Distances on a coordinate grid
Find horizontal and vertical distances, including paths crossing an axis.
Before you begin
- Coordinate pairs
- Absolute value and subtraction
Identify the coordinate that changes
Two points with the same y-coordinate lie on a horizontal line. Their separation is the absolute difference of their x-coordinates. With the same x-coordinate, use the y-coordinates instead. This gives an ordinary distance rather than a signed change. The method applies directly to horizontal or vertical segments, not to a diagonal segment where both coordinates change.
Add distances across zero
For points on opposite sides of an axis, the distance is the sum of their distances from that axis. From x = −4 to x = 3, travel four units to zero and then three more, giving seven. This is consistent with |3 − (−4)| = 7. Include the scale of the grid: seven coordinate units could represent seven metres or a different stated distance.
Worked example
Find the distance from (−5, 2) to (3, 2) on a grid measured in metres.
Show the worked solution
- Both y-coordinates equal 2, so the segment is horizontal.
- The x-coordinate changes from −5 to 3: |3 − (−5)| = 8.
- Check by counting five metres to the y-axis and another three metres beyond it.
8 m.
Try it yourself
What is the distance between (−2, −4) and (−2, 3)?
Common mistakes
- Do not apply a one-coordinate distance rule to a diagonal.
- A route with a turn may be longer than the straight-line distance.
What can you explain now?
A path goes from (−3, −2) to (4, −2), then to (4, 5). Find its total length.
Compare with the explanation
14 units.
The horizontal part is 7 units and the vertical part is 7 units. Add them for this two-part path; 14 is not the direct diagonal distance.
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Bring the ideas together
Signed numbers and coordinates
Try the unit reviewOriginal StudyForA content. AI-assisted checks completed; subject-teacher review is still pending. These lessons are not endorsed by an exam board.
Common Core mathematics standards ↗