Section 10.2 – Arcs and Chords Chapter 10 - Circles Section 10.2 – Arcs and Chords

How Do You Measure a Circle or Parts of a Circle? Area Circumference Arc length Arc measure

Central Angles A central angle is an angle whose vertex is the center of the circle and whose sides intersect the circle. A P B is a central angle

Measuring Arcs The measure of an arc is the same as the measure of its associated central angle. A P B

Major and Minor Arcs A major arc is an arc whose measure is more than 180º. A minor arc is an arc whose measure is less than 180º. A semicircle is an arc that measures exactly 180º.

Naming Minor Arcs Minor arcs are named by their endpoints. Arc

Naming Major Arcs Major arcs and semicircles are named by the two endpoints and a point on the arc. Arc

Example Find the measure of each arc: a. 70º b. c.

Example Find the measure of each arc: 90º 40º 60º a. b. c.

Example Find x and :

Investigate on Circle C Draw two distinct, congruent chords in circle C. In a different color construct the central angles formed by the endpoints of your chords. Find the measure of arc RJ and arc TK. What do you notice?

Congruent Chord Theorem In the same circle or congruent circles, two minor arcs are congruent iff their corresponding chords are congruent.

Example Find the measure of arc BD.

More with Circle C Construct line through C that is perpendicular to Name the point of intersection A Name the point of intersection E Measure

THEOREM In the same circle, or in congruent circles, two chords are congruent iff they are equidistant from the center.

Investigate with Circle G Construct a diameter Construct a chord that is perpendicular to your diameter. Name the point of concurrency K. Determine the measures of

Diameter Bisector Theorem of Congruency

Theorem If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. This can be used to locate a circle’s center.

Investigate on Circle E. Draw any two chords that are not parallel to each other Draw the perpendicular bisector of each chord. The perpendicular bisectors should intersect at the circles center. These are diameters.