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Basic geometry: language and labels

            What I want to do in this video is give an introduction           

            to the language or some of the characters

            that we use when we talk about geometry.

            And I guess the best place to start

            is even think about what geometry means,

            as you might recognize the first part of geometry right

            over here.

            You have the root word geo-- the same word

            that you see in things like geography and geology.

            And this refers to the Earth.

            My E looked like a C right over.

            This refers to the Earth.

            And then you see this metry part.

            And you see metry in things like trigonometry

            as well and metry or the metric system.

            And this comes from measurement or measure.

            So when someone's talking about geometry, the word

            itself comes from Earth measurement.

            And that's not so bad of a name, because it

            is such a general subject.

            Geometry really is the study and trying

            to understand how shapes and space and things that we see

            relate to each other.

            So when you start learning about geometry,

            you learn about lines and triangles and circles.

            And you learn about angles.

            And we'll define all of these things more and more

            precisely as we go further and further on.

            But it also encapsulates things like patterns

            and three-dimensional shapes.

            So it's almost everything that we see.

            All of the visually mathematical things that we understand

            can in some way be categorized in geometry.

            Now, with that out of the way, let's just

            start from the basics, a basic starting point from geometry.

            And then we can just grow from there.

            So if we just start at a dot, that dot right over there,

            it's just that little point on that screen right over there.

            We literally call that a point.

            And I'll call that a definition.

            And the fun thing about mathematics

            is that you can make definitions.

            We could have called this an armadillo.

            But we decided to call this a point, which, I think,

            makes sense, because it's what we would call it

            in just everyday language as well.

            That is a point.

            Now, what's interesting about a point

            is that it is just a position.

            You can't move on a point.

            If you were at this point and if you moved in any direction

            at all, you would no longer be at that point.

            So you cannot move on a point.

            Now, there are differences between points.

            For example, that's one point there.

            Maybe I have another point over here.

            And then I have another point over here

            and then another point over there.

            And you want to be able to refer to the different points.

            And not everyone has the luxury of a nice colored

            pen like I do.

            Otherwise, they could refer to the green point

            or the blue point or the pink point.

            And so in geometry to refer to points,

            we tend to give them labels.

            And the labels tend to have letters.

            So for example, this could be point A.

            This could be point B. This would be point C.

            And this right over here could be point D.

            So someone says, hey, circle point C.

            You know which one to circle.

            You know that you would have to circle that point

            right over there.

            Well, that so far is kind of interesting.

            You have these things called points.

            You really can't move around on a point.

            All they do is specify a position.

            What if we want to move around a little bit more?

            What if we want to get from one point to another?

            So what if we started at one point

            and we wanted all of the points, including that point, that

            connect that point at another point, so all

            of these points right over here.

            So what would we call this thing, all of the points

            that connect A and B along a straight--

            and I'll use everyday language here--

            along a straight line like this?

            Well, we'll call this a line segment.

            In everyday language, you might call it a line.

            But we'll call it a line segment,

            because we'll see when we talk in mathematical terms

            a line means something slightly different.

            So this is a line segment.

            And if we were to connect D and C,

            this would also be another line segment.

            And once again, because we always

            don't have the luxury of colors, this one

            is clearly the orange line segment.

            This is clearly the yellow line segment.

            We want to have the labels for these line segments.

            And the best way to label the line segments

            is with its end points.

            And that's another word here.

            So a point is just literally A or B.

            But A and B are also the end points of these line segments,

            because it starts and ends at A and B. So let me write this.

            A and B are end points, another definition right over here.

            Once again, we could have called them

            aardvarks or end armadillos.

            But we, as mathematicians, decided

            to call them end points, because that

            seems to be a good name for it.

            And once again, we need a way to label these line segments that

            have the end points.

            And what's a better way to label a line segment

            than with its actual end points?

            So we would refer to this line segment over here,

            we would put its end points there.

            And to show that it's a line segment,

            we would draw a line over it just like that.

            This line segment down here, we would write it like this.

            And we could have just as easily written it like this.

            CD with a line over it would've referred to that same line


            BA with a line over it would refer to that same line


            Now, you might be saying, well, I'm not satisfied just

            traveling in between A and B.

            And this is actually another interesting idea.

            When you were just on A, when you were just on a point

            and you couldn't travel at all, you

            couldn't travel in any direction without staying

            on that point, that means you have zero options to travel in.

            You can't go up or down, left or right, in or out of the page

            and still be on that point.

            And so that's why we say a point has zero dimensions.

            Now, all of a sudden, we have this thing,

            this line segment here.

            And this line segment we can at least

            go to the left and the right all along this line segment.

            We can go towards A or towards B.

            So we can go back or forward in one dimension.

            So the line segment is a one-dimensional idea almost

            or a one-dimensional object.

            Although these are more abstract ideas.

            There is no such thing as a perfect line segment,

            because you can't move up or down on this line segment

            while being on it, while in reality, anything that we think

            is a line segment, even a stick of some type, a very

            straight stick or a string that has taught,

            that still will have some width.

            But the geometrical pure line segment has no width.

