Language and Notation of the Circle Formal definition of a circle. Tangent and secant lines. Diameters and radii. major and minor arcs
Language and Notation of the Circle
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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 that are exactly
- 2cm away from "Point A." So 2cm on my screen
- is about that far. So clearly if I start at "A" and I go 2cm
- in that direction, this point is 2cm from "A." If I call that
- "Point B" then I could say line segment AB is 2cm
- the length is 2cm. Remember, this would refer to
- the actual line segment. I could say this looks nice but if
- I talk about it's 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 would say 2cm.
- 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 2cm away from "A." So I could go
- 2cm in the other direction, maybe get to point
- "C" right over here. So "AC" is also going to be equal
- to 2cm. But I could go 2cm in any direction. And so
- if I find that of all of the points that are exactly 2cm
- away from "A," I will get a very familiar looking
- shape, like this: (I'm drawing this free-hand,) 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. Let me clear out all of these and I will
- just draw a solid line. It could look something like
- that (my best attempt.) And this set of all the points
- that are exactly 2cm away from "A," is a circle, which
- I'm sure you are already familiar with. But that is
- the formal definition, the set of all points that are
- a fixed distance from "A." If I said, "The set of all points
- that are 3cm from "A," it might look something like this:"
- That would give us another circle. (I think you get the
- general idea.) Now, what I want to introduce to you
- in this video is ourselves to some of the concepts
- and words that we use when dealing with circles.
- So let me get rid of 3cm circle. So first of all, let's
- think about this distance, or one of these line segments
- that join "A" which we would call the center of the
- circle. So we will call "A" the center of the circle,
- which 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 the points that
- are equal distance from the center. So "AB," any point,
- 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 is 2cm. 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 at the
- high-school level is that it's probably the first
- class where you're introduced to a slightly more formal
- mathematics where we're a little 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
- more careful with our language over here. So "AB" is
- a radius, line segment "AB," and so is line segment,
- (let me put 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 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 and the only point on the circle that
- the only point in the set of all the points that are
- equal distance from "A," the only point on that
- circle that is also on that line is point "D." And we
- could call that line, "line L." So sometimes you will
- see lines specified by some of the points on them.
- So for example, if I have 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 "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 tangent. Tangent to the circle. So let me
- write it this way, "'line L' is tangent 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."
- 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 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." And the line that intersects at two points
- we call this a secant of circle "A." It is a secant line
- to this circle right 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 do, if we only
- spoke about this line segment, between "FG," and not
- thinking about going on forever, then all the 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 on a point of the circle, a point
- that is, in this case, 2cm away and then it finishes
- at a point on the circle. So it connects two points on
- the circle. Now, you can have cords 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 'H.'" And you have a straight line connecting
- "F" to "H" through "A." (That's about as straight as I could
- draw it.) So if you have a chord like that, that contains
- actual center of the circle, of course it goes from one
- point to another point of the circle, and it goes through
- the center of the circle, we call that a diameter of a
- circle. And you've probably seen this in tons of problems
- before when we were not talking about geometry as formally,
- but a diameter is made of two radiuses. We 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 you have another radius connecting "A" and "H,"
- a point connecting to the center of the circle. So the
- diameter is made of these two radiuses (or radii as 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 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 at "B." And I'm going to find
- some points, all 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 this circle. Let's call this, "J," "K," "S," "T," and "U."
- Let me center "B" a little bit more in the center here...
- Now, one interesting thing is, "what do you call
- the length of the circle that goes between two points?"
- Well, you could imagine in every language, we would
- call something like that an "arc," which is also what it's
- called in geometry. We would call this "JK," the two end
- points of the arc, the two points on the circle that are
- the inputs of the arc, and you would use a little notation like
- that, a little curve on top instead of a strait line.
- Now, you can also have another arc that connects "J"
- and "K," this is called the 'minor arc,' it is the shortest
- way upon 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 around the circle. And that is
- called the 'major arc.' And usually when we specify the
- major arc, just to show that you're going kind of the long
- way around, it's not the shortest way to go between
- "J" and "K," you will often specify another point that you're
- going through. So for example, this major arc we could
- specify. We started "J," we went through, we could have
- said "U," "T," or "S," but I will 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 are
- multiple ways to specify this major arc. The one thing I want
- to make clear is that the minor arc is the shortest distance,
- so this is the minor arc, and the longer distance around is
- the major arc. I will leave you there. Maybe the next few
- videos we will starts playing with some of this notation.
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