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The distance between two points

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Position and transformation

The distance between two points

A(1, 2) and B(4, 2) sit on the same horizontal line, so you do not need to draw anything. Subtract the x-coordinates and you have the distance.

12345123450(3, 2)

Points on a horizontal line

A(1, 2) and B(4, 2) have the SAME y-coordinate, so they lie on the same horizontal line. The distance between them is the difference between their x-coordinates: 4 āˆ’ 1 = 3 units Always subtract the smaller from the larger ā€” a distance is never negative.

Points on a vertical line

E(2, 4) and F(2, 7) have the SAME x-coordinate, so they lie on the same vertical line. Now it is the y-coordinates that differ: 7 āˆ’ 4 = 3 units Same idea, other coordinate.

Which coordinate do I use?

Look at which coordinate MATCHES. ā€¢ Same y ā†’ horizontal line ā†’ subtract the x-coordinates. ā€¢ Same x ā†’ vertical line ā†’ subtract the y-coordinates. It is the coordinate that is DIFFERENT that gives you the distance.

Negative coordinates

The rule does not change when coordinates go negative ā€” just be careful with the subtraction. P(āˆ’3, 5) and Q(4, 5): the distance is 4 āˆ’ (āˆ’3) = 7 units. Subtracting a negative adds. If you get a smaller answer than you expected, check that step.

Worked examples

A(1, 2) and B(4, 2)

  1. The y-coordinates match, so this is a horizontal line.
  2. Subtract the x-coordinates: 4 āˆ’ 1 = 3.

Answer: 3 units

C(9, 3) and D(5, 3)

  1. Same y-coordinate, so horizontal.
  2. 9 āˆ’ 5 = 4.

Answer: 4 units

E(2, 4) and F(2, 7)

  1. Same x-coordinate, so vertical.
  2. 7 āˆ’ 4 = 3.

Answer: 3 units

P(āˆ’3, 5) and Q(4, 5)

  1. Same y-coordinate, so horizontal.
  2. 4 āˆ’ (āˆ’3) = 4 + 3 = 7.

Answer: 7 units

R(6, āˆ’2) and S(6, 5)

  1. Same x-coordinate, so vertical.
  2. 5 āˆ’ (āˆ’2) = 5 + 2 = 7.

Answer: 7 units

Deciding which coordinate to use

  1. Compare the two points.
  2. Whichever coordinate is the SAME tells you the direction.
  3. Subtract the coordinate that is different.

Answer: Subtract the coordinate that differs

Where you see this in real life

  • Ships and planes work out how far apart they are from coordinates, not by drawing a map each time.
  • A game engine checks the gap between two objects with exactly this subtraction, thousands of times a second.

Key words

Coordinate
a pair of numbers (x, y) fixing a point
x-coordinate
the first number ā€” how far across
y-coordinate
the second number ā€” how far up
Horizontal line
a line running left to right
Vertical line
a line running up and down
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