High school geometry
Explore geometry fundamentals, including points, line segments, rays, and lines. Understand dimensions and how these elements form shapes and patterns. Learn key geometric terms like colinear points, midpoints, and vertices, and enhance your knowledge of geometry. Improve your skills and discover the world of shapes and space. Created by Sal Khan.
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- This video has a number of terms defined to help understand geometry. Where can I find each term discussed in this video?(640 votes)
- The following terms can be found at these approximate time markers:
Definition of name Geometry –0:10
Point (0 dimensions) -1:40
Line segment -3:10
End Points –4:10
Plane (2 dimensions) -11:30
3 dimensions –12:25(1409 votes)
- 2:30Earth Measurement. can someone explain?(136 votes)
- He's just referring to the origin of the term; only the Greeks know precisely why, but it seems likely that it refers to the measuring of "earth", meaning "the land" or "the physical world" in its various aspects rather than an actual attempt to measure our planet.(251 votes)
- At12:53, Sal mentions that some things have more than three dimensions. Are there stuff on earth, in the solar system, or even in the world that have more than three deminsions, or are they just theoretical, like a point or a line etc.? If there are stuff, can we see them?(112 votes)
- Mostly we have to use our imaginations to think about things that have more than three dimensions. Sometimes theoretical scientists like to think of time being the fourth dimension, so if you think about an balloon being inflated over time, that's maybe a little bit like a four dimensional "hypercone" that is a sphere at every instant just like a normal cone is a circle anywhere you make a flat slice across it.(121 votes)
- Do scalars just have magnitude and vectors have magnitude and direction?(64 votes)
- who else is trying to learn geometry before you take it in order to get better grades(36 votes)
- Yes, I am trying to learn geometry, algebra 1 was easy for me but I have heard that geometry is difficult in high school.(8 votes)
- So , the line has no end?(17 votes)
- they really lied when they said geometry is jus shapes...(25 votes)
- well technically it is just shapes because even though it contains letters and numbers, these letters and numbers are drawn by connecting different shapes, like for example the letter b is made by combining a line and a semicircle and the number 7 is made by combining 2 lines.(1 vote)
- More than three dimensions? What would that look like?(10 votes)
- Imagine a 3D sphere and a 2D 'plane segment'. Now imagine the sphere passes through the plane segment and ask 'What does that sphere look like to the plane?' What we perceive as time is the 4th dimension expressing itself through the 3rd (or should I say we are 5th dimension objects passing through the fourth), and so on up the dimensions.(6 votes)
- Can someone summarize what terms were taught in this video and what each term means?(0 votes)
- Here are all the terms taught in the video, and the definitions for each (credit to Ed for the timestamps):
Definition of name Geometry –0:10
The definition of the name geometry is "Earth measurement", which basically defines all of what Geometry is about. Geometry is the study of understanding how shapes and space and things that we see relate to each other. This is why you learn about many different shapes such as Triangles, Squares, and much more.
Point (0 dimensions) -1:40
Sal explains that a point is 0 dimensions and how you cannot really move one point around and still call it the same point. In addition, in order to recognize a point, you would need to give it a name. In Geometry, points are often labeled to as letters in the English alphabet.
Line segment -3:10
Now what if you wanted to get from one point to another? Sal explains that if you wanted to do that, you would need to create a straight line in between them. A straight line connecting two points is referred to as a line segment. These are most used in Geometry because they have a finite length.
End Points –4:10
What if you wanted to label your line segment? You would do so by looking at its endpoints, which are the literal endpoints of a line segment. For example, if AB were the endpoints of your line segment, then the line segment would be known as AB (with a straight line on top of AB to show that it's a line segment). The order doesn't matter, so AB could also be referred to as BA (with a straight line still on top of it). Line segments, unlike points, are one-dimensional, because they can travel left and right from one point to another. However, they have no width, so that reason doesn't let it go any further then one-dimension. If you wanted to specify the length, you would measure it and write it like this. For example: AB = 5 (remember the straight line on top of AB!)
What if you wanted to keep going in one direction, and not have a finite length? You would call that a ray. For example: The line would start at A, and it would continue going straight, regardless of where the other point D is located. The starting point of a ray would be called a vertex, and in this case, would be the point A. To specify a ray, you would write the two points, which are AD in this case, and put an arrow on top of it to show that it continues in one direction. Unlike in a line segment, the order matters in which you put the points in. You cannot write DA with an arrow on top, because it looks as if the ray's vertex is D, and goes past A instead.
What if you wanted to keep going in both directions? That would be called a line. For example: You have two points; E and F. The line in between them would continue travelling outside of these points, regardless of where they are. To specify a line, you would write both the points down, which in this case is E and F, and then draw a double-sided arrow on top of it. This shows that the line does not stop and keeps going in both directions.
What happens if you have more then two points on any line? For example: You have three points; XYZ on a line. These points would be called collinear, since there are three points on it.
If there was a point directly in the middle of the other two endpoints in a line segment, we would refer to that as a midpoint. This is because this point is in the "middle" of the line segment.
Plane (2 dimensions) -11:30
Sal now explains the concept of something having two dimensions. Something has two dimensions if it can go backwards or forwards in two different directions. For example, your screen can go up and down + left and right. This means it is two dimensional. Something that is two dimensions is called a Planar.
3 dimensions –12:25
Something is three dimensions when it can not only move up and down + left and right, but it can also go in and out of something. This is sort of like our three-dimensional space.
Hope this helped!
Source: Video(24 votes)
- Playback speed x2 is very helpful, you get all the info in less time.(13 votes)
- [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.