
Tip: A point on an axis always has a zero in its pair. Look for the zero before you decide the quadrant.
A common exam question gives two end points and asks for the midpoint. The midpoint is exactly halfway, so you find the mean of the two x-coordinates and then, separately, the mean of the two y-coordinates. For A(2, 9) and B(8, 3) the midpoint is ((2+8)/2, (9+3)/2)=(5, 6). Add first, then halve.
Negative numbers are where marks are lost. For A(-5, 4) and B(3, -2) the x-sum is -5+3=-2 and half of that is -1. Do not halve the difference. A quick check is that the midpoint must lie between the two ends in both coordinates: here -1 is between -5 and 3, and 1 is between -2 and 4.
Method: 1. Write the two x-coordinates, add them and halve the total. 2. Write the two y-coordinates, add them and halve the total. 3. Put the two answers in brackets as (x, y). 4. Check each answer lies between the two matching coordinates of the end points.
Tip: If the line is horizontal or vertical you can count: the midpoint is half the length along from either end.
Some questions say that AB = BC = CD with all four points on one straight line. Because the line is straight and the lengths are equal, every step from one point to the next is the same. Find the step from A to B: how far across and how far up or down. Then repeat that step to reach C and again to reach D.
For A(1, 2) and B(4, 8) the step is +3 across and +6 up. So C=(7, 14) and D=(10, 20). Write the step on its own line so a method mark is safe. Check by confirming the step from C to D is the same as the step from A to B. The same idea locates a point that is a fraction of the way along a line.
Method: 1. Subtract the coordinates of A from those of B to get the step across and the step up. 2. Add the step to B to get C, then add it to C to get D. 3. Check that C to D uses the same step as A to B. 4. Write each new point as a pair in brackets.
Shapes are usually named in order, so ABCD means the corners go round the shape A to B to C to D and back to A. In a parallelogram the step from A to B equals the step from D to C. For A(1, 1), B(6, 2), C(8, 6), the step A to B is (+5, +1), so D=C-(5, 1)=(3, 5). Always say which step you are copying.
Lines parallel to the axes have simple equations. A vertical line through (4, -3) is x=4 because every point on it has x-coordinate 4. A horizontal line through the same point is y=-3. Two such lines cross at a point with both values, so x=4 and y=-3 cross at (4, -3). Exam questions often ask for the equation or for where the line meets an axis.
Method: 1. Write the shape in order and find the step between two corners that are next to each other. 2. Copy that step across the shape to find the missing corner. 3. For a vertical line use x= the shared x-value; for a horizontal line use y= the shared y-value. 4. Check the missing corner by testing that opposite sides have the same step.
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