Written by

Malcolm McKinsey

Fact-checked by

Paul Mazzola

## Line segment

A**line segment**is a portion or piece of a line that allows you to build polygons, determine slopes, and make calculations. Its length is finite and is determined by its two endpoints.

The line segment is a snippet of the line. No matter how long the line segment is, it is finite.

### Line segment symbol

You name a line segment by its two endpoints. The shorthand for a line segment is to write the line segments two endpoints and draw a dash above them, like$\overline{CX}$CX:

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## What is a line?

The definition of a**line**is the set of points between and beyond two points. A line is infinite in length. All points on a line arecollinear points.

### Straight line symbol

In geometry, the straight line symbol is a line segment with two arrowheads at its ends, like $\overleftrightarrow{CX}$CX. You identify it with two named points, indicated by capital letters. Pick a point on the line and give it a letter, then pick a second; now you have the name of your line:

## Rays

A**ray**is a part of a line that has one endpoint and goes on infinitely in only one direction. You cannot measure the length of a ray.

A ray is named using its endpoint first, and then any other point on the ray. In this example, we have **PointB**and **PointA**($\overset{\to }{BA}$BA→).

## Measuring line segments

A line segment is named by its endpoints, but other points along its length can be named, too. Each portion of the line segment can be labeled for length, so you can add them up to determine the total length of the line segment.

### Line segment example

Here we have line segment$\overline{CX}$CX, but we have added two points along the way, **PointG**and **PointR**:

To determine the total length of a line segment, you add each segment of the line segment. The formula for the line segment **CX **would be: **CG + GR + RX = CX**

7units line segmentCG

5units line segmentGR

3units line segmentRX

$7+5+3=15$7+5+3=15

That's a total of **15** units of length for $\overline{CX}$CX.

### Coordinate plane

A**coordinate plane**, also called a**Cartesian plane**(thank you, René Descartes!), is the grid built up from a x-axis and a y-axis. You can think of it as two perpendicular number lines, or as a map of the territory occupied by line segments.

To determine the length of horizontal or vertical line segments on the plane, count the individual units from endpoint to endpoint:

To determine the length of line segment$\overline{LM}$LM, we start at **PointL**and count to our right five units, ending at **PointM**. You can also subtract the x-values:

$8−3=5$8−3=5

. For vertical lines, you would subtract y-values.

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When working in or across **QuadrantsII**,**III**and**IV**, recall that subtracting a negative number really means adding a positive number.

## How to find the length of a diagonal line segment on a coordinate plane

Use the**Pythagorean Theorem**to calculate line segment lengths of diagonals on coordinate planes. Recall that the Pythagorean Theorem is${a}^{2}+{b}^{2}={c}^{2}$a2+b2=c2for any right triangle.

A diagonal on a coordinate grid forms the hypotenuse of a right triangle, so can quickly count the units of the two sides:

Count units straight down from **PointJ**to the x-value**2**(which lines up with **PointL**):

$8-2=6$8−2=6

So, line segment $\overline{JK}=6$JK=6

Count units straight across from **PointK**to **PointL**:

$6-(-3)=9$6−(−3)=9

So, line segment $\overline{KL}=9$KL=9. Now we have ${6}^{2}+{9}^{2}={c}^{2}$62+92=c2:

$36+81={c}^{2}$36+81=c2

$117={c}^{2}$117=c2

$10.816=c$10.816=c

The length of line segment

$\overline{JL}$JL

is approximately$10.816 units$10.816units.

## The distance formula

A special case of the Pythagorean Theorem is the**Distance Formula**, used exclusively in coordinate geometry. You can plug in the two endpoint x- and y- values of a diagonal line and determine its length. The formula looks like this:

$D=\sqrt{({x}_{2}-{x}_{1}{)}^{2}+({y}_{2}-{y}_{1}{)}^{2}}$D=(x2−x1)2+(y2−y1)2

To use the distance formula, take the squares of the change in x-value and the change in y-value and add them, then take that sum's square root.

The expression$({x}_{2}-{x}_{1})$(x2−x1) is read as**the change in x**and$({y}_{2}-{y}_{1})$(y2−y1)is**the change in y**.

Imagine we have a diagonal line stretching from **PointP(6,9)**to **PointI(-2,3)**, and you want to measure the distance between the two points.

The distance formula makes this an easy calculation:

$D=\sqrt{(-2-6{)}^{2}+(3-9{)}^{2}}$D=(−2−6)2+(3−9)2

$D=\sqrt{{\left(-8\right)}^{2}+{\left(-6\right)}^{2}}$D=(−8)2+(−6)2

$D=\sqrt{64+36}$D=64+36

$D=\sqrt{100}$D=100

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$D=10$D=10

Using the Distance Formula, we find that line segment$\overline{PI}=10units$PI=10units.

## Examples of line segments in real life

Real-world examples of line segments are a pencil, a baseball bat, the cord to your cell phone charger, the edge of a table, etc. Think of a real-life quadrilateral, like a chessboard; it is made of four line segments. Unlike line segments, examples of line segments in real life are endless.