### Video Transcript

Central Angles and Arcs

In this video, we will learn how to
identify central angles, use their measures to find the measures of arcs, identify
adjacent arcs, find arc lengths, and identify congruent arcs in congruent
circles.

Letβs start by defining exactly
what we mean by an arc of a circle. An arc of a circle is a section of
the circumference of the circle between two radii. For example, consider the following
circle centered at the point π with two radii included. The section marked in orange is an
example of an arc of a circle. Itβs a section of the circumference
of the circle between two radii. And itβs worth noting the exact
same thing is true for the part marked in blue. Itβs also a section of the
circumference of the circle between two radii. Therefore, if weβre given two radii
of a circle, we canβt just say the arc of a circle. We need to differentiate between
these two cases. We call the longer of the two arcs
the major arc. And we call the shorter of the two
arcs the minor arc.

There is one more possibility. The two arcs could have the same
length. In this case, they make up half the
circle each. So we call these semicircular
arcs. And these will only occur when our
two radii form a diameter of the circle.

Now that weβre familiar with the
concept of arcs, letβs discuss how we denote arcs in a circle. First, because arcs are defined
between two radii of a circle, we can also talk about the points of intersection
with the radii and the circle. For example, we can talk about the
points π΄ and π΅ in this diagram. Then, we could say that the arc
marked in orange is the minor arc from π΄ to π΅ and the arc marked in blue is the
major arc from π΄ to π΅. We denote the arc from π΄ to π΅ by
using the following notation. And unless otherwise specified,
this means the minor arc.

Finally, it will be useful to
introduce a concept of adjacent arcs. We say that two arcs are adjacent
arcs if they only share a single point in common or they share both endpoints in
common, where the endpoints of an arc are the points where the arcs end. For example, in our diagram, both
the major and minor arc have endpoints π΄ and π΅. And since both of these arcs share
both of their endpoints in common, the major and minor arc between π΄ and π΅ are
adjacent. But this is not the only possible
way to have adjacent arcs. We could introduce another point πΆ
not on the minor arc from π΄ to π΅ as shown. Then, we can see in our diagram the
minor arc from π΄ to π΅ and the minor arc from π΅ to πΆ share a single point in
common, the point π΅. So the minor arc from π΄ to π΅ and
the minor arc from π΅ to πΆ are adjacent. Letβs now see an example where
weβre asked to identify the adjacent arcs in a circle.

For the given circle, which of the
following arcs are adjacent? Option (A) the minor arc from π΄ to
π΅ and the minor arc from πΆ to π·. Option (B) the minor arc from π΄ to
π΅ and the minor arc from π΅ to πΆ. Option (C) the minor arc from π΄ to
π· and the minor arc from π΅ to πΆ. Or option (D) the minor arc from π΄
to πΆ and the minor arc from π· to π΅.

In this question, weβre given a
circle and we need to determine which of the given pairs of arcs in this circle are
adjacent. To answer this question, letβs
start by recalling what it means for two arcs in our circle to be adjacent. We say that two arcs are adjacent
if they share only a single point or only both endpoints. Therefore, we can determine which
pair of arcs are adjacent by sketching them and determining how many points they
share in common. Letβs start with option (A). Weβll sketch the minor arc from π΄
to π΅. Remember, thereβs two arcs in our
circle from π΄ to π΅, and we want the shorter one. Next, weβll sketch the minor arc
from πΆ to π·. Once again, this is the shorter
section of the circumference of our circle from πΆ to π·. And we can see that these two arcs
share no points in common. So theyβre not adjacent.

We can do the same for option
(B). Letβs sketch the minor arc from π΄
to π΅ and the minor arc from π΅ to πΆ. The minor arc from π΄ to π΅ is the
shorter section of the circumference of our circle between π΄ and π΅. And the minor arc from π΅ to πΆ is
the shorter section of the circumference of our circle between π΅ and πΆ. We can see in our sketch both of
these arcs contain the point π΅. In fact, itβs the only point they
share in common. And this means that theyβre
adjacent. So the answer to this question is
option (B). We could stop here. However, for due diligence, letβs
check the other two options.

