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Current time:0:00Total duration:11:12

CCSS.Math:

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 two centimeters away from a so two centimeters on my screen is about is about that far so clearly if I start at a and I go to 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 a B is 2 centimeters the length is 2 centimeters remember this would refer 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 a B is equal to 2 if I wanted to put units I could say 2 centimeters but I'm not curious just about B I want to think about all the points the set of all of the points that are exactly 2 centimeters away from a so I could go to 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 familiar looking shape like this and I'm trying to draw it freehand so I would get a shape that looks like this and actually let me draw it in I don't want to make you think it's only the points where that there's white it's all of these points right over here you want to draw a dashed line over there maybe I should just let me clear out all of these and I'll just draw a solid line so this it could look something like that my best attempt and this set of all of the points that are exactly two 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 it 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 three centimetres circle so first of all let's think about this distance this distance or 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 a B is so a B a B connects the center and it connects a point on the circle itself remember the circle itself is all the points that are equal distance from the center so a B any point any line segment I should say that connects the center to a point on the circle we would call a radius is a radius and so the length of the radius a B over here is equal to two 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 a B is a radius line segment a B and so is line segment let me draw another so let me put another point on here let's say this is X so line segment a X is also a radius line segment a X is also a radius 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 and let's call that D and let's say you have a line let's say you have a line and the only point on the circle that the only point in the set of all the points that are equidistant from a the only point on that circle that is also on that line is point D and we could call that line let's call that line L so sometimes you'll see lines especially by some of the points on them so for example if I'd another point right over here called e we could call this line line de or we could just put a little 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 tangent to the circle so let me write it this way line L line L is tangent and 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 is 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 or the points equal distant from point a now L is tangent because it only intersects the circle on one point you could just as easily imagine a line you could 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 is at two points we call this a secant of circle a is a secant 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 lions 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 cord of the Circle is a chord a chord of circle a it starts at a point on the circle a point that is exactly in this case two 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 say call this point let's call this point H and you have a straight line connecting f2h through a so that's about a straight Ike as I could draw it so if you have a chord like that so if you have a chord like this that contains the actual center of the circle of course it goes from one point on the circle to another point of the circle and it goes through the center of the circle we call that a diameter of a is a diameter is a diameter of circle a and you've probably seen this in 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 know we already know that a radius connects a point to the center so you have one radius right over here that connects F and H sorry 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 a diameter is made up of these two radiuses or radii I should call 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 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 plus the length of segment aah now there's less 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 can also 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 some points all the points that are 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 this circle so let's call this let's call this I don't know what let's call that J let's call that K let's call this let's call that 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 be a little bit better let me erase that and put be 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 we would you call what would you call 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 we would call this would say 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 a little curve on top instead of a straight line now you can also have another arc that connects J and K this is the minor arc it is the shortest way along the circle to connect J and K but you can 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 kind of the long way around the way that you you know 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 UT 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 J UK that these are specifying the same thing or J SK so there's multiple ways to specify this major arc right over here but the one thing I want to make clear is is that the minor arc is the shortest distance so this is the minor arc minor arc and the longer distance around is the major arc is the major arc and I'll leave you there and maybe in the next few videos we'll start playing with some of this notation a little bit more