chapter 7 lesson 6 objective: to find the measures of central angles and arcs and the circumference
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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.
Centr
al
Centr
al
Angles
Angles
and A
rcs
and A
rcs
• 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 m
5 m
5 m
5 m
Central Angle is an angle whose vertex is the center of the circle. AA
BBCCDDABD ABC
Example 1Finding Central Angles
**Remember a circle measures **Remember a circle measures 360360°.**°.**Sleep:Sleep: 31% of 360
.31•360=111.6
FoodFood:: 9% of 360 .09•360=32.4Work:Work: 20% of 360 .20•360=72
Must DoMust Do:: 7% of 360 .07•360=25.2Entertainment:Entertainment: 18% of 360 .18•360=64.8
Other: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 arcs
AD, 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 Postulate
The 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• ABC
ABC 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 + mDY
mXY = 40 + 56 = 96
mDXM = mDX + 180
mDXM = 40 + 180
mDXM = 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