Coordinates and Midpoints

Go across first, then up: the x-coordinate is written first.

Points below the x-axis have a negative y-coordinate.

A point on the x-axis has not moved up or down at all.

The first number tells you how far to go across, and the second how far to go up or down.

The midpoint is halfway between A and B. Count the squares from A to B.

Count the squares from A down to B and go half of the way.

1. A is at (-2, 4) and B is at (-2, -2). Both have x-coordinate -2, so the midpoint does too.
2. From y=4 down to y=-2 is 6 squares. Half of 6 is 3.
3. 4-3=1, so the midpoint is (-2, 1).

The midpoint is halfway across and halfway up: find the number halfway between the two x-coordinates, then do the same for the y-coordinates.

1. Add the x-coordinates and halve: (2+8)\div2=5
2. Add the y-coordinates and halve: (3+11)\div2=7
3. The midpoint is (5, 7).

Add the two x-coordinates and halve, then do the same for the y-coordinates. Take care with the negatives.

1. x-coordinate: (-6+2)\div2=-4\div2=-2
2. y-coordinate: (5+(-1))\div2=4\div2=2
3. The midpoint is (-2, 2).

The sides of this rectangle go straight across and straight up and down. Which corner is D level with?

1. A is (-4, 3), B is (2, 3) and C is (2, -2).
2. D is directly below A, so it has the same x-coordinate as A: -4
3. D is level with C, so it has the same y-coordinate as C: -2
4. D is (-4, -2).

Which number tells you how far to go across?

1. The first number in (5, -3) is the x-coordinate: go 5 squares to the right.
2. The second number is the y-coordinate: -3 means go 3 squares down.
3. Sam went up 5 and left 3, which is the point (-3, 5). He has used the numbers the wrong way round.

A midpoint does not always have whole-number coordinates.

1. x-coordinate: (1+4)\div2=5\div2=2.5
2. y-coordinate: (2+6)\div2=8\div2=4
3. The midpoint is (2.5, 4).

M is halfway from A to B. Work out the move from A to M, then do the same move again.

1. From A to M: x goes up by 4-1=3 and y goes up by 5-3=2
2. M is halfway, so B is the same distance again past M.
3. B is (4+3, 5+2)=(7, 7).
4. Check: (1+7)\div2=4 and (3+7)\div2=5

Opposite sides of a parallelogram are parallel and the same length. Compare the move from B to C with the move from A to D.

1. A is (-4, -2), B is (1, -2) and C is (3, 2).
2. In a parallelogram, AD is parallel to BC and the same length.
3. From B to C is 2 to the right and 4 up.
4. Do the same move from A: D is (-4+2, -2+4)=(-2, 2).

The centre of a rectangle is halfway along the line from one corner to the opposite corner.

1. The centre of a rectangle is the midpoint of a diagonal, such as PR.
2. x-coordinate: (-3+5)\div2=1
3. y-coordinate: (1+7)\div2=4
4. The centre is (1, 4).

Halfway between k and 9 is 4. How far is it from 9 down to 4?

1. The midpoint's y-coordinate is halfway between k and 9.
2. So k+9=2\times4=8
3. k=8-9=-1
4. Check: (-1+9)\div2=4

The square is tilted. Work out the move from B to C, then make the same move from A.

1. A is (-3, 1), B is (1, 4) and C is (4, 0).
2. In the square, AD is parallel to BC and the same length.
3. From B to C is 3 to the right and 4 down.
4. Do the same move from A: D is (-3+3, 1-4)=(0, -3).

For part (a), M is halfway from A to B, so make the move from A to M again.

Part (a)

1. From A to M: x goes up by 3 and y goes down by 3
2. Do the same move again from M: B is (1+3, 2-3)=(4, -1)

Part (b)

1. B is (4, -1) and C is (6, 3).
2. x-coordinate: (4+6)\div2=5. y-coordinate: (-1+3)\div2=1
3. The midpoint of BC is (5, 1).

Work out how far across and how far up you go from A to B.

1. A is (-5, -4) and B is (-2, -2).
2. From A to B is 3 to the right and 2 up.
3. BC is twice as long as AB, so from B to C is 6 to the right and 4 up.
4. C is (-2+6, -2+4)=(4, 2).

A sketch helps. The centre is halfway along the diagonal AC.

Part (a)

1. The centre is the midpoint of AC.
2. From A to the centre: x goes up by 4 and y goes down by 3
3. Do the same move again: C is (1+4, 2-3)=(5, -1)

Part (b)

1. The sides are parallel to the axes, so the corners are (-3, 5), (5, 5), (5, -1) and (-3, -1).
2. B has the same y-coordinate as A, which is 5, and the same x-coordinate as C, which is 5
3. B is (5, 5).

Use N to find M first, then use M to find B.

1. From A to N is 2 to the right and 2 up.
2. N is halfway from A to M, so M is (4+2, 3+2)=(6, 5)
3. From A to M is 4 to the right and 4 up. M is halfway from A to B, so B is (6+4, 5+4)=(10, 9)