Lesson Video: Straight Lines, Line Segments, and Rays

In this video, we will learn how to identify points, lines, rays, line segments, and endpoints.

10:56

Video Transcript

In this video, we’ll learn how to identify points, lines, rays, line segments, and endpoints and the associated notation we use to describe these. As a standalone, this skill might seem somewhat unimportant. But once we can begin to describe these using mathematical language and notation, this opens up a whole world to us in geometry, from simply describing polygons through to trigonometry and geometrical proof. Let’s begin by looking at some of these definitions.

The definitions we’re going to look at all stem from this first definition. It’s the definition of a point. A point is simply a position or location. It has no size, such as width or length or depth, and we represent it using a dot. Here, we’ve drawn a dot representing point 𝐴. And once we have the definition of a point, let’s form the definition of a line.

A line is a straight set of points that extend infinitely in two directions. It only has one dimension and that’s its length. We represent it as shown. These two arrows show that it extends infinitely in those directions. If we represent this line as passing through the points 𝐴 and 𝐵, we see we define the line as shown with an arrowhead. We say that points 𝐴 and 𝐵 are colinear; they’re on the same line.

Now, what do we mean by a ray? A ray is a portion of a line. It starts at a given point, and we call this the endpoint, and then it goes off in a particular direction to infinity. A ray whose endpoint is 𝐴 and then passes through 𝐵 is represented as shown with a single arrow defining its direction. The endpoint that we could also alternatively consider as the starting point of our ray is 𝐴. Formerly, though, we say that the endpoint is the point at which a ray ends.

There’s one further definition we need, and that’s the definition of a line segment. A line segment is a part of a line that’s bounded by two distinct endpoints. And it contains every point on the line between those points. It’s always the shortest distance between these points. In this line segment, 𝐴 and 𝐵 are the endpoints. And so we define the line segment 𝐴𝐵 by using a bar as shown. Now that we have some definitions, we’ll look at some questions.

What has been drawn?

Now, if we look carefully, we can informally say it looks like we have part of a line. We’ll be able to identify its formal name by looking at what’s happening at the end of this line drawn. We have two arrows. Now, these arrows tell us that this line extends in both directions. In fact, it extends infinitely in both directions. And so we recall that the formal mathematical definition of a line is a straight set of points that extend infinitely in both directions. And so we can say that, here, we have a line.

What has been drawn?

It looks like we’ve been given a line or certainly a portion of a line. The key here is to consider what’s happening at the ends of our line. At the ends of our line, we have these two solid dots. They show us that this line ends in both these places. Now, in fact, a line is a straight set of points that extend infinitely in both directions. We can see that our straight set of points have been bounded at both ends.

And so we recall a second definition. We say that a line segment is a part of a line bounded by two endpoints. We can see that our straight set of points is indeed bounded by two endpoints. And so we have a line segment.

In our next example, we’ll consider how these definitions can help us to define polygons.

How many line segments does this shape have?

We begin by recalling the definition of a line segment. We know that a line segment is a part of a line that’s bounded by two distinct endpoints. And it contains every point of the line between those endpoints. Since a line is a straight set of points that extend infinitely in two directions, we know that, by definition, a line segment must be straight also.

And so to answer this question, we’re simply going to count the number of straight portions of line. We’ll highlight them as we go. There’s one here. Then there’s another here, so that’s two. We have a smaller one down here; that’s three. And we have another one here. That gives us a total of four.

It doesn’t actually matter that the line segments are different lengths. We have four distinct line segments in our shape. In fact, this allows us to define our shape. It’s a polygon with four sides. Remember, polygons have straight edges, and so we can say that this shape is a quadrilateral.

In our next two examples, we’ll look at how the definitions we’ve seen so far gonna help us to answer questions about points.

Using the given figure, determine whether the following is true or false. The line segment that is passing through points 𝐵 and 𝐷 is also passing through point 𝐶.

