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- [Instructor] What I wanna
do in this video is give
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an introduction to the language
or some of the characters
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that we use when we talk about geometry.
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And I guess the best place to start
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is to even think about
what geometry means,
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'cause you might recognize
the first part of geometry.
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Right over here you
have the root word geo,
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the same word that you see in things
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like geography and geology,
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and this refers to the earth.
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This refers, my E look
like a C right over there,
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this refers to the earth.
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And then you see this metry part.
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And you see metry in things
like trigonometry as well.
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And metry, or the metric system,
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and this comes from measurement.
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This comes from measurement,
or measure, measurement.
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So when someone's talking about geometry,
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the word itself comes
from earth measurement
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and that's kind of not so bad of a name,
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because it is such a general subject.
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Geometry really is the study
and trying to understand
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how shapes and space
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and things that we see
relate to each other.
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So, you know, when you start
learning about geometry,
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you learn about lines
and triangles and circles
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and you learn about angles
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and we'll define all of these
things more and more precisely
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as we go further and further on,
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but it also encapsulate
things like patterns
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and three-dimensional shapes,
so it's almost everything
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that we see, all of the
visually mathematical things
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that we understand can in some way
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be categorized in geometry.
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Now, with that out of the way,
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let's just start from the basics,
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a basic starting point from geometry,
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and then we can just grow from there.
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So if we just start at a dot.
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That dot right over there is just a point,
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it's just that little point on
that screen right over there.
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We literally call that a point.
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And I'll call that a definition.
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And the fun thing about mathematics
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is that you can make definitions.
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We could have called this an armadillo,
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but we decided to call this a point,
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which I think makes sense because
it's what we would call it
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in just everyday language as well.
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That is a point.
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Now, what's interesting about a point
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is that it is just a position
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that you can't move on a point.
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If you were on this point and you moved
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in any direction at all,
you would no longer be
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at that point, so you
cannot move on a point.
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Now, there are differences between points,
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for example, that's one point there.
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Maybe I have another point over here
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and then I have another point over here
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and then another point over there.
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And you want to be able to
refer to the different points,
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and not everyone has the luxury of a nice,
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colored pen like I do,
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otherwise they could
refer to the green point
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or the blue point or the pink point,
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and so in geometry, to refer to points,
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we tend to give them labels,
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and the labels tend to have letters.
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So, for example, this could be point A,
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this could be point B,
this would be point C,
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and this right over here could be point D.
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So if someone says, "Hey, circle point C,"
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you know which one to circle.
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You know that you would have to circle
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that point right over there.
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Well, that so far, it's
kind of interesting.
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You have these things called points.
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You really can't move around on a point,
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all they do is specify a position.
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What if we wanna move
around a little bit more?
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What if we wanna get from
one point to another?
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So what if we started at one point
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and we wanted all of the
points, including that point
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that connect that point and another point?
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So all of these points right over here.
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So what would we call this thing,
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all of the points that connect
A and B along a straight,
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and I'll use everyday language here,
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along kind of a straight line like this?
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Well, we'll call this a line segment.
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In everyday language,
you might call it a line,
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but we'll call it a line segment,
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'cause we'll see when we
talk in mathematical terms,
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a line means something slightly different.
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So this is a line segment.
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And if we were to connect D and C,
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this would also be another
line segment, a line segment.
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And once again, because
we always don't have
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the luxury of colors,
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this one is clearly the
orange line segment.
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This is clearly the yellow line segment.
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We want to have labels
for these line segments.
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And the best way to
label the line segments
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are with its endpoints, and
that's another word here.
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So a point is just literally A or B,
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but A and B are also the
endpoints of these line segments,
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'cause it starts and ends at A and B.
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So let me write this A and B.
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A and B are endpoints, another
definition right over here.
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Once again, we could have
called them aardvarks
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or end armadillos, but
we, as mathematicians,
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decided to call them endpoints,
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because that seems to
be a good name for it.
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And once again, we need a way
to label these line segments
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that have the endpoints,
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and what's a better way
to label a line segment
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than with its actual endpoints?
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So we would refer to this
line segment, over here,
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we would put its endpoints there.
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And to show that it's a line segment,
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we would draw a line
over it, just like that.
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This line segment down here,
we would write it like this.
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And we could have just as
easily written it like this,
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CD with a line over it would have referred
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to that same line segment.
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BA, BA with a line over it would refer
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to that same line segment.
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And now you might be saying,
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"Well, I'm not satisfied just
traveling in between A and B."
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And this is actually
another interesting idea.
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When you were just on A,
when you were just on a point
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and you couldn't travel at all,
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you couldn't travel at
all in any direction
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while staying on that point,
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that means you have zero
options to travel in.
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You can't go up or down, left or right,
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in or out of the page and
still be on that point.
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And so that's why we say a
point has zero dimensions,
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zero dimensions.
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Now all of the sudden we have this thing,
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this line segment here,
and this line segment,
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we can at least go to the left
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and the right along this line segment.
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We can go towards A or towards B.
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So we can go back or
forward in one dimension.
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So the line segment is a one-dimensional,
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it is a one-dimensional idea almost,
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or a one-dimensional object,
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although these are more
kind of abstract ideas.
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There is no such thing as
a perfect line segment,
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because a line segment,
you can't move up or down
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on this line segment while being on it,
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while in reality, anything that
we think is a line segment,
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even you know, a stick of some type,
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a very straight stick,
or a string that is taut,
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that still will have some width,
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but the geometrical pure
line segment has no width,
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it only has a length here.
