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In this video, we’ll learn how to identify points, lines, rays, line segments, and endpoints and the associated notation we use to describe these.
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As a standalone, this skill might seem somewhat unimportant.
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But once we can begin to describe these using mathematical language and notation, this opens up a whole world to us in geometry, from simply describing polygons through to trigonometry and geometrical proof.
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Let’s begin by looking at some of these definitions.
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The definitions we’re going to look at all stem from this first definition.
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It’s the definition of a point.
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A point is simply a position or location.
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It has no size, such as width or length or depth, and we represent it using a dot.
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Here, we’ve drawn a dot representing point 𝐴.
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And once we have the definition of a point, let’s form the definition of a line.
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A line is a straight set of points that extend infinitely in two directions.
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It only has one dimension and that’s its length.
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We represent it as shown.
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These two arrows show that it extends infinitely in those directions.
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If we represent this line as passing through the points 𝐴 and 𝐵, we see we define the line as shown with an arrowhead.
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We say that points 𝐴 and 𝐵 are colinear; they’re on the same line.
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Now, what do we mean by a ray?
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A ray is a portion of a line.
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It starts at a given point, and we call this the endpoint, and then it goes off in a particular direction to infinity.
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A ray whose endpoint is 𝐴 and then passes through 𝐵 is represented as shown with a single arrow defining its direction.
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The endpoint that we could also alternatively consider as the starting point of our ray is 𝐴.
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Formerly, though, we say that the endpoint is the point at which a ray ends.
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There’s one further definition we need, and that’s the definition of a line segment.
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A line segment is a part of a line that’s bounded by two distinct endpoints.
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And it contains every point on the line between those points.
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It’s always the shortest distance between these points.
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In this line segment, 𝐴 and 𝐵 are the endpoints.
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And so we define the line segment 𝐴𝐵 by using a bar as shown.
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Now that we have some definitions, we’ll look at some questions.
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What has been drawn?
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Now, if we look carefully, we can informally say it looks like we have part of a line.
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We’ll be able to identify its formal name by looking at what’s happening at the end of this line drawn.
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We have two arrows.
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Now, these arrows tell us that this line extends in both directions.
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In fact, it extends infinitely in both directions.
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And so we recall that the formal mathematical definition of a line is a straight set of points that extend infinitely in both directions.
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And so we can say that, here, we have a line.
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What has been drawn?
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It looks like we’ve been given a line or certainly a portion of a line.
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The key here is to consider what’s happening at the ends of our line.
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At the ends of our line, we have these two solid dots.
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They show us that this line ends in both these places.
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Now, in fact, a line is a straight set of points that extend infinitely in both directions.
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We can see that our straight set of points have been bounded at both ends.
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And so we recall a second definition.
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We say that a line segment is a part of a line bounded by two endpoints.
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We can see that our straight set of points is indeed bounded by two endpoints.
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And so we have a line segment.
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In our next example, we’ll consider how these definitions can help us to define polygons.
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How many line segments does this shape have?
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We begin by recalling the definition of a line segment.
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We know that a line segment is a part of a line that’s bounded by two distinct endpoints.
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And it contains every point of the line between those endpoints.
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Since a line is a straight set of points that extend infinitely in two directions, we know that, by definition, a line segment must be straight also.
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And so to answer this question, we’re simply going to count the number of straight portions of line.
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We’ll highlight them as we go.
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There’s one here.
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Then there’s another here, so that’s two.
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We have a smaller one down here; that’s three.
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And we have another one here.
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That gives us a total of four.
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It doesn’t actually matter that the line segments are different lengths.
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We have four distinct line segments in our shape.
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In fact, this allows us to define our shape.
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It’s a polygon with four sides.
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Remember, polygons have straight edges, and so we can say that this shape is a quadrilateral.
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In our next two examples, we’ll look at how the definitions we’ve seen so far gonna help us to answer questions about points.
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Using the given figure, determine whether the following is true or false.
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The line segment that is passing through points 𝐵 and 𝐷 is also passing through point 𝐶.
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In order to answer this question, we’re going to need to recall what we actually mean by a line segment.
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A line segment is a part of a line, and that part is bounded by two distinct endpoints.
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We’re told that our line segment passes through points 𝐵 and 𝐷.
