Lines, line segments, and rays are found everywhere in geometry. Using these simple tools, you can create parallel lines, perpendicular bisectors, polygons, and so much more. In this lesson, you will learn the definitions of lines, line segments, and rays, how to name them, and few ways to measure line segments.

- What are Lines, Line Segments, and Rays?
- Line Segment
- What is a line?
- Rays
- Measuring Line Segments
- How to Find the Length of Diagonal Segment on a Coordinate Plane
- The Distance Formula
- Examples of Line Segments In Real Life

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.

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 $\overline{CX}$:

You symbolize a line segment on drawing paper by using a straightedge to make a line and placing two dots at its ends, identified with capital letters; these are the endpoints of a line segment:

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 are collinear points.

In geometry, the straight line symbol is a line segment with two arrowheads at its ends, like $\overleftrightarrow{CX}$. 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:

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 Point $B$ and Point $A$ ($\overrightarrow{BA}$).

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.

Here we have line segment $\overline{CX}$, but we have added two points along the way, Point $G$ and Point $R$:

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

$7$ units line segment $CG$

$5$ units line segment $GR$

$3$ units line segment $RX$

$7+5+3=15$ units of length for $\overline{CX}$

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 line segment $\overline{LM}$'s length, we start at Point $L$ and count to our right five units, ending at Point $M$. You can also subtract the x-values: $8-3=5$. For vertical lines, you would subtract y-values.

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

Use the **Pythagorean Theorem** to calculate line segment lengths of diagonals on coordinate planes. Recall that the Pythagorean Theorem is ${a}^{2}+{b}^{2}={c}^{2}$ for 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 Point $J$ to the x-value $2$ (which lines up with Point $L$):

$8-2=6$, so line segment $\overline{JK}=6$

Count units straight across from Point $K$ to Point $L$:

$6-(-3)=9$, so line segment $\overline{KL}=9$

Now we have ${6}^{2}+{9}^{2}={c}^{2}$:

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

$117={c}^{2}$

$10.816=c$

The length of line segment $\overline{JL}$ is approximately $10.816$ units.

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:

$D=\sqrt{({x}_{2}-{x}_{1}{)}^{2}+({y}_{2}-{y}_{1}{)}^{2}}$

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 $({x}_{2}-{x}_{1})$ is read as **the change in x** and $({y}_{2}-{y}_{1})$ is **the change in y**.

Imagine we have a diagonal line stretching from Point $P$ $(6,9)$ to Point $I$ $(-2,3)$, and you want to measure the distance between the two points.

The Distance Formula makes this an easy calculation:

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

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

$D=\sqrt{64+36}$

$D=\sqrt{100}$

$D=10$

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

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.

After working your way through this lesson and video, you should be able to:

- Define and identify a straight line, line segments, and rays
- Construct line segments
- Measure the distance between two points
- Work with line segments on a Cartesian or coordinate plane
- Apply the Pythagorean Theorem and Distance Formula to calculate lengths on a coordinate plane

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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