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Studying for a test? Prepare with these 15 lessons on Circles.
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Let's start again with a point. Let's call that point point A. And what I'm curious about is all of the points on my screen right over here, that are exactly 2 centimeters away from A. So 2 centimeters on my screen is about that far. So clearly if I start at A and I go 2 centimeters in that direction, this point right over there, is 2 centimeters from A. If I call that point, point B, then I could say line segment AB is 2 centimeters. The length is 2 centimeters. Remember this would refer to the actual line segment. I could say this looks nice, but if I talk about its length, I would get rid of that line on top and I would just say AB is equal to 2. If I wanted to put units I could say 2 centimenters. But I'm not curious just about B, I want to think about all of the points, the set of all of the points that are exactly 2 centimeters away from A. So I could go 2 centimeters in the other direction, maybe get to point C right over here. So AC is also going to be equal to 2 centimeters, but I could go 2 centimeters in any direction. And so if I find the set of all of the points that are exactly 2 centimeters away from A, I will get a very familiar looking shape, like this, and I'm trying to draw it freehand. So I would get a shape that looks like this. Actually, let me draw it in. I don't want to make you think that it's only the points where there's white, it's all of these points right over here. I don't want to draw a dashed line over there, which maybe I should just, let me clear out all of these and I'll just draw it as a solid line. So this could look something like that. My best attempt. And this set of all of the points that are exactly 2 centimeters away from A, this is a circle, as I'm sure you are already familiar with. But that is the formal definition-- the set of all points that are a fixed distance, or that have a fixed distance, from A that are a given distance from A. If I said the set of all points that are 3 centimeters from A, it might look something like this. It might look something like that. That would give us another circle. I think you get the general idea. Now what I want to do in this video is introduce ourselves into some of the concepts and words that we use when dealing with circles. So let me get rid of that 3-centimeter circle. So first of all, let's think about this distance. This distance, or one of these line segments that join A, which we would call the center of the circle. So we'll call A the center of the circle, and it makes sense just from the way we use the word center in everyday life. What I want to do is think about what line segment AB is. AB connects the center and it connects a point on the circle itself. Remember, the circle itself is all of the points that are equal distance from the center. So AB, any line segment, I should say, that connects the center to a point on the circle, we would call a radius. And so the length of the radius, AB over here, is equal to 2 centimeters. And you're probably already familiar with the word radius, but I'm just being a little bit more formal. And what's interesting about geometry, at least when you start learning it at the high school level, is it's probably the first class where you're introduced into a slightly more formal mathematics, where we're a little bit more careful about giving our definitions and then building on those definitions to come up with interesting results and proving to ourselves that we definitely know what we think we know. And so that's why we're being a little bit more careful with our language over here. So AB is our radius, line segment AB, and so is line segment-- let me draw another point on here, let's say this is X-- so line segment AX is also a radius. Now you can also have other forms of lines and line segments that interact in interesting ways with the circle. So you could have a line that just intersects that circle at exactly one point. So let's call that point right over there, let's call that D. And let's say you have a line, and the only point on the circle-- the only point in the set of all of the points that are equal distant from A, the only point on that circle that is also on that line is point D. And we could call that line l. So sometimes you'll see lines specified by some of the points on them. So for example, if I had another point right over here called E, we could call this line line DE, or we could just put a little script letter here with an l and say this is line l. But this line that only has one point in common with our circle, we call this a tangent line. So line l is a tangent. It is tangent to the circle. So let me write it this way, line l is tangent-- you normally wouldn't write it in caps like this, I'm just doing that for emphasis-- is tangent to-- instead of writing the circle centered at A, you'll sometimes see this notation-- to the circle centered at A. So this tells us that this is the circle we're talking about. Because who knows, maybe we had another circle over here that is centered at M, another circle. So we have to specify it's not tangent to that one, it's tangent to this one. So this circle with a dot in the middle tells we're talking about a circle, and this is a circle centered at point A. I want to be very clear, point A is not on the circle, point A is the center of the circle. The points on the circle are the points equal distant from point A. Now, l is tangent, because it only intersects the circle in one point. You could just as easily imagine a line that intersects the circle at two points. So we could call, maybe this is F and this is G, you could call that line FG. So we could write it like this, line FG. And this line that intersects it at two points, we call this a secant of circle A. It is a secant line to this circle right over here, because it intersects it in two points. Now, if FG was just a segment, if it didn't keep on going forever, like lines like to do, if we only spoke about this line segment-- let me do this in a new color-- if we only spoke about this line segment between F and G, and not thinking about going on forever, then all of a sudden we have a line segment, which we would specify there. And we would call this a chord of the circle, a chord of circle A. It starts at a point on the circle, a point that is exactly in this case 2 centimeters away, and then it finishes at a point on the circle. So it connects two points on the circle. Now, you could have chords like this, and you can also have a chord, as you can imagine, a chord that actually goes through the center of the circle. So let's call this point, point H, and you have a straight line connecting F to H through A. So that's about as straight as I could draw it. So if you have a chord like that, that contains the actual center of the circle, of course it goes from one point of the circle to another point of the circle, and it goes through the center of the circle, we call that a diameter of A. And you've probably seen this in tons of problems before, when we were not talking about geometry as formally, but the diameter is made up of exactly two radiuses. We already know that a radius connects a point to the center. So you have one radius right over here that connects F and A, that's one radius, and then you have another radius connecting A and H, the center to a point on the circle. So the diameter is made up of these two radiuses, or radii I should call it, I think that's the plural for radius. And so the length of a diameter is going to be twice the length of a radius. So we could say the length of the diameter, so the length of FH-- and once again I don't put the line on top of it when I'm just talking about the length-- is going to be equal to FA, the length of segment FA, plus the length of segment AH. Now there's one last thing I want to talk about when we're dealing with circles, and that's the idea of an arc. So we also have the parts of the circle itself. So let me draw another circle over here. Let's center this circle right over here at B. And I'm going to find all of the points that are a given distance from B. So it has some radius, I'm not going to specify it right over here, and let me pick some random points on the circle. So let's call this J, let's call that K, let's call this S, let's call this T, let's call this U right over here. And I know it doesn't look that centered, let me try to center B a little bit better. Let me erase that. And if I put B a little bit closer to the center of the circle, so that's my best shot, so let's put B right over there. Now, one interesting thing is, what do you call the length of the circle that goes between two points? So what would you call-- let me get a good color here-- what would you call this? Well, you could imagine in everyday language we would call something that looks like that an arc, and we would also call that an arc in geometry. To specify this arc we would call this JK, the two endpoints of the arc, the two points on the circle that are the endpoints of the arc, and you use a little notation like that. So you put a little curve on top instead of a straight line. Now you could also have another arc that connects JK, this is the minor arc. It is the shortest way along the circle to connect J and K, but you could also go the other way around. You could also have this thing that goes all the way around the circle. And we would call that the major arc. And normally when you specify a major arc, just to show that you're going the long way around, it's not the shortest way to get between J and K, you'll often specify another point that you're going through. So for example, this major arc we could specify we start at J, we went through-- we could have said U, T, or S, but I'll put T right over there-- we went through T and then we went all the way to K. And so this specifies the major arc and this thing could have been the same thing as if I wrote JUK, these are specifying the same thing, or JSK. So there's multiple ways to specify this major arc right over here. But the one thing I make clear is, is that the minor arc is the shortest distance, so this is the minor arc, and the longer distance around is the major arc. And I'll leave you there and maybe the next few videos we'll start playing with some of this notation a little bit more.