11.1.2 Arcs and Chords
Key Objectives • Apply properties of arcs. • Apply properties of chords. Key Terms • A central angle is an angle whose vertex is the center of a circle. • An arc is an unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them. • A minor arc is an arc whose points are on or in the interior of a central angle. • A major arc is an arc whose points are on or in the exterior of a central angle. • If the endpoints of an arc lie on a diameter, the arc is a semicircle . • Adjacent arcs are the arcs of the same circle that intersect at exactly one point. • Within a circle or congruent circles, congruent arcs are two arcs that have the same measure. Theorems, Postulates, Corollaries, and Properties • Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measure of the two arcs. • Theorem In a circle or congruent circles: (1) Congruent central angles have congruent chords. (2) Congruent chords have congruent arcs. (3) Congruent arcs have congruent central angles. • Theorem In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc. • Theorem In a circle, the perpendicular bisector of a chord is a radius (or diameter). Example 1 Data Application
The measure of an arc is determined in this application example. The percent value of the sectors in a pie chart are given. To find m ABC , recognize that the percents given for the pie chart can represent the percent area of the sector of the circle, the percent measure of the central angle that forms the sector, or the percent measure of the length of the arc. The measure of an arc is equal to the measure of the central angle between the radii that form its endpoints. The measure of the central angle that forms ABC is 14 percent of the whole circle, or (0.14)360 ° . The measure of the arc is 50.4 ° .
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