            It only has a length here.

            So you can only move along the line.

            And that's what we say it's one-dimensional.

            A point you can't move at all.

            A line segment you can only move in that back and forth

            along that same direction.

            Now, I just hinted that it can actually have a length.

            How do you refer to that?

            Well, you refer to that by not writing that line on it.

            So if I write AB with a line on top of it like,

            that means I'm referring to the actual line segment.

            Let me do this in a new color.

            If I say that AB is equal to 5 units-- it might be centimeters

            or meters or whatever, just the abstract units 5-- that

            means that the distance between A and B

            is 5, that the length of line segment AB is, actually, 5.

            Now, let's keep on extending it.

            Let's say we want to just keep going in one direction.

            Let me do this in a new color.

            Let's say I start at A. And I want to go to D.

            But I want the option to keep on going.

            So I can't go further in A's direction than A.

            But I can go further in D's direction.

            So this little idea that I just showed, essentially,

            it's like a line segment, but I can

            keep on going past this endpoint, we call this a ray.

            And the starting point for a ray is called the vertex, not

            a term that you'll see too often.

            You'll see vertex later on in other contexts.

            But it's good to know.

            This is the vertex of the ray.

            It's not the vertex of this line segment.

            So maybe I shouldn't label it just like that.

            And what's interesting about a ray

            is it's once again a one-dimensional figure.

            But you can keep on going past one of the endpoints.

            And the way that we would specify a ray

            is we would call it AD.

            And we would put this little arrow over on top of it

            to show that it is an array.

            And in this case, it matters the order

            that we put the letters in.

            If I put DA as a ray, this would mean a different ray.

            That would mean that we're starting at D.

            And then we're going past A. So this is not ray DA.

            This is ray AD.

            Now, the last idea that I'm sure you're thinking about

            is, well, what if I could keep on going in both directions?

            My diagram is getting messy.

            So let me introduce some more points.

            So let's say I have point E. And then

            I have point F right over here.

            And let's say that I have this object that

            goes through both E and F but just keeps

            on going in both directions.

            When we talk in geometry terms, this is what we call a line.

            Now, notice a line never ends.

            You can keep going in either direction.

            A line segment does end.

            It has end points.

            A line does not.

            And actually, a line segment can sometimes

            be called just a segment.

            And so you would specify line EF with these arrows just

            like that.

            Now, the thing that you're going to see most typically when

            we're studying geometry are these right over here,

            because we're going to be concerned with sides

            of shapes, distances between points.

            And when you're talking about any of those things, things

            that have finite length, things that

            have an actual length, things that don't go off forever

            in one or two directions, then you

            are talking about a segment or a line segment.

            Now, just to keep talking about new words

            that you might confront in geometry, if we go back

            talking about a line, I was drawing a ray.

            So let's say I have point X and point Y.

            And so this is line segment XY.

            So I could denote it just like that.

            If I have another point, let's say

            I have another point right over here.

            Let's call that point Z. And I'll introduce another word.

            X, Y, and Z all lie on the same line,

            if you would imagine that a line could

            keep going on and on forever and ever.

            So we can say that X, Y, and Z are collinear.

            So those three points are collinear.

            They all sit on the same line.

            And they also all sit on line segment XY.

            Now, let's say we're told that XZ is equal to ZY

            and they are all collinear.

            So that means this is telling us that the distance between X

            and Z is the same as the distance between Z and Y.

            So sometimes we can mark it like that.

            This distance is the same as that distance over there.

            So that tells us that Z is exactly halfway

            between X and Y. So in this situation,

            we would call Z the midpoint of line segment XY,

            because it's exactly halfway between.

            Now, to finish up, we've talked about things

            that have zero dimensions-- points.

            We've talked about things that have one dimension-- a line,

            a line segment, or a ray.

            You might say, well, what has two dimensions?

            Well, in order to have two dimensions,

            that means I can go backwards and forwards

            in two different directions.

            So this page right here or this video

            or this screen that you're looking at

            is a two-dimensional object.

            I can go right-left.

            That is one dimension.

            Or I could go up-down.

            And so this surface of the monitor you're looking at

            is actually two dimensions.

            You can go backwards or forwards in two directions.

            And things that are two dimensions,

            we call them planar.

            Or we call them planes.

            So if you took a piece of paper that extended forever,

            it just extended in every direction forever,

            that in the geometrical sense was a plane.

            The piece of paper itself, the thing

            that's finite-- and you'll never see

            this talked about in a typical geometry class.

            But I guess if we were to draw the analogy,

            you could call a piece of paper maybe a plane segment,

            because it's a segment of an entire plane.

            If you had a third dimension, then you're

            talking about our three-dimensional space.

            In three-dimensional space, not only could you

            move left to right along the screen or up and down,

            you could also move in and out of the screen.

            You also have this dimension that I'll try to draw.

            You could go into the screen.

            Or you could go out of the screen like that.

            And as we go into higher and higher mathematics,

            although it becomes very hard to visualize,

            you'll see that we can even start

            to study things that have more than three dimensions.

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