The Point Dividing a Line in a Ratio

Find the point that divides a line segment in a given ratio, find a missing end point, and find the ratio when the point is known.

In the exam for Edexcel Further Pure Maths 4PM1 · 11 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

A fraction of the way along

Two friends share a 12 km walk in the ratio 2 : 1. The first walks 8 km, the second 4 km. They swap over at the point two thirds of the way along. Points on a line work the same way.

Rule

If P divides AB in the ratio $m : n$ , then $A P : P B = m : n$ . The line is cut into $m + n$ equal parts, and P is $\frac{m}{m + n}$ of the way from A to B.

So you can find P with steps. Work out the step from A to B, take the right fraction of it, and add it to A.

Worked example

1
$A (1, 2)$ and $B (10, 8)$ . P divides AB in the ratio $2 : 1$ . Find P.
2
$B - A : 10 - 1 = 9, 8 - 2 = 6$ — The full step from A to B: 9 across and 6 up.
3
$\frac{2}{3} \times 9 = 6, \frac{2}{3} \times 6 = 4$ — P is 2 parts out of 2 + 1 = 3, so take two thirds of each step.
4
$P = (1 + 6, 2 + 4) = (7, 6)$ — Add these steps to A.

The midpoint is the special case $1 : 1$ : P is half of the way from A to B.

Common mistake

Reading “AP : AB = 2 : 5” as “AP : PB = 2 : 5”. If AP is 2 parts of the whole 5, then PB is the other 3, so $A P : P B = 2 : 3$ .

Key idea

René Descartes published the idea of describing points with coordinates in 1637, in his book La Géométrie. That is why x, y coordinates are called Cartesian coordinates.

Key idea

$m : n$ means P is $m$ parts from A and $n$ parts from B, out of $m + n$ parts in total.

The section formula

The steps can be packed into one formula. It gives each coordinate of P in one line.

Formula

P = (\frac{n x _{1} + m x _{2}}{m + n}, \frac{n y _{1} + m y _{2}}{m + n})

$(x_{1}, y_{1})$ — the first point, A.
$(x_{2}, y_{2})$ — the second point, B.
$m$ — the part of the ratio next to A, as in $A P : P B = m : n$ . It multiplies B's coordinates.
$n$ — the part of the ratio next to B. It multiplies A's coordinates.

Why the cross-over? If m is big, P is far from A and close to B. So B's coordinates must get the bigger weight, m.

Rule

4PM1 does not print this formula on the formula sheet. It is on the list of formulae to learn.

Worked example

1
$A (- 3, 4)$ and $B (9, - 4)$ . P lies on AB with $A P : P B = 3 : 1$ . Find the coordinates of P.
2
$m = 3, n = 1$ — Read the ratio in the order A to P to B.
3
$x = \frac{1 ( - 3 ) + 3 ( 9 )}{3 + 1} = \frac{24}{4} = 6$ — n times A's x, plus m times B's x, over m + n.
4
$y = \frac{1 ( 4 ) + 3 ( - 4 )}{4} = \frac{- 8}{4} = - 2$ — The same pattern with the y-coordinates.
5
$P = (6, - 2)$ — Check: P should be three quarters of the way to B. Steps 12 and −8, three quarters of each is 9 and −6, and A + (9, −6) = (6, −2).

Common mistake

Putting m with A's coordinates. That gives the point dividing AB in the ratio $n : m$ , which is near the wrong end. Each point is multiplied by the ratio number of the other point.

Rule

Show the substitution. Mark schemes give a method mark for a correct formula with your numbers in it, then an accuracy mark for the point.

Key idea

Bigger share, further from A. Check that your answer sits closer to the correct end.

Finding a missing end

Sometimes you know A and the dividing point P, and the end point B is missing. Think in parts: the step from A to P is m parts. Find one part, then walk n more parts from P.

Worked example

1
$A (2, - 1)$ and $P (8, 3)$ . P lies on AB with $A P : P B = 2 : 3$ . Find B.
2
$P - A : (6, 4)$ — The step from A to P. This step is 2 parts.
3
$one part = (3, 2)$ — Halve the step, because AP is 2 parts.
4
$three parts = (9, 6)$ — PB is 3 parts.
5
$B = (8 + 9, 3 + 6) = (17, 9)$ — Walk 3 parts on from P.
6
$\frac{3 ( 2 ) + 2 ( 17 )}{5} = 8, \frac{3 ( - 1 ) + 2 ( 9 )}{5} = 3$ — Check with the formula: it gives back P(8, 3).

The same idea works when a question says “C divides BD in the ratio 4 : 3” and D is unknown. Find one part from B to C, then walk 3 parts on from C.

Common mistake

Walking from A instead of from P. PB starts at P, so the extra parts are added to P.

Key idea

Missing end: find the size of one part, then count parts from the point you know.

Finding the ratio

If all three points are given, you can find the ratio. Compare the steps. You do not need lengths or Pythagoras.

Worked example

1
$A (- 2, 1)$ , $P (6, 5)$ and $B (10, 7)$ lie on a straight line. Find $A P : P B$ .
2
$A \to P : 6 - (- 2) = 8$ — The x-step from A to P.
3
$P \to B : 10 - 6 = 4$ — The x-step from P to B.
4
$A P : P B = 8 : 4 = 2 : 1$ — Simplify the ratio.
5
$5 - 1 = 4, 7 - 5 = 2$ — Check with the y-steps: 4 : 2 = 2 : 1 as well.

This works because all three points are on the same straight line, so the x-steps and y-steps are in the same proportion. If the line is vertical, use the y-steps instead.

Ratio questions are often the first part of a longer question. Next you may need the line through P perpendicular to AB, from the perpendicular lines lesson.

Common mistake

Writing the ratio backwards. $A P : P B$ starts with the step from A, so put the A to P step first.

Key idea

Same line, same proportion. Compare x-steps (or y-steps) to read the ratio.

Which method? Read the question

Every ratio question is one of three types. The question tells you which piece is missing.

You see	You do
A, B and the ratio; find P	Section formula, or steps: P is $\frac{m}{m + n}$ of the way
“Find the midpoint”	Ratio $1 : 1$ : average the coordinates
A, P and the ratio; find B	Find one part from A to P, walk n parts on from P
A, P and B; find the ratio	Compare the x-steps (or y-steps)
“AP : AB = …” or “AP = $\frac{1}{4}$ AB”	Turn it into $A P : P B$ first

Common mistake

Mixing up the order of the ratio. Always say it in the order of the letters: A to P, then P to B.

Key idea

A ratio on a line is just a fraction of the way along. Steps or formula, both give the same point.

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