Year 10: Coordinates of points on a line

Many Year 10 coordinate geometry questions ask you to work out where a point must be if it lies on a given line. That might mean finding a missing coordinate, finding points a fixed distance away, or building a simple shape from an existing segment.

On a horizontal line the y-coordinate stays the same. On a vertical line the x-coordinate stays the same. That is the first thing to check in these questions.

Missing coordinates on a line

If a point lies on a horizontal line through another point, it shares the same y-value. If it lies on a vertical line, it shares the same x-value.

Example 1: A point lies on the horizontal line through

(3, 7)

. Its x-coordinate is

11

. Find the point.

The line is horizontal, so the y-coordinate stays $7$ .
The point is $(11, 7)$ .

Point on a line given a length

If you know the point must be a certain distance away on the same line, add or subtract that distance from the changing coordinate.

Example 2: Find the points that are 4 units from

(2, 1)

on the same horizontal line.

The y-coordinate stays $1$ .
Move 4 units right to get $(6, 1)$ .
Move 4 units left to get $(- 2, 1)$ .

Using a line segment to build a shape

Another common Australian Year 10 pattern is to use a known horizontal or vertical side to build a rectangle or simple planar shape. The new points usually come from repeating the same shift.

Example 3: The points

(1, 2)

and

(5, 2)

form one side of a rectangle. The opposite side is 3 units above. Find the other two vertices.

The original side is horizontal because both points have y-coordinate $2$ .
Moving 3 units above means adding 3 to each y-coordinate.
The new vertices are $(1, 5)$ and $(5, 5)$ .

Checking your answer

A quick sketch usually helps. If the point is meant to stay on the same horizontal line, the final y-coordinate should match the original. If the problem gives a distance, count the units mentally to see whether the answer is too far or too close.

Common mistakes

Changing both coordinates when only one should change.
Moving in the wrong direction when distance is given.
Forgetting that a distance question can have two possible answers.
Mixing up horizontal and vertical.

Study routine

Underline words like “horizontal”, “vertical”, “distance”, and “midpoint” before you start. Those words tell you which coordinate rule to use. A simple labelled sketch is often enough to prevent sign mistakes.