Question Video: Finding the Measure of an Arc given the Measure of the Other Arcsβ Central Angles Mathematics • 11th Grade

Consider the given circle. Find the measure of the arc πΆπ·.

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Video Transcript

Consider the given circle. Find the measure of the arc πΆπ·.

We recall first that the arc πΆπ· means the minor arc between points πΆ and π·. Thatβs this arc here on the diagram. We recall also that the measure of an arc is the measure of its central angle. Thatβs the angle between the two radii connecting the endpoints of the arc to the center of the circle. For the arc πΆπ·, the two radii are the line segments πΆπ and π·π. So the central angle for this arc is angle πΆππ·. If we want to determine the measure of the arc πΆπ· then, we need to determine the measure of the angle πΆππ·.

Now, in the figure, weβre given the measures of two other angles: angle π΅ππΆ, which is 56 degrees, and angle π΄ππ·, which is 32 degrees. Now, as the line segment π΄π΅ is a diameter of the circle, so it is a straight line, the measures of the three angles π΅ππΆ, πΆππ·, and π΄ππ· must sum to 180 degrees. We can therefore form an equation. 56 degrees plus the measure of angle πΆππ· plus 32 degrees equals 180 degrees. We can find the measure of angle πΆππ· by subtracting 56 degrees and 32 degrees from each side of the equation. The measure of angle πΆππ· is 180 degrees minus 56 degrees minus 32 degrees, which is 92 degrees. Angle πΆππ· is the central angle for the arc πΆπ·. So the measure of the arc πΆπ· is also 92 degrees.