8B - Dividing a line in a ratio

How can you locate a point one-third of the way along a sloping segment without measuring the diagram? You will find midpoints, divide segments in a stated ratio and work backwards to find a missing endpoint or ratio.

Midpoints

Where is the halfway point between $(-3,2)$ and $(7,8)$ ? It must be halfway through both coordinate changes.

The total horizontal change is $10$ and the vertical change is $6$ . Starting at $(-3,2)$ , half those changes take you to $(-3+5,2+3)=(2,5)$ .

Midpoint

The midpoint of a segment is the point on the segment at equal distances from its endpoints.

For one coordinate, starting at $x_1$ and adding half the change gives $x_1+\tfrac12(x_2-x_1)=\tfrac12(x_1+x_2)$ . The same reasoning applies to the other coordinate.

Midpoint coordinates

The midpoint of $A(x_1,y_1)$ and $B(x_2,y_2)$ is

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

Not on the formulae sheet: learn it as the equal-ratio case of the division formula.

Average the two $x$ -coordinates together and the two $y$ -coordinates together. Do not mix a horizontal coordinate with a vertical one.

Worked example 1: halfway through each change

Question. Find the midpoint of $A(-3,2)$ and $B(7,8)$ .

Plan. Average the coordinates in matching pairs.

\begin{aligned} x_M&=\frac{-3+7}{2}=2 &&\text{average the horizontal positions}\\ y_M&=\frac{2+8}{2}=5 &&\text{average the vertical positions} \end{aligned}

Answer. $M=(2,5)$ .

Check. From $A$ to $M$ the changes are $(5,3)$ , and from $M$ to $B$ they are also $(5,3)$ . Thus both half-segments have length $\sqrt{34}$ .

Now change one endpoint so that both coordinate sums are odd. A midpoint need not have integer coordinates.

Worked example 2: fractional midpoint coordinates

Question. Find the midpoint of $A(-3,2)$ and $C(8,9)$ .

Plan. Use the same averages and retain exact fractions.

\begin{aligned} x_M&=\frac{-3+8}{2}=\frac52\\ y_M&=\frac{2+9}{2}=\frac{11}{2} \end{aligned}

Answer. $M=\left(\tfrac52,\tfrac{11}{2}\right)$ .

Check. Each half of the segment has horizontal change $\tfrac{11}{2}$ and vertical change $\tfrac72$ , so the two lengths agree.

The formula also works for horizontal and vertical segments: averaging a coordinate that stays constant gives that same constant.

Internal division in a given ratio

A ratio tells you what fraction of the entire segment to travel from the first endpoint.

If $AP:PB=1:2$ , the whole segment contains $3$ equal parts, so $P$ is $\tfrac13$ of the way from $A$ to $B$ . It is nearer $A$ , because $AP$ is the shorter sublength.

[DIAGRAM: asset_name: One part and two parts along a segment; asset_slug: internal-ratio; caption: AP to PB is 1 to 2, so P is one-third of the way from A to B; description: Equal-scale axes show A at 1 comma 2, P at 5 comma 4 and B at 13 comma 8 on one line. Three equal steps along AB are indicated. The first step AP is pale blue and the two-step PB is pale peach.]

For $AP:PB=m:n$ , with $m,n>0$ , travel the fraction $\tfrac{m}{m+n}$ of each coordinate change. For example,

x_P=x_1+\frac{m}{m+n}(x_2-x_1)=\frac{nx_1+mx_2}{m+n}.

The weights look reversed because the starting coordinate contributes to both the starting position and the change.

Internal division coordinates

If the endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$ are distinct and $AP:PB=m:n$ , where $m,n>0$ , then

P\left(\frac{nx_1+mx_2}{m+n},\frac{ny_1+my_2}{m+n}\right).

This is internal division: $P$ lies strictly between $A$ and $B$ .

Not on the formulae sheet: learn it.

State the order $AP:PB$ before substituting. The coefficient of $B$ 's coordinates is $m$ , the numerator of the fraction travelled from $A$ .

Worked example 3: one-third of a segment

Question. $A(1,2)$ and $B(13,8)$ are joined by a segment. Find $P$ when $AP:PB=1:2$ .

Plan. Travel $\tfrac13$ of each change from $A$ .

\begin{aligned} x_P&=1+\frac13(13-1)=5\\ y_P&=2+\frac13(8-2)=4 \end{aligned}

Answer. $P=(5,4)$ .

Check. The changes from $A$ to $P$ are $(4,2)$ ; those from $P$ to $B$ are $(8,4)$ . The second length is twice the first, as required.

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8B - Dividing a line in a ratio

Midpoints

Internal division in a given ratio

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