Chapter 7 Lesson 6

1
Chapter 7 Lesson 6

• Objective To find the measures of central angles
  and arcs and the circumference.

2
Central Angles and Arcs

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

3
Congruent Circles have congruent radii.
5 m
5 m
Central Angle is an angle whose vertex is the
center of the circle.
A
C
D
B
4
Example 1 Finding Central Angles
Remember a circle measures 360.
Sleep 31 of 360
.31360111.6 Food 9 of 360
.0936032.4
Work 20 of 360
.2036072 Must Do 7 of 360
.0736025.2
Entertainment 18 of 360
.1836064.8 Other 15 of 360
.1536054
5

• An arc is a part of a circle.
• Types of arcs
• Semicircle is half of a circle.
•                                                 
                                                    
                                                    
                                                    
                

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

D
6
Example 2Identifying Arcs

• the minor arcs
• the semicircles
• 3. the major arcs that contain point A

7
Example 3Identifying Arcs
Identify the minor arcs, major arcs and
semicircles in O with point A as an
endpoint.  

D

A

O

• minor arcs
• AD, AE

B
E

• major arcs
• ADE, AED

• semicircles
• ADB, AEB

8
Adjacent arcs are arcs of the same circle that
have exactly one point in common.
Postulate 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.
mABC mAB mBC


C
B

A
9
Example 4Finding the Measures of Arcs
Find the measure of each arc.
58

• BC

D
C
B

• BD

32
O

• ABC

A

• AB

ABC is a semicircle.
10
Example 5Finding the Measures of Arcs
Find mXY and mDXM in C.
M
mXY mXD mDY
mXY 40 56
96
W
Y
C
56
mDXM mDX 180
D
40
X
mDXM 40 180
mDXM 220
11
The circumference of a circle is the distance
around the circle. The number pi (p) is the
ratio of the circumference of a circle to its
diameter.
Theorem 7-13  Circumference of a Circle The
circumference of a circle is p times the
diameter.
12
Circles that lie in the same plane and have the
same center are concentric circles.
13
Example 6 Concentric Circles
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
much farther does a tire travel than a tire on
the concentric inner circle? circumference of
outer circle C 2pr 2p(16.1) 32.2p 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 2pr 2p(11.4) 22.8p The
difference in the two distances is 32.2p - 22.8p,
or 9.4p.                                      
                                                  
                                      A tire on
the turning circle travels about 29.5 ft farther
than a tire on the inner circle.
14
The measure of an arc is in degrees while the arc
length is a fraction of a circle's circumference.
Theorem 7-14  Arc Length The length of an arc of
a circle is the product of the ratio
                 and the circumference of the
circle. length of         2pr
15
Example 7 Finding Arc Length
Find the length of each arc shown in red. Leave
your answer in terms of p.                     
                                                  
     
16
Example 8 Finding Arc Length
Find the length of a semicircle with radius of
1.3m. Leave your answer in terms of p.
17
Example 9 Finding Arc Length
Find he length of ADB in terms of p.
18
Congruent arcs are arcs that have the same
measure and are in the same circle or in
congruent circles.
19
Assignment
Pages 389-392 1-39