In order to answer this question, we’re going to need to recall what we actually mean by a line segment. A line segment is a part of a line, and that part is bounded by two distinct endpoints. We’re told that our line segment passes through points 𝐵 and 𝐷. In fact, its two endpoints, as we can see from the diagram, are 𝐵 and 𝐷. So we represent it as shown. It’s 𝐵𝐷 with a line above it.

So let’s compare point 𝐶 to points 𝐵 and 𝐷. It sits on the line passing through these points and halfway roughly between points 𝐵 and 𝐷, halfway between the endpoints of the line segment. By definition, a line segment must be straight and contain every point on the line between its endpoints. So we can say this is true. The line segment 𝐵𝐷 passes through point 𝐶.

Let’s consider another example like this.

Does the given figure allow you to conclude that the ray starting at 𝐶 and passing through 𝐷 passes through point 𝐵. In order to answer this question, we need to recall what we mean by a ray. We say that a ray is a portion of a line. Unlike a line segment, though, it extends from a single endpoint and goes off in a particular direction to infinity. Our ray starts at 𝐶, so 𝐶 is the endpoint, and it passes through 𝐷. We can represent this using a single arrow 𝐶𝐷 as shown.

So does this ray pass through point 𝐵? Well, no, we said that point 𝐶 is the endpoint. It’s the point at which a line segment or ray ends. It does not extend back out past this endpoint in the direction of 𝐵. And so we can say no, the given figure does not allow us to conclude that the ray that starts at 𝐶 and passes through 𝐷 also passes through point 𝐵.

Using the given figure, answer the following: Does point 𝐶 lie on the straight line?

We begin by recalling the definition of the word line. We say mathematically that a line is a straight set of points that extend infinitely in two directions. As shown in the diagram, it’s represented with the two arrows. Those arrows indicate to us that the line continues in both directions. And so if we were to continue this line downwards on our diagram, we could deduce that this line most likely passes through point 𝐸. But can it pass through point 𝐶? Well, no, we can see quite clearly that points 𝐶 and 𝐷 are not on that straight line. And so the answer is no.

In our final example, we’ll look at how to apply the mathematical notation.

The diagram shows a number of points, 𝐴, 𝐵, 𝐶, and 𝐷. Use mathematical notation to describe the straight set of points from 𝐵 and through 𝐷, the straight set of points from 𝐴 to 𝐶, the straight set of points through 𝐴 and 𝐶 that extend infinitely in both directions.

We’ll begin by looking at the straight set of points from 𝐵 and through 𝐷. That’s all of these. This set of points has an endpoint, but it extends infinitely through 𝐷 and beyond. We can therefore say that this set of points is a ray. We represent a ray using a one-sided arrow as shown, and we use the endpoint as the first letter. And so we can say that the straight set of points from 𝐵 and through 𝐷 is shown. It’s 𝐵𝐷 with an arrow.

Next, we’re told to describe the straight set of points from 𝐴 to 𝐶. That’s this line shown. We know that this point has a very specific start point and endpoint. But in fact, we call these both endpoints. And so the straight set of points between these endpoints must be a line segment. We use a bar to represent a line segment. And so the straight set of points from 𝐴 to 𝐶 is represented by 𝐴𝐶 with a bar.

Finally, we look at the straight set of points through 𝐴 and 𝐶 that extend infinitely in both directions. And so we see that this is all of this line here. The two arrows tell us that this line extends infinitely. A straight set of points that extends infinitely in two directions is a line. And we use a two-sided arrow, much like in the picture, to represent this. It’s 𝐴𝐶 with a two-sided arrow.

In this video, we learned first that a point is a location. It has no size like width nor length, and it’s represented by a dot. We learned that a line is a straight set of points that extend infinitely in both directions. We used two specific points to describe it and a double-sided arrow above these points to represent that we have a line. When points lie on the same line, they’re called colinear points.

We saw that a ray is a specific portion of a line. It has one endpoint — in our diagram, that’s 𝐴 — and it extends infinitely in a given direction. Here, that’s through point 𝐵. We represent this with a single-sided arrow. Then a line segment is also part of a line. But this time it has two distinct endpoints, and it contains every point on the line between those points. We represent this using a bar, as shown.