Video Transcript
Straight Lines, Line Segments, and
Rays
In this lesson, we’re going to
learn how to identify points, lines, rays, line segments, and endpoints. Let’s start with a line and a
question. When is a line not a line? Well, this might seem like quite a
strange question to be asking. But hopefully by the end of this
video, you’ll understand that the answer to this question is “when we’re speaking
mathematically.” Because you know maths has its own
vocabulary, its own language of words and terms that mean very special things.
A good example of this is the word
“sum.” Perhaps you’ve heard people use the
word “sum” to mean calculation. I did a whole page of sums
today. Well, we know that if we’re
speaking mathematically, the word “sum” means something very definite. When we find the sum of some
numbers, it’s to do with addition; we add them together. So, if we’re speaking
mathematically, we can’t really use the word “sum” to mean any calculation. It has to mean addition.
And did you know, if we want to use
the language of maths properly, what we would normally describe as a line in
everyday life isn’t actually a line. Let’s use an example to find out
more. And as we do, we’re going to learn
some other words to do with lines, and each one is going to be part of this language
of maths.
This is Archie. He’s waiting for a bus. And as you do when you wait for a
bus, he’s standing still at an exact position. We could even label this
position. We’ll call it 𝐴. And in the language of maths, do
you know the word we use to describe an exact position like this? It’s a point. There are lots of points we could
draw, but only one point that Archie’s standing at. And that’s point 𝐴. Now let’s imagine time goes on and
a few more people join the queue. One of them is Archie’s good friend
Bert. Bert’s standing at an exact
position too. It’s another point, B for Bert, so
let’s call this point 𝐵.
Now what if we get a ruler and we
join up point 𝐴 with point 𝐵? It sort of represents our queue of
people, doesn’t it? Now normally you might say that
something that looks like this or this or this is a line. As we’ve said already, in this
video, we’re going to be talking the language of maths. And so this line isn’t a line. It’s part of a line. It’s what we call a line
segment.
Because this can be tricky to
understand, let’s write down what we actually mean by a line segment. A line segment, as we said already,
is part of a line with two endpoints. Here’s another new word for us. What do you think an endpoint
is? Well, it won’t surprise you to
learn that an endpoint is a point on the end. These are at the end of line
segments. So point 𝐴 and point 𝐵 are
endpoints. And this teaches us something very
important about lines in maths. A line goes on forever. The idea of something going on and
on and on forever is quite difficult. But in maths, this is what we mean
by a line. And because we can see that what
we’ve drawn here starts and then stops at two different endpoints, we can see that
it doesn’t go on forever. This is just part of a line that
goes on and on and on. It’s a line segment.
Now let’s imagine that this bus
doesn’t turn up and more and more people join the queue after Bert. The queue just goes on and on and
on and on. In fact, they never ever stop. It just keeps on going forever. Now we’ve reached the end of our
screen here. But how can we show that this goes
on and on and on because at the moment it looks like we’ve got another endpoint. Why don’t we draw an arrowhead at
the end? There we go. Now we’ve shown that it doesn’t
end. It carries on going forever.
But do you know, in maths, we still
can’t call this a line. Although the arrow at one end shows
that it does carry on forever in one direction, we can see that what we’ve drawn
does actually have an endpoint. It’s the point where Archie is
standing at the front of the queue. He’s still standing at a fixed
endpoint. And there’s a word for what we’ve
got here, another word to add to our maths vocabulary. What we can see here is a ray. Let’s think about the Sun’s rays
for a moment. They start at an exact
position. They come from the Sun, don’t
they? But the Sun’s rays travel in
straight lines and go on and on and on from that starting point. It’s a useful way to remember what
the word “ray” in maths means. We start from a point and then keep
on going on and on and on and on.
Now so far everyone’s standing very
politely in this queue waiting for the bus. So Archie turns around to say hello
to Bert. But unfortunately, just as he does
this, while his back’s turned, somebody pushes in and then someone else and someone
else. We’ve run out of room again. So, to show that our queue is going
on and on and on in this direction, let’s draw another arrowhead. And you’ll be pleased to know we’ve
just drawn a line, because in maths a line is a straight path. You can see what we’ve drawn is
straight, can’t you? It’s a straight path that continues
in both directions, not just in one direction like a ray, which doesn’t end. It just goes on and on and on and
on and on and on. We’ll stop there, but you know what
we’re trying to say.
Just before we answer some
questions based on these words that we’ve learned, let’s just go over them
again. Let’s get rid of the queue and just
draw this on a blank page. If we draw two dots on a page and
label them 𝐶 and 𝐷, what have we got? These are two points, aren’t
they? They’re two exact positions: point
𝐶 and point 𝐷.
Now if we take a ruler and we join
point 𝐶 up with point 𝐷, what have we got? Well, what we’ve got has two
endpoints, doesn’t it, one here and one here? What we’ve drawn starts and
stops. We’ve drawn a line segment. It’s part of a line, and it’s got
two endpoints. We could call it line segment
𝐶𝐷. It goes from 𝐶 and stops at
𝐷.
What else can we do? Well, what if we extend our line
segment past point 𝐷 on and on and on? And to show that it goes on and on
and on, we’ll put an arrow. What we’ve drawn now only has one
endpoint. We can’t really say what’s
happening at the other end because there is no other end. It just keeps on going on and on
and on. What we’ve got here is still part
of a line. It’s a ray. We know that rays start from a
point and continue in one direction. And because we start at point 𝐶,
we go through point 𝐷 and carry on going, we could call this ray 𝐶𝐷. So, if this is ray 𝐶𝐷, what’s
this? We can see it’s still a ray, but
this time the endpoint is point 𝐷. And our ray continues through point
𝐶 on and on and on. So it’s still a ray; it’s just
continuing in the opposite direction.
