Year 12 Coordinate Geometry: Coordinates of points on a line

This subtopic is about locating an unknown point when you already know two points on a line and some information about where the new point sits between them. In Year 12, these questions often look simple, but they reward clean structure. If you jump straight into guessing coordinates from a picture, you can lose the direction of the line or use the wrong fraction. A better approach is to treat the line segment as a movement from one point to another.

A (x_{1}, y_{1})

and

B (x_{2}, y_{2})

, then the direction vector is

vector A B = (x_{2} - x_{1}, y_{2} - y_{1})

. A point part-way from

A

B

is found by taking a fraction of that vector.

What this subtopic includes

Points that divide a segment in a known fraction.
Midpoints and quarter-points.
Checking whether a point really lies on the segment.
Moving along a line without using a full line equation.

Main method

Choose a starting point, usually the one named first in the question.
Find the direction vector from the starting point to the other endpoint.
Convert the wording into a fraction of the whole segment.
Multiply the direction vector by that fraction.
Add the result to the starting point.

This method is more reliable than working separately with x-coordinates and y-coordinates because it preserves the geometry of the segment. It also scales well when the fraction is unusual, such as two-fifths or seven-eighths.

Worked example 1

Problem: Point

P

lies one quarter of the way from

A (- 2, 3)

B (10, - 1)

. Find the coordinates of

P

Find the direction vector: $vector A B = (10 - (- 2), - 1 - 3) = (12, - 4)$ .
One quarter of this vector is $(3, - 1)$ .
Start at $A$ and add the quarter-vector: $P = (- 2, 3) + (3, - 1)$ .
So $P = (1, 2)$ .
Check the answer. From $A$ to $P$ the movement is $(3, - 1)$ , which is one quarter of $(12, - 4)$ , so the result is consistent.

Answer:

P = (1, 2)

Worked example 2

Problem: Point

Q

lies three fifths of the way from

M (4, - 6)

N (- 1, 9)

. Find

Q

Find the direction vector: $vector M N = (- 1 - 4, 9 - (- 6)) = (- 5, 15)$ .
Take three fifths of that vector: $(3 / 5) x (- 5, 15) = (- 3, 9)$ .
Add this to the starting point: $Q = (4, - 6) + (- 3, 9)$ .
This gives $Q = (1, 3)$ .
A quick reasonableness check helps. The x-coordinate should move from 4 toward -1, so 1 is sensible. The y-coordinate should move from -6 toward 9, so 3 is also sensible.

Answer:

Q = (1, 3)

Useful special case: midpoint

The midpoint is just the halfway point, so it uses the fraction $1 / 2$ . That gives the familiar midpoint formula: $((x_{1} + x_{2}) / 2, (y_{1} + y_{2}) / 2)$ . It is worth recognising that this is not a separate idea. It is the same vector method with half the movement.

Common mistakes

Using the wrong direction, such as moving from $B$ when the question says from $A$ .
Taking the wrong fraction, especially when the wording is "three fifths from A to B".
Forgetting that negative movement is normal when a coordinate decreases.
Assuming the answer must have integer coordinates.

Quick checklist

Did you write the direction vector correctly?
Did you use the fraction from the correct starting point?
Does the final point lie between the endpoints when it should?
Do the coordinates make sense visually?

Practice links

Coordinates of points on a line