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So you can only move along the line,
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and that's why we it's one-dimensional.
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A point, you can't move at all.
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A line segment, you can only
move in that back and forth
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along that same direction.
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Now, I just hinted that it
can actually have a length.
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How do you refer to that?
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Well, you refer to that by
not writing that line on it.
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So if I write AB with a
line on top of it like that,
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that means I'm referring
to the actual line segment.
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If I say that, let me
do this in a new color,
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if I say that AB is equal to five units,
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it might be centimeters,
or meters, or whatever,
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just the abstract units five,
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that means that the distance
between A and B is five,
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that the length of line
segment AB is actually five.
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Now, let's keep on extending it.
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Let's say we wanna just
keep going in one direction.
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So let's say that I start at A,
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let me do this in a new color,
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let's say I start at
A and I wanna go to D,
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but I want the option of keep
on, I wanna keep on going,
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so I can't go further
in A's direction than A,
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but I can go further in D's direction.
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So this little, this
idea that I just showed,
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essentially, it's like a like segment,
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but I can keep on going
past this endpoint,
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we call this a ray.
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And the starting point for
a ray is called the vertex,
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not a term that you'll see too often.
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You'll see vertex later
on in other contexts,
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but it's good to know, this
is the vertex of the ray.
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It's not the vertex of this line segment,
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so maybe I shouldn't
label it just like that.
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And what's interesting about a ray,
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it's once again a one-dimensional figure,
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but you could keep on going
in one of the (murmurs),
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you can keep on going to or
past one of the endpoints.
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And the way that we would
specify a ray is we would say,
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we would call it AD and we
would put this little arrow
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over on top of it to show that is a ray.
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And in this case, it matters the order
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that we put the letters in.
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If I put DA as a ray, this
would mean a different ray.
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That would mean that we're stating at D
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and then we're going past A,
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so this is not ray DA, this is ray AD.
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Now, the last idea that I'm
sure you're thinking about is,
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well, what if I could keep
on going in both directions?
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So let's say I can keep going in, let me,
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my diagram is getting messy.
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So let me introduce some more points.
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So let's say I have point E
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and then I have point F right over here.
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And let's say that I have this object
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that goes through both E and F,
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but just keeps on going
in both directions.
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This is, when we talk in geometry terms,
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this is what we call a line.
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Now, notice, a line never ends.
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You can keep going in either direction.
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A line segment does end, it
has endpoints, a line does not.
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And actually, a line segment
can sometimes be called
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just a segment.
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And so you would specify line EF,
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you would specify line EF
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with these arrows just like that.
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Now, the thing that you're
gonna see most typically
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when we're studying geometry
are these right over here,
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because we're gonna be
concerned with sides of shapes,
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distances between points.
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And when you're talking
about any of those things,
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things that have finite length,
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things that have an actual length,
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things that don't go off forever
in one or two directions,
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then you are talking about
a segment or a line segment.
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Now, if we go back to a line segment,
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just to kind of keep
talking about new words
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that you might confront in geometry.
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If we go back talking about a line,
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that time I was drawing a ray,
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so let's say I have point X and point Y.
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And so this is line segment XY,
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so I could denote it just like that.
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If I have another point,
let's say I have another point
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right over here, let's call that point Z,
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and I'll introduce another word,
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X, Y, and Z are on the same,
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they all lie on the same line
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if you would imagine that
a line could keep going on
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and on forever and ever.
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So we can say that X,
Y, and Z are colinear.
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So those three points are
co, they are colinear.
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They all sit on the same line
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and they also all sit on line segment XY.
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Now, let's say we know, we're told that XZ
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is equal to ZY and they are all colinear.
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So that means this is
telling us that the distance
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between X and Z is the same as
the distance between Z and Y.
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So sometimes we can mark it like that.
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This distance is the same
as that distance over there.
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So that tells us that Z is
exactly halfway between X and Y.
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So in this situation we
would call Z the midpoint,
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the midpoint of line segment XY,
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'cause it's exactly halfway between.
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Now, to finish up, we've
talked about things
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that have zero dimensions, points.
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We've talked about things that
have one dimension, a line,
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a line segment, or a ray.
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You might say, well,
what has two dimensions?
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Well, in order to have two dimensions,
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that means I can go backwards and forwards
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in two different directions.
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So this page right here, or this video,
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or this screen that you're looking at
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is a two-dimensional object.
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I can go right, left,
that is one dimension,
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or I can go up, down.
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And so this surface of the
monitor you're looking at
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is actually two
dimensions, two dimensions.
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You can go backwards or
forwards in two directions.
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And things that are two
dimensions, we call them planar,
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or we call them planes.
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So if you took a piece of
paper that extended forever,
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it just extended in
every direction forever,
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that in a geometrical sense was a plane.
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The piece of paper itself,
the thing that's finite,
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and you'll never see this talked about
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in a typical geometry class,
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but I guess if we were
to draw the analogy,
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you could call a piece of
paper maybe a plane segment,
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because it's a segment of an entire plane.
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If you had a third dimension,
then you're talking about
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kind of our three-dimensional space.
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In three-dimensional space,
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not only could you move left
or right along the screen,
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or up and down, you could also move in
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and our of the screen.
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You could also have this
dimension that I'll try to draw.
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You could go into the screen
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or you could go out of
the screen like that.
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And as we go into higher
and higher mathematics,
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although it becomes
very hard to visualize,
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you'll see that we can
even start to study things
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that have more than three dimensions.