To check option (C), we need to
sketch the minor arc from π΄ to π· and the minor arc from π΅ to πΆ. If we do this, we get the
following. We can see that these two arcs
share no points in common, so theyβre not adjacent. Finally, letβs look at option
(D). We need to determine whether the
minor arc from π΄ to πΆ and the minor arc from π· to π΅ are adjacent. We can sketch both of these arcs
onto our circle, and we notice something interesting. Every single point between π΅ and
πΆ on our circle lies in both arcs. So although they do share a point
in common, they actually share an infinite number of points in common. So these arcs are not adjacent. Therefore, of the given options,
only the minor arc from π΄ to π΅ and the minor arc from π΅ to πΆ are adjacent, which
was option (B).

Weβre now almost ready to determine
the length of an arc. However, thereβs a few more
definitions we need first. We start by noting that an arc is a
portion of the circumference of a circle. Therefore, if we knew this
proportion, we would be able to use this to determine the arc length. It would be the circumference
multiplied by the proportion. And to determine this proportion,
we need to introduce the concept of a central angle.

The central angle of an arc between
two radii is the angle at the center of the circle between the two radii subtended
by the arc. So, in our diagram, the angle π is
the central angle for the minor arc π΄π΅. Similarly, 360 degrees minus π
will be the central angle for the major arc π΄π΅. And we can see the larger the
central angle, the larger the arc of the circle will be. In fact, this allows us to
determine major or minor arcs by their central angle. If the central angle of an arc is
between zero and 180 degrees, the arc will be a minor arc. If the central angle is equal to
180 degrees, we will have a semicircular arc. And if itβs bigger than 180
degrees, we will have a major arc.

Before we move on to arc length,
thereβs one more thing we need to define, the measure of an arc. The measure of the minor arc from
π΄ to π΅, written measure of minor arc π΄ to π΅, is equal to its central angle. For example, if the central angle
of the minor arc from π΄ to π΅ was 60 degrees, then we could say the measure of this
arc is 60 degrees.

Weβre now ready to determine the
arc length. Letβs start with an example. Letβs say we have a circle of
radius π and an arc with central angle 90 degrees. First, we recall the circumference
of our circle is two π multiplied by the radius. The circumference is two ππ. We want to determine the length of
the arc which has central angle 90 degrees. If we add the points π΄ and π΅,
this is the minor arc from π΄ to π΅.

We can see in our diagram that this
makes up one-quarter of the circle. So we could just multiply the
circumference by one-quarter to find the length of this arc. However, itβs good practice to
remember exactly why this makes up one-quarter of the circle. A full turn around the circle is
360 degrees, and our central angle is 90 degrees. 90 degrees divided by 360 degrees
is one-quarter. This confirms the length of this
arc will be one-quarter of the circumference of the circle, one-quarter times two
ππ, which we can of course simplify to ππ divided by two.

In general though, the central
angle or measure of our arc will not be 90 degrees. Instead, it will be some angle π
degrees. However, we can find the length of
the arc in exactly the same way. The proportion of the circumference
of the circle represented by the arc will be π degrees divided by 360 degrees. So the length of the arc will be
the circumference of the circle multiplied by this proportion. It will be π degrees divided by
360 degrees multiplied by two ππ. We can write these formulae as
follows. If the central angle or measure of
an arc in a circle of radius π is π degrees, then the length of the arc πΏ is
given by the following formula. πΏ is equal to π degrees divided
by 360 degrees multiplied by two ππ. Letβs see an example of applying
this formula.

Letβs say we have a circle of
radius two and an arc of measure 30 degrees. Then, the length of the arc πΏ is
equal to the measure of the arc 30 degrees divided by 60 degrees multiplied by two
π times the radius, which is two. And if we simplify this expression,
we see itβs equal to π by three. And remember, this represents a
length, so we can say this has length units.

Letβs now see an example of
applying some of these definitions to a question.

Find the measure of the minor arc
from π΄ to π·.

In this question, weβre asked to
determine the measure of the minor arc from π΄ to π·. Letβs start by sketching this arc
onto our diagram. Remember, the minor arc from π΄ to
π· is the shorter section of the circle between π΄ and π·. And we were asked to find the
measure of this arc. And we recall the measure of an arc
is equal to the measure of its central angle. And the central angle of an arc is
the angle at the center of a circle which is subtended by the arc. In this case, the central angle of
the minor arc from π΄ to π· is the angle π·ππ΄. And weβre told that the measure of
this angle is 33 degrees. And the measure of the minor arc
from π΄ to π· must be equal to this value. Therefore, the measure of the minor
arc from π΄ to π· is 33 degrees.

