KCSE 2025 Maths P2 Q13 — Ratio in Which a Point Divides a Line

KCSE 2025 Form 4 Coordinate Geometry

Published

The Question

“The points A(-4, 7), B(4, 1) and C(16, -8) lie on a straight line. Determine the ratio in which B divides AC.”

1

Find the vector AB

Because B lies on the line from A to C, the step from A to B is just a fraction of the whole step from A to C. Start by finding the vector AB, which is the position of B minus the position of A. Subtract the coordinates entry by entry to see how far you move across and down to get from A to B.

AB⃗=(41)−(−47)=(8−6)\vec{AB} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} - \begin{pmatrix} -4 \\ 7 \end{pmatrix} = \begin{pmatrix} 8 \\ -6 \end{pmatrix}
2

Find the vector AC

Now find the vector for the whole journey, AC, in exactly the same way: the position of C minus the position of A. This is the full displacement from A all the way to C, which you will compare against AB.

AC⃗=(16−8)−(−47)=(20−15)\vec{AC} = \begin{pmatrix} 16 \\ -8 \end{pmatrix} - \begin{pmatrix} -4 \\ 7 \end{pmatrix} = \begin{pmatrix} 20 \\ -15 \end{pmatrix}
3

Compare AB with AC component by component

Divide each component of AB by the matching component of AC. If the points are truly on one straight line, both ratios must be equal. Here both simplify to the same fraction, which confirms the points are collinear and tells you what fraction of AC the vector AB represents.

820=25and−6−15=25\frac{8}{20} = \frac{2}{5} \quad \text{and} \quad \frac{-6}{-15} = \frac{2}{5}
AB⃗=25 AC⃗\vec{AB} = \tfrac{2}{5}\,\vec{AC}
4

Turn the fraction into a ratio

Since AB is two-fifths of the whole line AC, AB takes up two parts out of five. That leaves the remaining three parts for the rest of the journey, from B to C. So AB to BC is two parts to three parts.

AB:BC=2:(5−2)=2:3AB : BC = 2 : (5 - 2) = 2 : 3

Final Result

B divides AC in the ratio 2 to 3.

Why this method works

A point on a line splits the line so that the piece before it and the piece after it are fixed fractions of the whole. Writing AB and AC as vectors captures those pieces exactly, and because AB comes out as a single fraction (two-fifths) of AC, that same fraction fixes the split. Two-fifths of the way along means two parts are behind B and the other three parts of the five are ahead of it, which is precisely the ratio 2 to 3.

✓

Two parts plus three parts make five parts, and AB being two of those five parts agrees with AB = two-fifths of AC. The matching fractions 8/20 and -6/-15 also confirm A, B and C really are collinear.