Year 12 Coordinate Geometry: Line segment ratios

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Ratio questions tell you how a point divides a segment. The important part is reading the ratio correctly. If a point $P$ divides $AB$ in the ratio $2:1$, the usual interpretation is $\text{AP}:PB=2:1$. 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.

Work through the sections in order, then try the linked Year 12 skill quizzes after each section.

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

Worked example 2: using fractions of the whole segment

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.