Chapter 7 Chapter 7 Lesson 6Lesson 6

Objective:Objective: To find the To find the measures of central angles measures of central angles

and arcs and the and arcs and the circumference.circumference.

Centra

l

Centra

l

Angles

Angles

and Arcs

and Arcs

• In a plane, a In a plane, a circlecircle is the set of all points. is the set of all points. • The set of all points equidistant from a given point is the The set of all points equidistant from a given point is the centercenter..• A A radiusradius is a segment that has one endpoint at the is a segment that has one endpoint at the center and the other endpoint of the circle.center and the other endpoint of the circle.• A A diameterdiameter is a segment that contains the center of a is a segment that contains the center of a circle and has both endpoints on the circle.circle and has both endpoints on the circle.

Congruent CirclesCongruent Circles have congruent radii. have congruent radii.

5 m5 m

5 m5 m

Central Angle is an angle whose vertex is the center of the circle. AA

BB CCDDABD ABC

Example 1Finding Central Angles

**Remember a circle measures **Remember a circle measures 360360°.**°.**Sleep:Sleep: 31% of 360

.31•360=111.6FoodFood:: 9% of 360 .09•360=32.4Work:Work: 20% of 360 .20•360=72Must DoMust Do:: 7% of 360 .07•360=25.2Entertainment:Entertainment: 18% of 360 .18•360=64.8Other:Other: 15% of 360 .15•360=54

• An arcarc is a part of a circle. Types of arcsTypes of arcs

• Semicircle is half of a circle.

                                                                                                                                                                                          

                          

•AA

DAEDAE

• A minor arc is smaller than a semicircle.• A major arc is greater than a semicircle.

ABAB

Minor arcMinor arc

•DD ADBADB

Major arcMajor arc

Example 2:Identifying Arcs

Identify the following in O.  1.the minor arcs

2.the semicircles

3. the major arcs that contain point A

•OOAA CC

DD EE

Example 3:Identifying Arcs

Identify the minor arcs, major arcs and semicircles in O with point A as an endpoint.  

O••

•

•

•

A

B

D

E

• minor arcsAD, AE• major arcs ADE, AED

• semicircles ADB, AEB

Adjacent arcs are arcs of the same circle that have exactly one point in common.

Postulate 7-1Postulate 7-1: Arc Addition PostulateThe measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.

•• •

AA

BB C

mABC = mAB + mBC

Example 4:Finding the Measures of Arcs

Find the measure of each arc.Find the measure of each arc.

•

58°

32°

A

B

CD

O

• BC

mBOCmBC 32• BD

180mABC

5832mBD 90• ABCABC is a semicircle.

mCDmBCmBD

• AB 32180mAB 148

Example 5:Finding the Measures of Arcs

56°

40°

M

C W

XD

Y

Find mXY and mDXM in C.

mXY = mXD + mDYmXY = 40 + 56 = 96

mDXM = mDX + 180mDXM = 40 + 180mDXM = 220

The circumference of a circle is the distance around the circle.

The number pi (π) is the ratio of the circumference of a circle to its diameter.

Theorem 7-13  Circumference of a CircleThe circumference of a circle is π times the diameter.

Circles that lie in the same plane and have the same center are concentric circles.

A car has a turning radius of 16.1 ft. The distance between the two A car has a turning radius of 16.1 ft. The distance between the two front tires is 4.7 ft. In completing the (outer) turning circle, how front tires is 4.7 ft. In completing the (outer) turning circle, how

much farther does a tire travel than a tire on the concentric inner much farther does a tire travel than a tire on the concentric inner circle?circle?

circumference of outer circle = C = 2πr = 2π(16.1) = 32.2π

To find the radius of the inner circle, subtract 4.7 ft from the turning radius. radius of the inner circle = 16.1 − 4.7 = 11.4 circumference of inner circle = C = 2πr = 2π(11.4) = 22.8π

The difference in the two distances is 32.2π − 22.8π, or 9.4π.

                                                                                                                            A tire on the turning circle travels about 29.5 ft farther than a tire on the inner

circle.

Example 6:Concentric Circles

The measure of an arc is in degrees while the arc length is a fraction of a circle's circumference.

Theorem 7-14Theorem 7-14  Arc LengthThe length of an arc of a circle is the product of the ratio

                 and the circumference of the circle. length of    =      • 2πr

Example 7: Finding Arc Length

Find the length of each arc shown in red. Leave your answer in terms of π.                                                                            

Example 8: Finding Arc Length

Find the length of a semicircle with radius of 1.3m. Leave Find the length of a semicircle with radius of 1.3m. Leave your answer in terms of your answer in terms of π.π.

r2360180

)3.1(221

6.221

3.1

Example 9: Finding Arc Length

Find he length of ADB in terms of π.π.

••

•

A

B

M D•150° 18 cm

rmADB 2360

)18(2360210

)18(2360210

21

Congruent arcs are arcs that have the same measure Congruent arcs are arcs that have the same measure andand are in the same circle or in congruent circles. are in the same circle or in congruent circles.

AssignmenAssignment:t:

Pages 389-392 Pages 389-392 #1-39#1-39