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## High school geometry

# Circles glossary

CCSS.Math:

Formal definition of a circle. Tangent and secant lines. Diameters and radii. major and minor arcs. Created by Sal Khan.

## Video transcript

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.