Arcs in Circles
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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:


   =  15.35889742 = 15


 

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.