Welcome to a lesson on circles.
The goals of this video are to define a circle
and the key features of a circle, as well as describe
the features of two coplanar circles.
A circle is a set of points in a plane that are
the same distance from a point, called the center.
So here we see a blue circle, where the center
would be this point here.
The radius is a segment from the center to the circle,
which we see here in green.
The length of the radius would represent the distance
from the center to all of the points on the circle.
Now let's define several other key features of a circle.
A chord is a line segment with the endpoints on the circle.
So in this diagram here, notice how this purple segment
has both endpoints on the circle, and therefore
this is is a chord.
A diameter is a chord that passes through the center.
So we know from before, this is the center of the circle,
and since this red segment has its endpoints on the circle,
and passes through the center, this red segment
is the diameter.
And you can probably notice that the diameter is
twice the length of the radius.
Next, a secant line intersects the circle in two points.
So we take a look at this orange line here, assuming that
it's a line extending in both directions forever.
It intersects the circle in two points, and therefore this
would be a secant line.
Now there's two more things we want to define.
The next is a tangent line.
A tangent line is a line that intersects
the circle in one point.
Remember, a secant line intersects the circle in two points,
but a tangent line intersects the circle in one point.
Looking at this brown line here, notice how it
only intersects the circle at this point here,
therefore this brown line is a tangent line.
And the point of tangency is the line where the tangent line
touches or intersects the circle.
So this black point right here, is the point of tangency.
Now let's talk about when we have two circles graphed
on the same plane.
Congruent circles are circles that have the same radius,
but different centers.
So looking at these two circles, if the length
of this radius is equal to the length of this radius,
and the centers are obviously at different points,
then these are congruent circles.
Concentric circles are two circles that have
the same center, notice how they share this point
here as the center, but have different radii.
And radii is the plural form of radius.
So the length of this radius would not be the same
as length of the radius of the larger circle.
So concentric circles may remind you of a bulls-eye.
And when we graph two circles, on the same plane
and they intersect, there are a couple things
that can happen.
One possibility is that they intersect in two points,
as we see here.
Here's one point of intersection, and here's another
point of intersection.
But it's also possible that two circles intersect
at one point, which means that two circles would
be tangent to one another.
And when circles are tangent to each other,
there are two ways they can be tangent.
They can be externally tangent, as we see here,
and they could also possibly be internally tangent,
as we see here on this graph.
So here we have two externally tangent circles,
and here we have two internally tangent circles.
Now let's finish by talking about some characteristics
when two circles don't intersect.
They can still share a common tangent line, as we see
here on the left, and here on the right.
Here on the left see that these two circles have
a common external tangent line, in blue.
And here on the right say that these two circles
have a common internal tangent line.
Notice how for the external tangent, the line does not
cross between the two circles, but for the
internal tangent line it does.
Thank you for watching.