3.4 - Dividing a line in a ratio

Use equal steps along a straight line to work out a midpoint, a point in a given ratio or a missing endpoint. You will distinguish a point inside a segment from one outside it, so that the ratio and the stated point order select the right answer.

Midpoints and missing endpoints

How can both $(4,0)$ and $(12,0)$ have distances from $A(0,0)$ and $B(6,0)$ in the ratio $2:1$ ? The first has distances $4$ and $2$ ; the second has distances $12$ and $6$ . We will see why “on the line” and “between the points” are different instructions.

Begin with the most familiar ratio, $1:1$ . Half the change in each coordinate takes you halfway along the segment. This is equivalent to averaging the two $x$ -coordinates and averaging the two $y$ -coordinates.

Midpoint

The midpoint of a segment $AB$ is the point on the segment equally distant from its endpoints. For distinct endpoints, it divides the segment in the ratio $AM:MB=1:1$ . A point merely sharing the same $x$ -coordinate as the midpoint need not be the midpoint: its $y$ -coordinate must also be halfway.

Averages work for negative coordinates too. The midpoint between $-4$ and $6$ on a number line is $-4+\frac12(10)=1$ , not the average of their positive distances from zero.

Midpoint coordinates

For $A(x_1,y_1)$ and $B(x_2,y_2)$ , the midpoint is

M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

If $A$ and $M$ are known, the missing endpoint is

B(2x_M-x_A,\,2y_M-y_A)

Not on the formulae sheet: learn it.

The two questions below use the same halfway relationship in opposite directions.

Worked example 1: averaging both coordinates

Question. Work out the midpoint of $A(-4,3)$ and $B(6,7)$ .

Plan. Average the coordinates separately, keeping each sign.

\begin{aligned} x_M&=\frac{-4+6}{2}=1 && \text{average the horizontal coordinates}\\ y_M&=\frac{3+7}{2}=5 && \text{average the vertical coordinates} \end{aligned}

Why this step? Halfway along a straight segment means taking half of both coordinate changes at the same time.

Answer. $M=(1,5)$ .

Check. The move from $A$ to $M$ is $(5,2)$ , and the move from $M$ to $B$ is also $(5,2)$ . Equal moves give equal lengths in the same direction.

Worked example 2 changes only which point is unknown. Knowing the midpoint lets you repeat the first half of the journey.

Worked example 2: recovering an endpoint

Question. $A(-4,3)$ and $B$ are endpoints of a segment with midpoint $M(1,5)$ . Work out the coordinates of $B$ .

Plan. Make each unknown endpoint coordinate the subject of its midpoint equation.

\begin{aligned} \frac{-4+x_B}{2}&=1 && \text{horizontal midpoint condition}\\ x_B&=2(1)+4=6 && \text{double and isolate}\\ \frac{3+y_B}{2}&=5 && \text{vertical midpoint condition}\\ y_B&=2(5)-3=7 && \text{double and isolate} \end{aligned}

Why this step? The midpoint is halfway from the endpoint, not halfway from the origin. Subtracting the known endpoint after doubling accounts for its position.

Answer. $B=(6,7)$ .

Check. From $A$ to $M$ is $5$ right and $2$ up. Repeating that move from $M$ reaches $(6,7)$ .

You can recover a missing endpoint by algebra or by equal steps. Both express the same fact, so choose the form that makes your signs clearest.

Watch out

Wrong: double the midpoint to obtain $B=(2,10)$ . This assumes the other endpoint is the origin. Right: $B=(2x_M-x_A,2y_M-y_A)$ , which gives $(6,7)$ here.

Now both halves cross coordinate axes. Keep the subtraction of the known endpoint visible.

Equal steps inside a segment

If $P$ lies between $A$ and $B$ , the ratio $AP:PB=m:n$ divides the whole segment into $m+n$ equal parts.

You already know

In a positive ratio $m:n$ , the total contains $m+n$ equal parts. The first share is $\frac{m}{m+n}$ of the total, not $\frac mn$ . Along one straight line, the same fraction applies to both coordinate changes.

Start at the endpoint named first in the ratio. For $AP:PB=2:3$ , move two of the five equal steps from $A$ , or three steps backwards from $B$ . The first route is often easier to keep consistent.

A point inside a segment

For distinct points $A$ and $B$ , with $P$ between them and $AP:PB=m:n$ , where $m,n>0$ ,

\begin{aligned} x_P&=x_A+\frac{m}{m+n}(x_B-x_A)\\ y_P&=y_A+\frac{m}{m+n}(y_B-y_A) \end{aligned}

Use signed coordinate changes; the lengths in the ratio are positive.
Not on the formulae sheet: learn it.

Coordinate changes scale together because all the points lie on the same straight segment. You do not divide the absolute coordinates by the ratio entries.

Worked example 3: two parts out of five

Question. $A(-2,1)$ , $P$ and $B(8,6)$ lie in that order on a straight line. $AP:PB=2:3$ . Work out the coordinates of $P$ .

Plan. Calculate the whole move from $A$ to $B$ , take $\frac25$ of it, then add it to $A$ .

\begin{aligned} \text{whole change}&=(8-(-2),6-1)=(10,5) && \text{five equal parts}\\ \text{one part}&=(2,1) && \text{divide both changes by }5\\ \text{two parts}&=(4,2) && \text{the move from }A\text{ to }P\\ P&=(-2+4,1+2)=(2,3) && \text{start at }A \end{aligned}

Why this step? The ratio gives a fraction of the change from $A$ , not a fraction of the final coordinates of $B$ .

Answer. $P=(2,3)$ .

Check. The moves $AP=(4,2)$ and $PB=(6,3)$ are respectively $2$ and $3$ copies of $(2,1)$ , so their lengths have ratio $2:3$ .

Worked example 4 changes only the unknown: the interior point is known and the second endpoint is missing. Use the number of parts represented by the known move.

Worked example 4: an endpoint from an interior point

Question. $A(-2,1)$ , $P(2,3)$ and $B$ lie in that order on a straight line. $AP:PB=2:3$ . Work out the coordinates of $B$ .

Plan. Divide the move from $A$ to $P$ by its two parts, then take three more parts from $P$ .

\begin{aligned} AP\text{ change}&=(2-(-2),3-1)=(4,2) && \text{two parts are known}\\ \text{one part}&=(2,1) && \text{divide by }2\\ PB\text{ change}&=3(2,1)=(6,3) && \text{three remaining parts}\\ B&=(2+6,3+3)=(8,6) && \text{continue from }P \end{aligned}

Why this step? The known segment represents $2$ parts, not all $5$ . Dividing it by $5$ would make every step too short.

Answer. $B=(8,6)$ .

Check. The whole move is $(10,5)$ ; its first $\frac25$ is $(4,2)$ , which reaches the given point $P$ .

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Number

Algebra

Coordinate geometry

Calculus

Matrix transformations

Geometry

3.4 - Dividing a line in a ratio

Midpoints and missing endpoints

Equal steps inside a segment

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