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.
