Year 12 Coordinate Geometry: Line segment ratios

Ratio questions tell you how a point divides a segment. The important part is reading the ratio correctly. If a point $P$ divides $A B$ in the ratio $2 : 1$ , the usual interpretation is $AP : P B = 2 : 1$ . That means the point is closer to $B$ than to $A$ , because the smaller part sits near $B$ . Many mistakes come from reversing those weights.

P

divides

A B

internally in the ratio

m : n

, then

P = (n A + m B) / (m + n)

. The opposite endpoint gets paired with each weight.

Why the formula works

A ratio point is a weighted average of the endpoints. If the point is closer to $B$ , then $A$ must receive the smaller weight in the final average. Thinking of the point as a balance point helps keep the formula consistent. You can also derive the same answer using vectors and fractions of the total segment, so the formula is not a trick. It is just a compact version of the same geometry.

Worked example 1: internal division

Problem: Point

P

divides the segment joining

A (- 4, 1)

and

B (8, 7)

in the ratio

2 : 1

, with

AP : P B = 2 : 1

. Find

P

Use the section formula: $P = (1 A + 2 B) / (2 + 1)$ .
For the x-coordinate: $x = (1 (- 4) + 2 (8)) / 3 = 12 / 3 = 4$ .
For the y-coordinate: $y = (1 (1) + 2 (7)) / 3 = 15 / 3 = 5$ .
So $P = (4, 5)$ .
Check the placement. The point should be closer to $B$ because the final part is one unit of the ratio, and $(4, 5)$ is indeed nearer to $B (8, 7)$ than to $A (- 4, 1)$ .

Answer:

P = (4, 5)

Worked example 2: using fractions of the whole segment

Problem: Point

Q

divides the segment from

M (3, - 2)

N (15, 10)

in the ratio

1 : 3

, with

M Q : Q N = 1 : 3

. Find

Q

The whole segment is split into 4 equal parts, so $Q$ is one quarter of the way from $M$ to $N$ .
Direction vector: $vector M N = (12, 12)$ .
One quarter of that vector is $(3, 3)$ .
Add to the starting point: $Q = (3, - 2) + (3, 3) = (6, 1)$ .

Answer:

Q = (6, 1)

Internal versus external division

Most school questions use internal division, where the point lies between the endpoints. External division means the point lies beyond one endpoint on the same line. The algebra changes sign in that case, and a diagram becomes even more important. If the answer lies outside the segment, do not panic. That can be correct if the question is describing extension of the line, not just the segment itself.

Common mistakes

Reversing the weights in the section formula.
Reading $2 : 1$ as one third from A instead of two thirds from A.
Forgetting that the ratio parts add to give the whole segment.
Skipping the diagram and losing track of which point should be closer.

Exam habit

Before calculating, decide where the point should roughly sit. If the ratio is $4 : 1$ , the point should be much closer to the second endpoint named in the ratio statement. That expectation acts as a built-in error check once you compute the coordinates.