Starter Given: Circle O – radius = 12 – AB = 12 Find: OP O A B P

Arcs of a Circle Advanced Geometry 10.3

The Circle and their Arcs ARC An arc consists of two points on a circle and all of the points on the circle needed to connect the points by a single path

The Circle and their Arcs Center of an ARC The center of an arc is the center of the circle of which the arc is a part.

The Circle and their Arcs Central Angle A central angle is an angle whose vertex is at the center of a circle. The measure of a central angle is equal to the measure of it’s intercepted arc! Note: the measure of a central angle is less than 180 degrees 63 degrees 131 degrees ?

The Circle and their Arcs Minor ARC A minor arc is an arc whose points are on or between the sides of a central angle The measure of a minor arc is always less than 180 degrees A minor arc is named with its endpoints.

The Circle and their Arcs Major ARC A major arc is an arc whose points are outside the sides of a central angle The measure of a major arc is always greater than 180 degrees A major arc must be named with three points on the circle!

The Circle and their Arcs Semicircle A semicircle is an arc whose endpoints are on the diameter of a circle The measure of a semicircle is always equal to 180 degrees A semicircle must be named with three points on the circle!

The Circle and their Arcs Measure of a Minor ARC The measure of a minor arc is equal to the measure of its central angle The measure of a minor arc is always less than 180 degrees The measure of the minor arc is indicated by

The Circle and their Arcs Measure of a Major ARC The measure of a major arc is equal to the 360 minus the measure of its central angle The measure of a major arc is always more than 180 degrees The measure of the major arc is indicated by

The Circle and their Arcs Congruent Arcs Two arcs are congruent whenever they have the same measure and are parts of the same circle or congruent circles.

The Circle and their Arcs Arcs with the same measure that are NOT Congruent Arcs 3cm 5 cm Arcs are not in the same or congruent circles!

If two central angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are congruent. If two arcs of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent. (Converse) Theorems

Congruent central angles congruent chords

Theorems Congruent arcs congruent chords

Theorems (a summary) In the same or congruent circles…

Example 1 A D C B

Example 1 A D C B

Example 2 O A B 102°

Example 3 D P Q R A BC Given: Circles P and Q Prove:

Example 3 D P Q R A BC Given: Circles P and Q Prove:

Example 4 A) What fractional part of a circle is an arc of 36°? Of 200°? B) Find the measure of an arc that is 7/12 of its circle.

Matching Problem On the whiteboard: p. 454 #1

Homework p. 455 #6 – 13, 15, 19 Read p , Notecard: Arc length (formula) Read p , Notecards: Definition of sector Area of sector formula Definition of segment

EXIT SLIP Given: Circle Q Find: AB A Q B