Letβs now see an example involving
two arcs with the same measure.

Given circle π with two arcs from
π΄ to π΅ and πΆ to π· that have equal measures and that the arc from π΄ to π΅ has a
length of five centimeters, what is the length of the arc from πΆ to π·?

In this question, weβre given a
circle. And weβre told that two of its arcs
have the same lengths, the minor arc from π΄ to π΅ and the minor arc from πΆ to
π·.

We can add both of these to our
diagram. Remember, an arc of a circle is a
section of the circumference of a circle, and the minor arc will be the shorter arc
between the two points. And weβre told that the minor arc
from π΄ to π΅ has a length of five centimeters, so we can also add this to our
diagram. We need to use this to determine
the length of the arc from πΆ to π·. To answer this question, letβs
start by recalling what the measure of an arc means. The measure of an arc is the
measure of its central angle. Thatβs the angle at the center of
the circle, which is subtended by the arc. For example, the angle π΄ππ΅ is
the central angle of the minor arc from π΄ to π΅. And the angle π·ππΆ is the central
angle of the minor arc πΆπ·. And since the measure of these two
arcs are equal, the measures of their central angles must also be equal.

Letβs then say that these angles
have a measure of π degrees. Now, we can determine an expression
for the lengths of both of these arcs. First, we recall the following
formula for finding the length of an arc πΏ. If its central angle is π degrees
and the radius of the circle is π, then πΏ is equal to π degrees divided by 360
degrees multiplied by two ππ. We can apply this formula to the
arc π΄π΅. We know its length is five
centimeters; its central angular measure is π degrees. However, we donβt know the radius
of this circle. Weβll just call this value π. We get five is π degrees divided
by 360 degrees multiplied by two ππ.

We can do the same for the length
of the minor arc from πΆ to π·. Weβll call this value πΏ. The central angle of this arc is
also π degrees. So we get πΏ is π degrees divided
by 360 degrees multiplied by two ππ. We can then see the right-hand side
of both of these equations are equal. Therefore, the left-hand sides must
also be equal. Therefore, the length of the minor
arc from πΆ to π· is five centimeters. And in fact, this result is true in
general. If two arcs in congruent circles
have the same measure, then their lengths are equal. And the reverse result is also
true. If two arcs in congruent circles
have the same length, then their measures must also be equal. But in this question, we were able
to show the length of the arc from πΆ to π· is five centimeters.

We can show a very similar result
to the one in the previous question. We want to show if the chords
between two points π΄π΅ and πΆπ· have equal length, then the arc lengths between
these points also have equal length. In fact, the exact same result will
be true in reverse. If two arcs π΄π΅ and πΆπ· have
equal length, then the chords between them will also have equal length. Weβll prove these results in both
directions.

Letβs start by assuming the arc
from π΄ to π΅ and the arc from πΆ to π· have equal length. By using the results in the
previous question, since these arcs have equal lengths, their central angles must be
equal. We also know that π΄π, π΅π, πΆπ,
and π·π are radii. They all have the same length. We can now see that triangles
π΄ππ΅ and πΆππ· are congruent by the side-angle-side criterion. Therefore, the side π΄π΅ and the
side πΆπ· must have equal length. Therefore, if the arcs have equal
lengths then the chords between them have equal lengths, now letβs start by assuming
that the chords have equal lengths. The radii of the circle still all
have the same length. And once again, we see triangles
π΄ππ΅ and πΆππ· are congruent, this time by the side-side-side rule. And congruent triangles have the
same angle. So the measure of angle π΄ππ΅ and
the measure of angle πΆππ· are equal. And therefore, the arc length of
π΄π΅ is equal to the arc length πΆπ· because their central angles are the same. Letβs now see an example of how we
can apply this property.

Given a circle π with two chords
π΄π· and π΅πΆ that have equal lengths and the arc from π΄ to π· with a length of
five centimeters, what is the length of the arc from π΅ to πΆ?

