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# Terms & labels in geometry

Sal discusses geometry terms like point, line, and ray. He also shows to label them. Created by Sal Khan.

Video transcript

What I wanna 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 comes this refers to the earth This refers My E looks like a C right over there 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 This comes from measurement or measure Measurement So when someone's talking about Geometry, The word itself comes from earth measurement And that's kind of not so bad of a name Because it is such of 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 you know 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 also encapsulated things like patterns And three dimensional shapes So it's almost everything that we see All the Math- visually Mathematical things that we understand Can in someway be categorized in Geometry Now with that out of the way, let's just start from the basics So 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 a point It's just that little point on that screen right over there we'd literally call that a point And I'd call that a definition And the fun thing about Mathematics is That you can make definitions We could've called this We could've called this an armadillo But we decide to call this a point which I think make sense Because it's just what we would call it in everyday language as well That is a point Now what's interesting about a point is that it is just a position; that you can't move on a point If you moved if you were are 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 B; this would be point C; and this right over here could be point D So if someone says hey circle point C- you know which one to circle You know that you would have to circle, You would have to circle that point right over there Well that so far it's kind of interesting You have these things called points You can't really move around on a point All they do is specify a position What if we wanna move around little bit more? What if we wanna get from one point to another? So what if we took We started at one point and we wanted we wanted all of the points including that point that connect that point in another point So all of these points right over here So what would we call 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 kind of a straight line like this, Well we'll call this a line segment In everyday language you might call it a line But well call it a line segment coz we'll see in ge 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 A 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 labels for these line segments And the best way to label the line segments are 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 Coz it starts and ends at A and B So let me write this A and B A and B are end points Another definition right over here We, once again, we could've called them aardvarks or end armadillos But we as mathematicians decide 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 With 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've just as easily written it like this: CD with a line over it Would've refer to the same line segment BA, BA with a line segment with a line over it would refer to that same line segment And now you might be saying well I'm not satisfied Just travelling 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 at all in any direction without, while 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 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 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 It is a one dimensional idea almost or one dimensional object Although these are more kind of abstract ideas There is no such thing as a perfect line segment Because everything a line segment you can't move 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 you know, a stick of some type- a very straight stick or a string that is thought 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 why 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 said to you 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 That means I'm referring to the actually line segment If I say that let me do this in a new color If I say that AB is equal to five units It might be centimeters or meters whatever I just the abstract unit is five, That means that the distance between A and B is five That the length of line segment AB is actually five Now let's keep on extending it Let's say we wanna just keep going in one direction So let's say that I start at A Let me do this in a new color Let's say I start at A and I wanna go to D But I wanna I want the option of keep on I wanna 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 this idea that I just showed; This essentially a cycle line segment But I can keep on going past this end point we call this 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 context 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 once again its one a dimensional figure But you could keep on going in one of the direct- You can keep on going toward past one of the end points And the way that we would specify a ray is we would say We would call it AD and we would put this little arrow over on top of it To show that is a ray And in this case it matters the order That we put the letters in If I put DA if I put DA as a ray This would mean a different ray That would mean 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 What if I could keep on going in every, in both directions? So let's say I can keep going in My diagram is getting messy So let's 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 can That goes through both E and F but this keeps on going in both directions This is, when we talk in geometry terms, this is what we call a line Now know this 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 you would specify line EF with these arrows just like that Now the thing you're gonna see most typically When we're studying Geometry are these right over here Because we're gonna be concerned With sides of shapes, distances between points And we'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're talking about a segment or a line segment Now if we go back to a line segment just to kind of keep talking about new words- That you might confront in Geometry If we go back talking about a line That's 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 specify it, 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 XY And Z are on the same they're all lined 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 XY and Z are collinear So those three points are co- that they are collinear They all sit on the same line or And they also all sit on the line segment XY Now let's say we know, 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 we can mark it like that This distance is the same as that distance over there So that tells us that Z is exactly half way between X and Y So in this situation we would call Z the midpoint The midpoint of line segment XY coz it's exactly half way 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 the screen that you're looking at is a two dimensional object I can go I can go right left that is one dimension; or I can go up down And so this surface of the monitor you're looking at Is actually two dimensions 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 if you took a piece of paper Then extend it forever; just extended in every direction forever That, in a 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 kind of A three dimensional space In three dimensional space, not only could you move left or right along the screen Or up and down, you could also move in and out of the screen You can 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 to 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