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Question Video: Finding the Measure of an Arc given the Measure of the Other Arcs’ Central Angles Mathematics • 11th Grade

Consider the given circle. Find the measure of the arc 𝐢𝐷.

01:53

Video Transcript

Consider the given circle. Find the measure of the arc 𝐢𝐷.

We recall first that the arc 𝐢𝐷 means the minor arc between points 𝐢 and 𝐷. That’s this arc here on the diagram. We recall also that the measure of an arc is the measure of its central angle. That’s the angle between the two radii connecting the endpoints of the arc to the center of the circle. For the arc 𝐢𝐷, the two radii are the line segments 𝐢𝑀 and 𝐷𝑀. So the central angle for this arc is angle 𝐢𝑀𝐷. If we want to determine the measure of the arc 𝐢𝐷 then, we need to determine the measure of the angle 𝐢𝑀𝐷.

Now, in the figure, we’re given the measures of two other angles: angle 𝐡𝑀𝐢, which is 56 degrees, and angle 𝐴𝑀𝐷, which is 32 degrees. Now, as the line segment 𝐴𝐡 is a diameter of the circle, so it is a straight line, the measures of the three angles 𝐡𝑀𝐢, 𝐢𝑀𝐷, and 𝐴𝑀𝐷 must sum to 180 degrees. We can therefore form an equation. 56 degrees plus the measure of angle 𝐢𝑀𝐷 plus 32 degrees equals 180 degrees. We can find the measure of angle 𝐢𝑀𝐷 by subtracting 56 degrees and 32 degrees from each side of the equation. The measure of angle 𝐢𝑀𝐷 is 180 degrees minus 56 degrees minus 32 degrees, which is 92 degrees. Angle 𝐢𝑀𝐷 is the central angle for the arc 𝐢𝐷. So the measure of the arc 𝐢𝐷 is also 92 degrees.

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