Video Transcript
In this video, we’ll learn how to
identify the basic parts of the circle, such as the radius, chord, and diameter, and
use their properties to solve problems. A circle, which is not a polygon
since it has no straight edges, is a one-sided, two-dimensional shape. It will have a center which is
usually defined by the letter 𝑜. And all points on the circle
itself, that’s this curve line, are equidistant from this center. In other words, the lines joining
the center to these points are equal in length. Let’s begin by having a look at
some of the other key parts of a circle.
Label each part of the circles. Then, we have a few terms. We have radius, diameter,
circumference, chord, tangent, arc, sector, minor segment, and major segment.
Let’s go through each of these
words term by term and give them a definition. We begin with the radius. The radius is any line which is
made by joining the point at the center of our circle to a point on the
circumference. If we look at the circle at the
bottom of our diagram, we see that there’s a radius marked here. In fact, there’s also a radius in
our top circle. Remember, it’s simply the line that
joins the point at the center to any point on the circumference. For this reason, we can draw an
infinite number of radii in our circles.
And then we move on to the
diameter. The diameter is closely linked to
the radius. This time, though, the diameter is
found by joining points on opposite sides of the circle with a straight line. And this straight line must pass
through the center. And so, if we look at our first
circle, we see we have a diameter here. We then move on to the
circumference. We say that the circumference is
the perimeter of the circle. It’s the outside. And so, we label the curve line
that encloses our circle in the upper diagram as the circumference.
Next, we move on to the chord. A chord is any line that joins two
points on the circumference of the circle. Now, notice we’ve already labeled
the chord on our first circle as a diameter. But on our second circle, we have a
chord here. And what about the tangent? A tangent to a circle is a straight
line which touches the circle at only one point. We only have one line. This is on the outside of our
circles, and it’s this one. So, this is the tangent. The next word is the arc. Essentially, the arc of a circle is
just part of its circumference. That’s this line here.
Next, we have the sector. The sector is the portion of the
circle that looks a little bit like a piece of pie. It’s this one here, and it’s the
entire shape. Finally, we have a minor segment
and a major segment. The minor segment is a part of the
circle that looks a little bit like an orange segment. It’s this slice here. Once again, we have the entire
shape, and it’s made by portioning up the circle using a chord. It follows that the major segment
is the rest of the circle. It’s this one here.
We must be able to identify each of
the parts of these circles by heart. And we’re next going to see how we
can use some of the properties of these parts of the circles to solve problems.
Are all the radii of a circle equal
in length?
Radii is just the plural of
radius. And the radius of the circle is any
line that joins the center of that circle to a point on its circumference. So, if we take a circle center 𝑜,
we could have a radius here. It joins the center to a point on
its circumference. We could have one here or one over
here. Note that since the line begins at
the center of the circle each time and ends on the circumference, this line must be
the same length as this line and this line. And so, we say yes that all radii
of a circle are equal in length.
And in fact, this is a really
useful fact. It means we can solve problems
involving angles in a circle by creating isosceles triangles with any two radii.
Is the diameter the longest chord
in a circle?
Remember, a chord is a line that
joins one point on the circumference of a circle to another. So, if we take a circle whose
center is 𝑜, we could have a chord here, one here, or one here. There are an infinite number of
chords that we can draw. Note that to create the longest
chord, that’s the longest line by joining two points on the circumference, we’re
going to need to find the widest part of the circle.
Now, this line must pass through
the center of the circle. But of course, we know that the
diameter is the line that passes through the center of the circle and joins two
points on its circumference. And this means that we can say that
yes, the diameter is the longest chord in a circle.
Next, we look at the relationship
between the radius and the diameter of a circle.
The length of the radius of a
circle is what the length of its diameter.
Remember, the radius of a circle is
the straight line that joins the point at the center to any point on the
circumference of the circle. So, if we take a circle whose
center is 𝑜, we could have a radius here. And what about the diameter? Well, the diameter also passes
through the center of the circle. But this time, it’s a straight line
that joins two points on the circumference. So, the diameter of this circle is
this line here.
But of course, If we look carefully
at the diameter, we can see that we can make up of one, two radii. And so, this means that a diameter
of the circle must be twice or double the length of the radius. And in turn, we can say that the
radius of the circle must therefore be half the length of its diameter.
We’ll now look at how to use these
features to solve problems involving circles.
If the diameters of circles 𝑀 and
𝑁 are two centimeters and six centimeters, respectively, determine the length of
the line segment 𝑀𝑁.
The line segment 𝑀𝑁 is shown. It’s a single straight line. And if we look carefully, we see
that each part of the straight line is made up by joining the point at the center of
each circle to a point on its circumference. That means this first part of the
line segment is the radius of 𝑀. The radius of a circle is the line
that joins a point on the circumference to its center. Similarly, the second part of our
line segment is the radius of our circle 𝑁. This means that the line segment
𝑀𝑁 is the sum of these. It’s the length of the radius of 𝑀
plus the length of the radius of 𝑁.
Now, the problem is we’re not
actually given information about the radii of our circles. We are, however, told that the
diameters of our circles are two centimeters and six centimeters, respectively. Now, we know that the diameter of a
circle is twice or double the length of the radius. And we can equivalently say that
this means that the radius must be half the length of the diameter. Of course, to find half of a
number, we divide it by two. So, this means the radius of 𝑀 is
two divided by two, which is equal to one centimeter.
Similarly, we halve the length of
the diameter of circle 𝑁. That’s six divided by two, which is
three centimeters. So, 𝑀𝑁 is the sum of these two
lengths. It’s one plus three, which is equal
to four or four centimeters. The length of line segment 𝑀𝑁 is
four centimeters.
We’ll consider one more
problem-solving example.
In the figure, 𝑀𝐴 is equal to 34
centimeters. Determine the length of 𝐶𝐸.
We’ve been given part of a circle
and some information about the length of the line 𝑀𝐴. Now, 𝑀𝐴 is this line here. And let’s imagine that we actually
had a full circle. If we do, we see that 𝑀𝐴 is the
radius. It’s the line that joins the center
to a point on the circumference of the circle. In fact, we have a radius here. This line here is also a
radius. Now, we’re told that 𝑀𝐴, the
radius of the circle, is 34 centimeters. So, each of these lines that we’ve
added must all be 34 centimeters.
Now, in fact, if we join 𝑀 to 𝐷
to create the line segment 𝑀𝐷, we see that that is also the radius of the
circle. So, 𝑀𝐷 must also be 34
centimeters. Now, you might be thinking that you
need to use something like Pythagoras’ theorem next. But no, we see if we look carefully
that 𝑀𝐶𝐷𝐸 is a rectangle. And we know that the diagonals of a
rectangle are equal in length. This means that 𝐶𝐸, which is also
a diagonal, must be equal in length to 𝑀𝐷. We established 𝑀𝐷 was equal to 34
centimeters. So, this means that 𝐶𝐸 is also
equal to 34 centimeters.
In this video, we’ve learned that
the key parts of a circle are the radius, diameter, circumference, chord, tangent,
arc, minor segment, major segment, and sector. We saw that all radii of a circle
are equal in length, that the length of the radius is exactly half the length of the
diameter, and that the diameter is the longest possible chord in a circle. We also saw that these features can
help us to solve problems involving parts of a circle.
