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.

Work through the sections in order, then try the linked Year 10 skill quizzes after each section.

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 $(11, 7)$ .

Skill quiz: Find a missing coordinate

Decide which coordinate stays fixed on a horizontal or vertical line.

Coordinates of points on a line

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 $(- 2, 1)$ .

Skill quizzes: Use a given distance

Start with Level 1, then consider both possible directions in the harder Level 2 questions.

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 $(5, 5)$ .

Skill quiz: Find vertices of a planar shape

Use the known line segment and repeat the required coordinate shift to find other vertices.

Coordinates of a planar shape from a line

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.

Skill quizzes: Check the line and distance conditions

Check the fixed coordinate and use the stated length to verify your new point.

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.

Skill quiz: Check directions and possible points

Revisit the harder distance questions and consider whether moving either way gives a valid point.

Level 2: Coordinate of a point on a line given the lengthPremium

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.

Skill quizzes: Choose the correct coordinate rule

Use the line direction and stated distance to find points, then apply the shifts to a shape.