Finding length and midpoint of a line segment

What is the midpoint of a line segment?

In this lesson we’ll look at how to find the length of a line segment algebraically when we’re given information and measurements about parts of the line segment.

Remember that a line segment is a finite piece of a line, named by its endpoints. For instance, the line segment $\overline{AB}$ might look like this:

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Line segments and distance

The distance between two points on a line segment is called the length of the segment. We usually use the same symbol for the length of the line segment that we use for the segment itself. So $\overline{AB}$ could be used to represent the segment itself, but also the length of the segment.

In this example, the distance between points $A$ and $B$ is

$\overline{AB}=|3-(-2)|$

$\overline{AB}=|3+2|$

$\overline{AB}=|5|$

$\overline{AB}=5$

In this example, you could also count from $A$ to $B$ and get a distance of $5$. As you can see, sometimes it may be helpful to draw a number line in order to visualize the length of a line segment.

Finding the length of a line segment, and then its midpoint

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Using a number line to solve midpoint problems

Example

Points $S$, $T$, $U$, $V$, and $W$ lie in order on a number line. Point $U$ is at $-2$. Where are the rest of the points located?

$\overline{ST}=2$

$\overline{TU}=1$

$\overline{UV}=3$

$\overline{VW}=2$

If we plot point $U$ at $-2$, then $S$, $T$, $U$, $V$, and $W$ must be plotted this way:

We know point $U$ is on $-2$ and $\overline{TU}=1$. This lets us locate point $T$ at $-3$. Now we can use $\overline{ST}=2$ to find that $S=-5$ and $\overline{UV}=3$ to find that $V=1$. Now we can locate point $W$ by using $\overline{VW}=2$, so $W=3$.

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Let’s look at another example.

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The distance between two points on a line segment is called the length of the segment.

Example

Find $\overline{AB}$, if $\overline{AC}=12$ and $\overline{BC}=7$.

We know that $\overline{AC}=12$ and $\overline{BC}=7$. From the diagram, we also know that $\overline{AB}$ is part of $\overline{AC}$.

We can see that $\overline{AC}-\overline{BC}=\overline{AB}$, so we have $\overline{AB}=12-7=5$.

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Let’s look at one last example.

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Example

The locations of four points on a number line are $A=2$, $B=4$, $C=-3$, and $D=-6$. What is the value of $\overline{AB}+\overline{CD}$?

We can draw a number line to help solve the problem.

Now we can see that $\overline{AB}=|4-2|=2$ units and $\overline{CD}=|-3-(-6)|=3$ units. So $\overline{AB}+\overline{CD}=2 + 3 = 5$ units.

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