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Terms & labels in geometry

Learn about geometry terms like point, line, and ray.  We will also learn how to label them. Created by Sal Khan.

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  • female robot grace style avatar for user Giles Hutchings
    Do scalars just have magnitude and vectors have magnitude and direction?
    (0 votes)
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  • mr pink red style avatar for user Sting-Rayy42
    Earth Measurement. can someone explain?
    (0 votes)
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  • starky sapling style avatar for user REHEHEHEH
    (5 votes)
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  • leafers tree style avatar for user Incygnius
    Can you technically move on a point because the point has a diameter/radius?
    (4 votes)
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  • boggle blue style avatar for user x.asper
    I still don't quite understand the prospects of a line. Why does it have no width? If a line (or line segment) has no width, then by prospect it wouldn't actually exist. Everything that "exists" has at least a length and width. Am I right?
    (3 votes)
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    • mr pink green style avatar for user David Severin
      So Geometry always starts with the three undefined terms A point has no length, no width, and no height. A line has length, but no width and no height. A plane has length and width, but no height. Using these undefined terms, we create all of Geometry from there. As you note, they really do not "exist," but that does not mean that we cannot use them for both theoretical and practical perposes. If you think about all of planar geometry, you are working in a 2 dimensional world which does not exists, for it to "exist" in the sense that you are talking about, we work in 3 dimensions. Water and other liquids do not have a length, width, and height per se because they take the shape of the container or spread out in path of least resistance and gases are even more loose, but they operate in 3 dimensions. So as soon as we draw a line on paper, it is not really a line because it has length, width, and height, but we can still represent it as a line. A triangle does not exist, but that does not me that we cannot have objects with a particular triangular shape.
      (2 votes)
  • purple pi pink style avatar for user You-Know-Who
    At , sal says that we can go in one dimension....
    therefore, is dimension the same thing as direction.
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  • hopper happy style avatar for user Megan D.
    What would it mean for an object to be beyond 3 dimensional?
    (1 vote)
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    • orange juice squid orange style avatar for user Tucker Quering
      In theory, an object can have any positive integer number of dimensions. To understand what an n-dimensional object would look like, let's start with something we can thoroughly understand.

      Imagine a 2-D plane, extending forever in all directions, and having no substance. Now imagine that it is covered with 2-D structures, and populated by 2-D people. Now imagine we have a 3-D shape, a sphere. If the sphere intersected the plane, someone from a 2-dimensional perspective would only be able to see a circle, because this would be the only portion of the sphere intersecting the plane they live on. And, if the sphere were to travel all the way through the plane, a 2-D observer would at first see a point, which would slowly expand into a circle, and grow larger and larger. After it passed the halfway mark, the circle would begin to shrink, become a point, and vanish.

      Now let's imagine the 3-D cell that we live in, extending out forever in all directions (we can't verify if our 3-D cell does extend infinitely, but we will assume it does for these purposes). Now let's imagine a hypothetical 4-D shape called a 3-sphere. A 3-sphere would extend in all 4 dimensions equally from a central point, just like a sphere does in 3 dimensions, and a circle does in 2 dimensions. If a 3-sphere were to intersect the 3-D cell we live in, we would only be able to see a sphere at the area of intersection. If the 3-sphere were to pass all the way through our 3-D cell, we would at first see a point, slowly expanding into a sphere. The sphere would begin to shrink after the 3-sphere passed the halfway mark, becoming ever smaller spheres, and finally a point before it vanished.

      Theoretical geometry about n dimensions is a very interesting topic. Just try searching google for "tesseract" or "3-sphere" if you want to know more. Also, there is a very interesting book about it called Flatland: A Romance of Many Dimensions (it involves no actual romance). It explores things like this from the perspective of a 2-D fictional character. If you find this intriguing, I highly recommend reading it. Here's the link:

      (5 votes)
  • aqualine ultimate style avatar for user Vin O'Shea
    An embodiment of 0 dimensions specify a "point;" 1 dimension specifies a "line;" 2 dimensions specify a "plane;" what would the specification for a 3 dimensional embodiment be called?
    (3 votes)
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  • male robot hal style avatar for user Albanianblur
    Can there be 5 or more dimensions?
    (2 votes)
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    • piceratops ultimate style avatar for user Lauren Mauricio
      In short: maybe.

      Most people are familiar with four dimensions: width (represented by the x-axis), height (represented by the y-axis), depth (represented by the z-axis), and time.

      Some physicists have theorized that there are more than these four dimensions. String Theory, for example, makes such a claim.

      It takes a lot of math to understand, and it has not been proven. And most people have a hard time comprehending and conceptualizing such theories. But that does not necessarily mean it's not true.

      Maybe you'll be the one to find out for sure!

      - Lauren
      (2 votes)
  • aqualine ultimate style avatar for user J Choi
    I've been told that nothing in the world is truly two-dimensional. Am I right?
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    • piceratops ultimate style avatar for user Just Keith
      That is both true and yet not true, depending on what you mean.
      It true that no material object is truly 2D. All of them have some sort of thickness, even if it is only one atom thick (as in the case of a form of carbon called graphene).

      However, there are phenomena, not objects, that are truly 2D. Shadows are 2D. The image on a computer, TV or movie screen (including those called 3D) are 2D. Your reflection in a standard mirror is 2D (note, there are types of mirrors that do have 3D images).
      (7 votes)

Video transcript

- [Instructor] 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 to even think about what geometry means, 'cause 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. This refers, my E look 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 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 it also encapsulate 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 is just a point, 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 that you can't move on a point. If you were on this point and 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 if 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, it's 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 wanna move around a little bit more? What if we wanna 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 and 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 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 we'll call it a line segment, 'cause 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, 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 endpoints, and that's another word here. So a point is just literally A or B, but A and B are also the endpoints of these line segments, 'cause it starts and ends at A and B. So let me write this A and B. A and B are endpoints, another definition right over here. Once again, we could have called them aardvarks or end armadillos, but we, as mathematicians, decided to call them endpoints, 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 endpoints, and what's a better way to label a line segment than with its actual endpoints? So we would refer to this line segment, over here, we would put its endpoints 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 have referred to that same line segment. BA, BA with a line over it would refer to that same line segment. And 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 at all in any direction 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 the 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 a one-dimensional object, although these are more kind of abstract ideas. There is no such thing as a perfect line segment, because a line segment, 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 taut, 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 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, that means I'm referring to the actual 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, or whatever, just the abstract units 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 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, essentially, it's like a like 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, it's once again a one-dimensional figure, but you could keep on going in one of the (murmurs), you can keep on going to or past one of the endpoints. 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 as a ray, this would mean a different ray. That would mean that we're stating 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? So let's say I can keep going in, let me, 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. This is, 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 endpoints, 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 that 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 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, 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 time 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 are on the same, they 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 colinear. So those three points are co, they are colinear. They all sit on the same line and they also all sit on line segment XY. Now, let's say we know, we're told that XZ is equal to ZY and they are all colinear. 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, the midpoint of line segment XY, 'cause 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 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 piece of paper that extended forever, it 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 our 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 our of the screen. You could 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.