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In fact, its two endpoints, as we can see from the diagram, are 𝐵 and 𝐷.
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So we represent it as shown.
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It’s 𝐵𝐷 with a line above it.
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So let’s compare point 𝐶 to points 𝐵 and 𝐷.
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It sits on the line passing through these points and halfway roughly between points 𝐵 and 𝐷, halfway between the endpoints of the line segment.
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By definition, a line segment must be straight and contain every point on the line between its endpoints.
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So we can say this is true.
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The line segment 𝐵𝐷 passes through point 𝐶.
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Let’s consider another example like this.
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Does the given figure allow you to conclude that the ray starting at 𝐶 and passing through 𝐷 passes through point 𝐵.
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In order to answer this question, we need to recall what we mean by a ray.
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We say that a ray is a portion of a line.
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Unlike a line segment, though, it extends from a single endpoint and goes off in a particular direction to infinity.
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Our ray starts at 𝐶, so 𝐶 is the endpoint, and it passes through 𝐷.
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We can represent this using a single arrow 𝐶𝐷 as shown.
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So does this ray pass through point 𝐵?
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Well, no, we said that point 𝐶 is the endpoint.
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It’s the point at which a line segment or ray ends.
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It does not extend back out past this endpoint in the direction of 𝐵.
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And so we can say no, the given figure does not allow us to conclude that the ray that starts at 𝐶 and passes through 𝐷 also passes through point 𝐵.
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Using the given figure, answer the following: Does point 𝐶 lie on the straight line?
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We begin by recalling the definition of the word line.
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We say mathematically that a line is a straight set of points that extend infinitely in two directions.
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As shown in the diagram, it’s represented with the two arrows.
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Those arrows indicate to us that the line continues in both directions.
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And so if we were to continue this line downwards on our diagram, we could deduce that this line most likely passes through point 𝐸.
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But can it pass through point 𝐶?
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Well, no, we can see quite clearly that points 𝐶 and 𝐷 are not on that straight line.
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And so the answer is no.
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In our final example, we’ll look at how to apply the mathematical notation.
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The diagram shows a number of points, 𝐴, 𝐵, 𝐶, and 𝐷.
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Use mathematical notation to describe the straight set of points from 𝐵 and through 𝐷, the straight set of points from 𝐴 to 𝐶, the straight set of points through 𝐴 and 𝐶 that extend infinitely in both directions.
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We’ll begin by looking at the straight set of points from 𝐵 and through 𝐷.
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That’s all of these.
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This set of points has an endpoint, but it extends infinitely through 𝐷 and beyond.
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We can therefore say that this set of points is a ray.
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We represent a ray using a one-sided arrow as shown, and we use the endpoint as the first letter.
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And so we can say that the straight set of points from 𝐵 and through 𝐷 is shown.
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It’s 𝐵𝐷 with an arrow.
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Next, we’re told to describe the straight set of points from 𝐴 to 𝐶.
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That’s this line shown.
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We know that this point has a very specific start point and endpoint.
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But in fact, we call these both endpoints.
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And so the straight set of points between these endpoints must be a line segment.
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We use a bar to represent a line segment.
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And so the straight set of points from 𝐴 to 𝐶 is represented by 𝐴𝐶 with a bar.
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Finally, we look at the straight set of points through 𝐴 and 𝐶 that extend infinitely in both directions.
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And so we see that this is all of this line here.
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The two arrows tell us that this line extends infinitely.
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A straight set of points that extends infinitely in two directions is a line.
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And we use a two-sided arrow, much like in the picture, to represent this.
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It’s 𝐴𝐶 with a two-sided arrow.
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In this video, we learned first that a point is a location.
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It has no size like width nor length, and it’s represented by a dot.
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We learned that a line is a straight set of points that extend infinitely in both directions.
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We used two specific points to describe it and a double-sided arrow above these points to represent that we have a line.
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When points lie on the same line, they’re called colinear points.
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We saw that a ray is a specific portion of a line.
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It has one endpoint — in our diagram, that’s 𝐴 — and it extends infinitely in a given direction.
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Here, that’s through point 𝐵.
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We represent this with a single-sided arrow.
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Then a line segment is also part of a line.
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But this time it has two distinct endpoints, and it contains every point on the line between those points.
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We represent this using a bar, as shown.