medium difficulty8 min

Distance between two points on a plane

This concept means finding the exact separation between any two points on the coordinate plane.

In one line: Same height means horizontal counting; same side means vertical counting.

Understand it

A point-to-point distance is not guessed from the sketch; it is calculated from the coordinate changes. If the points lie on a horizontal or vertical line, the distance is just the difference of one coordinate. In every other case, the straight-line distance is found by the distance formula.

Example

The points (-2, 3) and (4, 3) lie on the same horizontal line, so their distance is 6 units.

Formula and rule

Use d = √((x2 - x1)² + (y2 - y1)²).

If one coordinate is the same, the formula reduces to a simple difference along the other axis.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

For points (-2, 3) and (4, 3), the y-values are equal.

1So distance = |4 - (-2)| = 6 units.
2No square root is needed because the segment is horizontal.

Method you can reuse

1Read both coordinates carefully.
2Check whether one coordinate is the same.
3Use simple subtraction for horizontal or vertical lines.
4Otherwise use the full distance formula.
5Simplify the answer.

Sketch it in your answer

Plot both points carefully, then decide whether the segment is horizontal, vertical, or slanted before calculating.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is the distance between points (-2, 3) and (4, 3)?

Think first, then show answer

The distance is 6 units because the y-coordinates are the same, so only the horizontal change matters.

Common wrong answer

A common wrong answer is 1 unit for points (-2, 3) and (4, 3), because students look only at the y-difference and ignore the horizontal gap.

Don't mix this up

Students sometimes use the wrong coordinate change, or they treat a horizontal or vertical line as if it were a diagonal line.

Exam tip

First check whether the points are on the same horizontal or vertical line. That can save time and reduce calculation errors.