Coordinates and Midpoints

Coordinates give the position of a point on a grid. This page covers reading and plotting points in all four quadrants, finding the midpoint of a line segment, and finding a missing point, such as the last corner of a shape.

Early in a paper you are usually asked to write down the coordinates of a point, or to mark a point on a grid with a cross, for 1 mark each. Finding the midpoint of two points is worth 1 or 2 marks.

Later in the paper, a question worth 3 or 4 marks may give you two points on a straight line, with no grid, and ask you to find another point further along the line.

A grid has two axes. The x-axis goes across and the y-axis goes up. They cross at the origin, O, which is the point (0, 0)

A point is written (x, y). The first number, the x-coordinate, tells you how far to go across: right if it is positive, left if it is negative. The second number, the y-coordinate, tells you how far to go up or down: up if it is positive, down if it is negative.

Always go across first, then up or down. For (2, -3), start at O, go 2 squares right, then 3 squares down.

The axes split the grid into four parts, called quadrants. The signs of the two coordinates tell you which quadrant a point is in.

A point on an axis has a 0 in it. Every point on the y-axis has an x-coordinate of 0, such as (0, 4). Every point on the x-axis has a y-coordinate of 0, such as (-5, 0)

To plot a point, draw a small cross exactly where the two grid lines meet, then write the point's letter next to it.

A line segment is the straight line between two points. Its midpoint is the point exactly halfway along it.

To find the midpoint, add the two x-coordinates and halve the total. Then add the two y-coordinates and halve the total. If the ends of the line segment are (x_1, y_1) and (x_2, y_2), then

\text{midpoint}=\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)

For example, A is (-2, 3) and B is (4, 1). Add the x-coordinates: -2+4=2, and halve: 1. Add the y-coordinates: 3+1=4, and halve: 2. So the midpoint M is (1, 2)

A midpoint can have a half in it. The midpoint of (2, 1) and (7, 5) is (4.5, 3), because (2+7)\div2=4.5 and (1+5)\div2=3

A vertex is a corner of a shape. The letters of a shape go round it in order, so in ABCD, corner D is joined to A and to C, and is opposite B.

In a rectangle with its sides along the grid lines, each corner is straight above or below one corner, and level with another. So the missing corner takes its x-coordinate from one neighbour and its y-coordinate from the other.

In a parallelogram, opposite sides are equal and parallel. So in ABCD, the move from B to C is the same as the move from A to D. Describe the move, such as "2 right and 1 down", then do it from A.

Squares, rectangles and rhombuses are all parallelograms, so the same move works for them too, even when the shape is tilted.

Sometimes you are given one end of a line segment, A, and its midpoint, M, and asked for the other end, B. The midpoint is halfway, so the move from M to B is the same as the move from A to M.

For example, A is (1, 1) and M is (4, 3). The move from A to M is 3 right and 2 up. Do it again from M, and B is (4+3, 3+2)=(7, 5)

The same idea works for any points spaced equally along a straight line: each equal step along the line is the same move.

• Writing the coordinates the wrong way round. The x-coordinate comes first: go across, then up or down.
• Losing a minus sign. A point left of the y-axis has a negative x-coordinate, and a point below the x-axis has a negative y-coordinate.
• Not writing the midpoint as a pair. Give the answer as one pair of coordinates in brackets, such as (1, 2), not just a cross on the grid.
• Finding the midpoint of the wrong two points. Check which letters the question names, such as AC rather than AB.
• Stopping at the midpoint when the question asks for the other end. The other end is the same move again, on from the midpoint.
• Forgetting to label a plotted point. Mark it with a small cross and write its letter next to it.