Weβre given two chords in a circle
with equal lengths, π΄π· and π΅πΆ. We can highlight these chords on
our diagram and the fact that they have equal length. Weβre also told that the length of
the minor arc from π΄ to π· is five centimeters. We can also add this to our
diagram. We need to use this to determine
the length of the minor arc from π΅ to πΆ. We can answer this question
geometrically by noticing π΄π, π΅π, πΆπ, and π·π are radii of the circle, so
they have equal length. This means that triangles π΄ππ·
and π΅ππΆ are congruent. So the measures of the central
angles of these two arcs are equal.

Then, since the central angles of
these two arcs are equal, the lengths are equal, meaning that π΅πΆ has length five
centimeters. However, we couldβve also answered
this question by just recalling if the chords between two points on a circle are
equal, then their arc lengths are also equal. Using either method, we were able
to show the length of the minor arc from π΅ to πΆ is five centimeters.

Letβs now go through one final
example.

Given that the line segment π΄π΅ is
a diameter in circle π and the measure of angle π·ππ΅ is five π₯ plus 12 degrees,
determine the measure of the minor arc from π΄ to πΆ.

In this question, weβre given
information about a circle. We need to use this information to
determine the measure of the minor arc from π΄ to πΆ. Letβs start by adding the minor arc
from π΄ to πΆ onto our diagram. We can recall that an arc of a
circle between two points is the proportion of the circumference of the circle
between those two points. And unless stated otherwise, this
means the minor arc, which is the shorter of the two arcs. So the minor arc from π΄ to πΆ is
shown on our diagram. Weβre also told that the measure of
angle π·ππ΅ is five π₯ plus 12 degrees. So we can also add this value to
our diagram.

We want to determine the measure of
arc π΄πΆ. And we recall the measure of an arc
is equal to the measure of its central angle. And the central angle of an arc is
the angle at the center of a circle subtended by the arc. And weβre given the measure of this
angle in the diagram. The measure of arc π΄πΆ is four π₯
degrees. Therefore, to determine the measure
of arc π΄πΆ, we need to find the value of π₯. To do this, weβre going to use the
fact that the line segment π΄π΅ is a diameter of the circle. In particular, because π΄π΅ is a
diameter of the circle, the angles π΅ππ· and π·ππ΄ make up a straight line. The sum of their measures will be
180 degrees. Therefore, we have 180 degrees is
equal to two π₯ degrees plus five π₯ plus 12 degrees.

We can then solve this equation for
π₯. We start by removing the degrees
symbol, and then we simplify to get 168 is equal to seven π₯. Dividing both sides of the equation
through by seven gives us that π₯ is equal to 28. Finally, we can substitute this
value of π₯ into our equation for the measure of arc π΄πΆ. The measure of arc π΄πΆ is four
times 28 degrees, which we can calculate is 96 degrees. Therefore, given that the line
segment π΄π΅ in circle π was a diameter and the measure of angle π·ππ΅ was five π₯
plus 12 degrees, we were able to determine the measure of arc π΄πΆ is 96
degrees.

Letβs now go over the key points of
this video. First, we saw that an arc of a
circle is a section of the circumference between two radii. We also saw, given two radii, there
are two possible arcs between them. We call the longer of these two
arcs the major arc and the shorter of these two arcs the minor arc. And instead of thinking about an
arc between two radii, we can think of an arc between two points on the
circumference of a circle. And the minor arc from π΄ to π΅ is
denoted by the following notation. We also called the central angle of
an arc the angle at the center of the circle between the two radii, which is
subtended by the arc. And we also defined the measure of
an arc to be the measure of its central angle. And the measure of the minor arc
from π΄ to π΅ is written in the following notation.

We also proved if the central angle
or measure of an arc in a circle of radius π is π degrees, then the length πΏ of
the arc is given by πΏ is equal to π degrees divided by 360 degrees multiplied by
two ππ. Finally, we showed two useful
results. First, if two arcs in congruent
circles have the same length, then we showed that their central angles will have
equal measure. And we also showed that the same
result was true in reverse. If the central angles of two arcs
in congruent circles have equal measure, then the two arcs will have the same
length. And we showed a very similar
result. If two arcs in congruent circles
have the same length, then the chords between the respective endpoints of these two
arcs will also have equal length.

And finally, we showed the same
result was true in reverse. If two chords in congruent circles
had equal length, then the arcs between the respective endpoints of those two chords
will have equal length.