Year 10: Distance and midpoint on a line

Distance and midpoint questions are common in Australian Year 10 coordinate geometry. They connect graph reading, number lines, and the idea of average position. When points lie on a horizontal or vertical line, the calculations are especially clear and are a good entry point into more advanced geometry later on.

On a horizontal line, distance depends on the x-values. On a vertical line, distance depends on the y-values. The midpoint is found by averaging the changing coordinates.

Distance on a horizontal line

If two points have the same y-coordinate, the segment is horizontal. The distance is the absolute difference between the x-values.

Example 1: Find the distance between

(2, 5)

and

(9, 5)

The y-values match, so the segment is horizontal.
Distance = $| 9 - 2 | = 7$ .

Answer: the distance is

7

units.

Distance on a vertical line

If the x-values are the same, the segment is vertical. The distance is the absolute difference between the y-values.

Example 2: Find the distance between

(- 3, 1)

and

(- 3, 8)

The x-values match, so the segment is vertical.
Distance = $| 8 - 1 | = 7$ .

Finding the midpoint

The midpoint is the point exactly halfway between the endpoints. On a horizontal or vertical segment, keep the fixed coordinate the same and average the changing coordinate.

Example 3: Find the midpoint of

(2, 5)

and

(9, 5)

The y-coordinate stays $5$ .
Average the x-values: $(2 + 9) / 2 = 11 / 2 = 5.5$ .
The midpoint is $(5.5, 5)$ .

Working backwards

Year 10 questions often give the midpoint or distance and ask for a missing endpoint. In those cases, set up the average or distance equation first, then solve the resulting simple algebra.

Example 4: One endpoint is

(4, 2)

and the midpoint is

(10, 2)

. Find the other endpoint.

The missing point is on the same horizontal line, so its y-coordinate is $2$ .
Let the missing x-coordinate be $x$ .
Use the midpoint average: $(4 + x) / 2 = 10$ .
$4 + x = 20$ , so $x = 16$ .
The other endpoint is $(16, 2)$ .

Common mistakes

Forgetting to use absolute value in a distance question.
Adding coordinates instead of averaging them for the midpoint.
Changing the coordinate that should stay fixed.
Missing the fact that horizontal and vertical lines use different coordinates.

Study routine

First decide whether the segment is horizontal or vertical. That immediately tells you which coordinate stays fixed. Then use a difference for distance or an average for midpoint. This keeps the working organised and reduces sign errors.