Finally then, what if we show that
what we’ve drawn goes on and on in both directions? It doesn’t have any endpoints. We know that a straight path that
continues in both directions and never ends is called a line. And we could call this line 𝐶𝐷
because it passes through points 𝐶 and 𝐷.
Now how well do you think you’ve
learned these words? Let’s answer some questions now
where we have to practice our knowledge of what they mean.
Which of the following figures is a
straight line?
In this question, we’re given five
pictures or figures. And we’re told that we need to find
which one of them is a straight line. Now perhaps before we start, we can
see that one of these figures is definitely not a straight line. Figure (e) is curved, isn’t it? So we can be sure that (e) isn’t
the right answer. But what about the others? Let’s go through each one and look
at how they’re different.
Figure (a) is a shape. It looks like the letter H, doesn’t
it? Figure (b) definitely looks like we
could call it a straight line. And figure (c), in fact this looks
almost the same as figure (b), except it has two arrowheads, one on either end. But then (d) also looks like a
straight line. This time it’s only got one
arrowhead.
Well, to answer this question, we
need to think about what the word “straight line” means in maths because it means
something very definite. In maths, a line is a straight path
that continues in both directions and does not end. If we think about figure (b) for a
moment, we can see that it definitely starts and stops. It’s a certain length, isn’t
it? It has what we call two
endpoints. Now, if a line goes on and on and
on in both directions, we can see that this is only part of our line, doesn’t go on
and on at all. Figure (b) is a line segment. And although we might have said
that the shape in figure (a) is made up of lots of straight lines, we can see now
that each one starts and stops. This shape is made of lots of line
segments.
It looks like our answer is either
going to be figure (c) or (d). Which one of these is a straight
path that continues in both directions and doesn’t end? Well, if we look at figure (d) for
a moment, we can see that it does have a start point. At the other end, it goes on and
on. We can tell this because we can see
an arrowhead. But it doesn’t go on and on in both
directions. It’s only in one direction. This is what we call a ray. And so if we look at figure (c), we
can see arrowheads at both ends. This is a straight path that does
continue in both directions. It doesn’t matter which way we
travel. It’s going to go on and on and
on. The figure that is a straight line
is figure (c).
Select the correct name for this
object. Is it a point, a line, a ray, a
line segment, or an angle?
The object that this question is
talking about is this here. Notice how we can’t just say “this
line here,” because although we might look at this and say it’s a line, we’ve
actually got five different answers to choose from. So perhaps it isn’t a line. In everyday life, we’d call it a
line. But this is a maths question. And sometimes words in maths have
very definite meanings. Let’s go through each of our
possible answers and see what each one means. Then perhaps we’ll be able to say
what the name of the object is.
Our first word is “point.” We know that a point is an exact
position. There’s a point here, here,
here. There are points along our
object. And there are endpoints at either
end of our object. But we can’t use the word “point”
to describe the whole object. This isn’t a point.
Is it a line? Well, it definitely looks like a
line. And as we’ve said already, in
everyday life, we would describe it as a line. But in maths, the word “line” means
something very definite. It’s a path that has no
endpoints. It keeps on going in both
directions. Now, if our object had arrowheads
at either end, we’d see that it didn’t have any endpoints and it carried on going in
both directions. We then could say it was a
line. As we’ve said already, our object
does have two endpoints. We can’t use the word “line” to
describe it.
What about the word “ray”? A ray has one endpoint, and it
continues in one direction. It looks like this. Can you see the orange arrow that’s
pointing to the word “ray”? That’s a ray too. It has an endpoint, and it
continues in one direction. Our object doesn’t continue in any
direction. It has two endpoints. So we know it can’t be a ray.
Is it a line segment? A line segment is part of a
line. It has two endpoints. In other words, it starts and it
stops. Well, our object starts and stops,
doesn’t it?
Just to check the last word, we
know we don’t have an angle either. An angle is the number of degrees
of turn that are measured between two rays. And these two rays begin from the
same endpoint. Our object has two endpoints. It starts and it stops. Although we might be tempted to
call it a line, we know the correct name for this object is a line segment.
Which is the starting point of the
ray?
We have four points to choose from:
𝑁, 𝐴, 𝑋, or 𝐶. This question talks about a
ray. And in order to answer the
question, we need to understand what a ray is. In maths, a ray is the name we give
for part of a line. It has one endpoint, and it
continues in only one direction. So, on our diagram, can you see the
ray? It’s this path here. It’s part of a line with one
endpoint. And this arrowhead at the other end
shows that it continues in only one direction. This arrowhead is a way of showing
us that the ray is going to continue all the way to the edge of the video screen,
off the screen, and on and on and on.
So now that we know what a ray is,
which one of our four points is the starting point of the ray? Well, we know the answer can’t be
𝑁 or 𝐶. These points are nowhere near our
ray. Point 𝐴 is a point on the ray, but
it’s not where our ray begins. It might sound strange, but the
starting point of our ray is the endpoint. It’s point 𝑋. Let’s draw over it in blue to show
what’s happening. We’ll start at point 𝑋. And our ray continues in a straight
line through point 𝐴 and on and on and on. The starting point of the ray is
point 𝑋.
What have we learned in this
video? We’ve learned how to identify
points, lines, line segments, endpoints, and rays.
