Distance and Internal Division of Line Segments

Hello! In the last lesson, you used similarity to match corresponding parts of triangles and set up proportions carefully. Coordinate geometry uses the same habits—identify what corresponds, preserve signs, and check whether an answer makes sense—but now the information is given as coordinate pairs.

In this lesson, you will learn two core tools:

• the distance formula, for the straight-line distance between two points;
• the internal section formula, for locating a point that divides a line segment in a stated ratio.

Both are built from ideas you already know: Pythagoras’ theorem and proportional division.

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1. Distance on the coordinate plane

A point such as $A(x_1,y_1)$ has a horizontal position $x_1$ and a vertical position $y_1$. For two points,

$$A(x_1,y_1)$$

and

$$B(x_2,y_2),$$

the direct segment $AB$ is usually diagonal. We cannot find its length by simply subtracting coordinates. Instead, form a right triangle by moving horizontally and vertically between the points.

The horizontal leg has length equal to the change in $x$:

$$x_2-x_1$$

The vertical leg has length equal to the change in $y$:

$$y_2-y_1$$

The segment $AB$ is the hypotenuse. By Pythagoras’ theorem,

$$A{B}^2=(x_2-x_1{)}^2+(y_2-y_1{)}^2$$

Taking the positive square root gives the distance formula:

$$\boxed{d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}}$$

Distance cannot be negative, so we always use the non-negative square root.

A worked example

Find the distance between

$$A(-3,4)$$

and

$$B(5,-2).$$

Keep the coordinates paired correctly:

$$x_1=-3,y_1=4$$ $$x_2=5,y_2=-2$$

Now substitute, using parentheses around negative numbers:

$$d=\sqrt{(5-(-3){)}^2+(-2-4{)}^2}$$ $$d=\sqrt{8^2+(-6{)}^2}$$ $$d=\sqrt{64+36}$$ $$d=\sqrt{100}$$ $$\boxed{d=10\text{ units}}$$

Notice that the vertical change was negative because the second point lies lower. That is not a problem: squaring makes each leg length contribute positively.

When points are on the same horizontal or vertical line

If both points have the same $y$-coordinate, the segment is horizontal. For example, the distance from

$$(-7,2)$$

to

$$(4,2)$$

is simply

$$|4-(-7)|=11.$$

The distance formula still works:

$$d=\sqrt{(4-(-7){)}^2+(2-2{)}^2}$$ $$d=\sqrt{11^2+0^2}=11.$$

So the formula is reliable in every case; it just becomes simpler when one coordinate does not change.

A dependable distance routine

For an exam question, use this structure:

1. Write each point with its $x$- and $y$-coordinates clearly matched.
2. Subtract the two $x$-coordinates.
3. Subtract the two $y$-coordinates.
4. Square both differences.
5. Add them, then take the positive square root.
6. Simplify the square root exactly when possible. Use a decimal only if the question asks for an approximation.

The most common mistakes are:

• subtracting an $x$-coordinate from a $y$-coordinate;
• forgetting parentheses, especially in expressions such as $5-(-3)$;
• writing $\sqrt{(x_2-x_1{)}^2}+(y_2-y_1{)}^2$. The square root must cover the entire sum;
• giving a negative value for a distance.

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2. Dividing a line segment internally

Now consider a point $P$ that lies between endpoints $A$ and $B$. This is what internally means.

Suppose

$$A(x_1,y_1)$$

and

$$B(x_2,y_2),$$

and point $P$ divides the segment in the ratio

$$AP:PB=m:n.$$

For example, a ratio of $2:3$ means that the segment is split into five equal parts in total: $AP$ accounts for two parts and $PB$ accounts for three. Point $P$ is therefore $\frac{2}{5}$ of the way from $A$ to $B$.

The coordinates of $P$ are given by the internal section formula:

$$\boxed{P\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right)}$$

The order is important:

$$AP:PB=m:n.$$

The coefficient $m$ multiplies the coordinates of $B$, the opposite endpoint. The coefficient $n$ multiplies the coordinates of $A$.

