Arcs and Chords

1. Arcs and Chords In a plane, an angle whose vertex is the center of a circle is a central angle of the circle. Minor arc: Central angle < 180 Major arc: Central angle > 180 Semicircle: Central angle = 180

2. Naming of an Arc • Minor Arcs are named for their end points. • Major arcs and semicircles are named by their end points and a point on the arc

3. Measure of an Arc • The measure of a minor arc is defined to be the measure of its central angle. • The measure of a major arc is defined as the difference between 360 and the measure of its associated minor arc.

4. Arcs

5. B A C m ABC = m AB + m BC Arc Addition Postulate • Adjacent arcs have exactly one point in common. • The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs

6. Congruent Arcs • In a circle or in congruent circles, two minor arcs are congruent iff their corresponding central angles are congruent. W X 40 Q 40 Z Y

7. Arc Length • The arc length is different from the degree measure of an arc. • Suppose a circle was made of string. • The length of the arc would be the linear distance of that piece of string representing the arc.

8. Arc Length • The length of the arc is a part of the circumference proportional to the measure of the central angle when compared to the entire circle. • Arc Length/Circumference = Central Angle/360), or • Arc Length = Circumference * (Central Angle/360)

9. And the Theorems • In the same circle, or in congruent circles, two minor arcs are congruent iff their corresponding chords are congruent

10. If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

11. If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.