Warm-Up 1. Calculate the exact value of sine and cosine of 30° 2. Calculate the sum of the square of the sine and cosine of 30° 3. Explain what you think you would get if you did the same thing (find the sine and cosine of the angle, square them, and add them together) with 60°, 45°, or any other angle

Circles and Spheres Key Standards MM2G3. Students will understand the properties of circles. b. Understand and use properties of central, inscribed, and related angles.

Circle  What is the definition of a circle?  A circle is the locus of points that are a constant distance from a given point, called the center.  The circle is named for its center, ex  P  What is that constant distance called?  A radius is a segment whose endpoints are the center and any point on the circle.  How many radii does circle have?  An infinite number

Locus of Points  Look at the investigation on page 460 – 461 of the Geometry book.

Central Angle  Two radii form a central angle  A central angle of a circle is an angle whose vertex is the center of the circle.

Chords  A chord is a segment whose endpoints are on a circle  A diameter is a chord what contains the center of the circle.

Arcs  An arc is an unbroken part of a circle.  Minor Arcs are named for their end points.  The measure of a minor arc is defined to be the measure of its central angle.  Minor arc: Central angle < 180

Arcs  The measure of a major arc is defined as the difference between 360 and the measure of its associated minor arc.  Major arcs and semicircles are named by their end points and a point on the arc  Major arc: Central angle > 180  Semicircle: Central angle = 180

Nomenclature  Pay particular attention to the nomenclature as shown in the following slide.  The arc AB is designated: This same nomenclature will be used to designate the length of the arc later.  The measure of the arc in degrees is designated:

Example 1:

Ex. 2: Finding Measures of Arcs  Find the measure of each arc of  R. a. b. c. 80 °

Ex. 2: Finding Measures of Arcs  Find the measure of each arc of  R. a. b. c. Solution: is a minor arc, so m = m  MRN = 80 ° 80 °

Ex. 2: Finding Measures of Arcs  Find the measure of each arc of  R. a. b. c. Solution: is a major arc, so m = 360 ° – 80 ° = 280 ° 80 °

Ex. 2: Finding Measures of Arcs  Find the measure of each arc of  R. a. b. c. Solution: is a semicircle, so m = 180 ° 80 °

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 mABC = mAB+ mBC B C A

Ex. 3: Finding Measures of Arcs  Find the measure of each arc. a. b. c. m = m + m = 40 ° + 80° = 120° 40 ° 80 ° 110 °

Ex. 3: Finding Measures of Arcs  Find the measure of each arc. a. b. c. m = m + m = 120 ° + 110° = 230° 40 ° 80 ° 110 °

Ex. 3: Finding Measures of Arcs  Find the measure of each arc. a. b. c. m = 360 ° - m = 360 ° - 230° = 130° 40 ° 80 ° 110 °

W X 40 Q 40 Z Y Congruent Arcs  In a circle or in congruent circles, two minor arcs are congruent iff their corresponding central angles are congruent.  Need Congruent:  Central angles  Radii.

Ex. 4: Identifying Congruent Arcs  Find the measures of the blue arcs. Are the arcs congruent? and are in the same circle and m = m = 45 °. So, = 45 °

Ex. 4: Identifying Congruent Arcs  Find the measures of the blue arcs. Are the arcs congruent? and are in congruent circles and m = m = 80 °. So, = 80 °

Ex. 4: Identifying Congruent Arcs  Find the measures of the blue arcs. Are the arcs congruent? 65 ° m = m = 65°, but and are not arcs of the same circle or of congruent circles, so and are NOT congruent.

Application:  Determine each central angles to make a pie chart from the following data: CategoryNumber of each color %Number of Degrees in the Central Angle Blue25 Orange15 Green10

Application:  Determine each central angles to make a pie chart from the following data: CategoryNumber of each color %Number of Degrees in the Central Angle Blue25 Orange15 Green10 Total50

Application:  Determine each central angles to make a pie chart from the following data: CategoryNumber of each color %Number of Degrees in the Central Angle Blue2550 Orange1530 Green1020 Total50

Application:  Determine each central angles to make a pie chart from the following data: CategoryNumber of each color %Number of Degrees in the Central Angle Blue2550 Orange1530 Green1020 Total50100

Application:  Determine each central angles to make a pie chart from the following data: CategoryNumber of each color %Number of Degrees in the Central Angle Blue Orange Green Total50100

Application:  Determine each central angles to make a pie chart from the following data: CategoryNumber of each color %Number of Degrees in the Central Angle Blue Orange Green Total

Application:  What is the central angles if we wanted to combine Blue and Green? CategoryNumber of each color %Number of Degrees in the Central Angle Blue Orange Green Total °

Practice  Page 193, # 3 – 39 by 3’s and 19 (14 problems)