This “opposite endpoint” pattern can seem strange at first, but it makes sense. If $AP$ is large compared with $PB$, then $P$ must be closer to $B$, so $B$'s coordinates should have the stronger influence.

Understanding the formula by moving along the segment

There is a useful way to check the section formula. If

$$AP:PB=m:n,$$

then the whole segment has $m+n$ equal parts. The fraction of the journey from $A$ to $B$ that reaches $P$ is

$$\frac{m}{m+n}.$$

So we can start at $A$, find the change from $A$ to $B$, and take that fraction of the change:

$$x=x_1+\frac{m}{m+n}(x_2-x_1)$$ $$y=y_1+\frac{m}{m+n}(y_2-y_1)$$

These expressions simplify to the section formula. This interpretation is especially helpful for checking whether your point is in a sensible location.

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3. Applying the internal section formula

Let

$$A(-2,3)$$

and

$$B(8,13).$$

Find the coordinates of $P$ if

$$AP:PB=2:3.$$

Here,

$$x_1=-2,y_1=3$$ $$x_2=8,y_2=13$$ $$m=2,n=3.$$

Substitute into the section formula:

$$P\left(\frac{2(8)+3(-2)}{2+3},\frac{2(13)+3(3)}{2+3}\right)$$

Now simplify the $x$-coordinate:

$$\frac{16-6}{5}=\frac{10}{5}=2$$

Then simplify the $y$-coordinate:

$$\frac{26+9}{5}=\frac{35}{5}=7$$

Therefore,

$$\boxed{P(2,7)}$$

Check the answer

The ratio $2:3$ means $P$ should be $\frac{2}{5}$ of the way from $A$ toward $B$.

From $A(-2,3)$ to $B(8,13)$, the coordinate changes are:

$$8-(-2)=10$$

and

$$13-3=10.$$

Taking $\frac{2}{5}$ of each change gives:

$$\frac{2}{5}(10)=4.$$

Starting at $A$:

$$-2+4=2$$ $$3+4=7.$$

This confirms that

$$P=(2,7).$$

Because $AP<PB$, point $P$ should be closer to $A$, and $(2,7)$ is indeed closer to $(-2,3)$ than to $(8,13)$.

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4. The midpoint: an important special case

A midpoint divides a segment into two equal lengths, so its ratio is

$$1:1.$$

Put $m=1$ and $n=1$ into the internal section formula:

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

So the midpoint is found by averaging the two $x$-coordinates and averaging the two $y$-coordinates.

For the endpoints

$$A(-6,1)$$

and

$$B(4,9),$$

the midpoint is

$$M\left(\frac{-6+4}{2},\frac{1+9}{2}\right)$$ $$M\left(\frac{-2}{2},\frac{10}{2}\right)$$ $$\boxed{M(-1,5)}.$$

The midpoint should lie halfway between the endpoint coordinates. Here, $-1$ is halfway between $-6$ and $4$, while $5$ is halfway between $1$ and $9$.

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5. Choosing and checking the correct method

Use this quick guide.

If the question asks for…	Use…
The length of the segment joining two points	Distance formula
The point halfway between two endpoints	Midpoint formula
A point that splits a segment in a ratio such as $2:3$ or $5:1$	Internal section formula

For the section formula, make these checks before finalising your answer:

• Is the point internal? Its coordinates should lie between the corresponding endpoint coordinates.
• Is it closer to the correct endpoint? In $AP:PB=1:4$, $P$ is close to $A$, because $AP$ is the shorter piece.
• Did you multiply each endpoint by the correct ratio part? For $AP:PB=m:n$, pair $m$ with $B$ and $n$ with $A$.
• Did you preserve negative signs? Write parentheses when substituting a negative coordinate.

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Key takeaways

The distance formula is Pythagoras’ theorem applied to the horizontal and vertical coordinate changes:

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$

For a point $P$ that divides $AB$ internally in the ratio

$$AP:PB=m:n,$$

use:

$$P\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right).$$

The midpoint is the special $1:1$ case:

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

Next, the course moves to trigonometry, beginning with the standard values of trigonometric ratios and the reference triangles that explain them.