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Question Video: Finding Measures of Arcs Using the Measure of the Central Angle Mathematics • 11th Grade

Given circle 𝑀 with two arcs 𝐴𝐡 and 𝐢𝐷 that have equal measures and that the arc 𝐴𝐡 has a length of 5 cm, what is the length of the arc 𝐢𝐷?

02:27

Video Transcript

Given circle 𝑀 with two arcs from 𝐴 to 𝐡 and 𝐢 to 𝐷 that have equal measures and that the arc from 𝐴 to 𝐡 has a length of five centimeters, what is the length of the arc from 𝐢 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 𝐡 and the minor arc from 𝐢 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 𝐡 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 𝐢 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 𝐴𝑀𝐡 is the central angle of the minor arc from 𝐴 to 𝐡. And the angle 𝐷𝑀𝐢 is the central angle of the minor arc 𝐢𝐷. 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 πœƒ 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 πœƒ degrees and the radius of the circle is π‘Ÿ, then 𝐿 is equal to πœƒ degrees divided by 360 degrees multiplied by two πœ‹π‘Ÿ. We can apply this formula to the arc 𝐴𝐡. We know its length is five centimeters; its central angular measure is πœƒ degrees. However, we don’t know the radius of this circle. We’ll just call this value π‘Ÿ. We get five is πœƒ degrees divided by 360 degrees multiplied by two πœ‹π‘Ÿ.

We can do the same for the length of the minor arc from 𝐢 to 𝐷. We’ll call this value 𝐿. The central angle of this arc is also πœƒ degrees. So we get 𝐿 is πœƒ degrees divided by 360 degrees multiplied by two πœ‹π‘Ÿ. 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 𝐢 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 𝐢 to 𝐷 is five centimeters.

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