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

Given a circle ๐‘€ with two chords ๐ด๐ท and ๐ต๐ถ that have equal lengths and the arc ๐ด๐ท with a length of 5 cm, what is the length of the arc ๐ต๐ถ?

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Video Transcript

Given a circle ๐‘€ with two chords ๐ด๐ท and ๐ต๐ถ that have equal lengths and the arc from ๐ด to ๐ท with a length of five centimeters, what is the length of the arc from ๐ต to ๐ถ?

Weโ€™re given two chords in a circle with equal lengths, ๐ด๐ท and ๐ต๐ถ. We can highlight these chords on our diagram and the fact that they have equal length. Weโ€™re also told that the length of the minor arc from ๐ด to ๐ท 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 ๐ต to ๐ถ. We can answer this question geometrically by noticing ๐ด๐‘€, ๐ต๐‘€, ๐ถ๐‘€, and ๐ท๐‘€ are radii of the circle, so they have equal length. This means that triangles ๐ด๐‘€๐ท and ๐ต๐‘€๐ถ 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 ๐ต๐ถ has length five centimeters. However, we couldโ€™ve 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 ๐ต to ๐ถ is five centimeters.

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