Arcs in Circles
An arc is part of a circle’s circumference.
Definition:
In a circle, the degree measure of an arc is equal to the measure of the central angle that intercepts the arc.
Definition:
In a circle, the length of an arc is a portion of the circumference.
Remembering that the arc measure is the measure of the central angle, a definition can be formed as:
Example:
In circle O, the radius is 8, and the measure of minor arc is 110 degrees. Find the length of minor arc to the nearest integer.
Solution:
Understanding how an arc is measured makes the next theorems common sense.
Theorem:
In the same circle, or congruent circles, congruent central angles have congruent arcs.
Theorem: (converse)
In the same circle, or congruent circles, congruent arcs have congruent central angles.
Remember: In the same circle, or congruent circles, congruent arcs have congruent chords. Knowing this theorem makes the next theorems seem straight forward.
Theorem:
In the same circle, or congruent circles, congruent central angles have congruent chords.
Theorem: (converse)
In the same circle, or congruent circles, congruent chords have congruent central angles.