Video Transcript
Given circle 𝑀 with two arcs from
𝐴 to 𝐵 and 𝐶 to 𝐷 that have equal measures and that the arc from 𝐴 to 𝐵 has a
length of five centimeters, what is the length of the arc from 𝐶 to 𝐷?
In this question, we’re given a
circle. And we’re told that two of its arcs
have the same lengths, the minor arc from 𝐴 to 𝐵 and the minor arc from 𝐶 to
𝐷.
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 we’re told that the minor arc
from 𝐴 to 𝐵 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 𝐶 to 𝐷. To answer this question, let’s
start by recalling what the measure of an arc means. The measure of an arc is the
measure of its central angle. That’s the angle at the center of
the circle, which is subtended by the arc. For example, the angle 𝐴𝑀𝐵 is
the central angle of the minor arc from 𝐴 to 𝐵. And the angle 𝐷𝑀𝐶 is the central
angle of the minor arc 𝐶𝐷. And since the measure of these two
arcs are equal, the measures of their central angles must also be equal.
Let’s then say that these angles
have a measure of 𝜃 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 𝐿. If its central angle is 𝜃 degrees
and the radius of the circle is 𝑟, then 𝐿 is equal to 𝜃 degrees divided by 360
degrees multiplied by two 𝜋𝑟. We can apply this formula to the
arc 𝐴𝐵. We know its length is five
centimeters; its central angular measure is 𝜃 degrees. However, we don’t know the radius
of this circle. We’ll just call this value 𝑟. We get five is 𝜃 degrees divided
by 360 degrees multiplied by two 𝜋𝑟.
We can do the same for the length
of the minor arc from 𝐶 to 𝐷. We’ll call this value 𝐿. The central angle of this arc is
also 𝜃 degrees. So we get 𝐿 is 𝜃 degrees divided
by 360 degrees multiplied by two 𝜋𝑟. 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 𝐶 to 𝐷 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 𝐶 to 𝐷 is five centimeters.
