File

9.2 Angles and Arcs
STANDARDS
Objectives
Content: G.C.2, G.C.5
Practices: 1, 2, 3, 4, 6, 7, 8
Use with Lesson 10-2
• Derive and apply the formula for arc length.
• Derive the fact that the length of the arc intercepted by an angle
is proportional to the radius.
• Define and apply radian measure.
A central angle of a circle is an angle in which the vertex is the center of
the circle. In the figure, ∠ACB is a central angle.
An arc is a portion of a circle defined by two endpoints. A minor arc is the
shortest arc connecting two endpoints. A major arc is the longest arc
connecting two endpoints.
minor arc AB
B
A
C
major arc ADB
Arc length is the distance between endpoints along an arc.
D
EXAMPLE 1
Investigate Arc Length
G.C.2
EXPLORE An architect is designing the seating area for a theater.
The seating area is formed by a region that lies between two circles,
as shown in the figure. The architect is planning to place a brass rail in
front of the first row of seats. She wants to know the length of the rail.
a. COMMUNICATE PRECISELY In the figure, the arc that represents the
brass rail is marked 120°. Explain what this means. SMP 6
seating area
120°
R
S
T
15 ft
brass rail
b. CALCULATE ACCURATELY Explain how to find the circumference
of T. Express the circumference in terms of π. SMP 6
c. REASON ABSTRACTLY Explain how you can use your answer to part b and
proportional reasoning to find the length of the brass rail. Express the length in terms
of π and to the nearest tenth of a foot. SMP 2
294 CHAPTER 9 Circles
Copyright © McGraw-Hill Education
d. DESCRIBE A METHOD The
⏜architect is considering changing the radius of T or
RS. Describe a general method she can use to find the length
changing
the
measure
of
⏜
of RS . SMP 8
e. USE A MODEL The seating area is 15 feet deep, so the distance from the center of the
circle to the edge of the seating area is 30 feet. If the architect wants to surround the
entire seating area with a brass rail, what length of rail does she need? Explain your
reasoning. SMP 4
The Key Concept box summarizes the relationship you discovered in the previous exploration.
KEY CONCEPT
Arc Length
Complete the proportion and the equation.
The ratio of the length of an arc to the circumference of the circle
is equal to the ratio of the degree measure of the arc to 360.
2π r
Proportion: ______ = __________
EXAMPLE 2
x°
r
Equation: =
Apply Arc Length
G.C.2
A model train is set up on a circular track with a diameter of
8 feet. As the train travels from the station to the coal mine, it
moves 84° around the track. The train moves at a constant rate
of 6 inches per second. Follow these steps to determine how
long it takes the train to travel from the station to the coal mine.
a. INTERPRET PROBLEMS Sketch and label a figure that represents
this situation in the space at the right. SMP 1
b. CALCULATE ACCURATELY Explain how to find the length of the track
from the station to the coal mine to the nearest hundredth of a foot. SMP 6
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c. CALCULATE ACCURATELY Explain how to find the time it takes the train to travel from
the station to the coal mine to the nearest second. SMP 6
d. CRITIQUE REASONING A student said that if the radius of the track were doubled,
then the amount of time it takes the train to travel from the station to the coal mine
would also be doubled. Do you agree? Explain. SMP 3
9.2 Angles and Arcs 295
e. CRITIQUE REASONING A student said that if the angle were doubled, then the amount
of time it takes the train to travel from the station to the coal mine would also be
doubled. Do you agree? Explain. SMP 3
EXAMPLE 3
Investigate Proportionality
G.C.2, G.C.5
Follow these steps to investigate how the length of the arc intercepted by a central
angle of a circle is related to the radius.
45°
3
a. CALCULATE ACCURATELY The figure shows an arc intercepted by a central angle
of 45°. Show how to find the length of the arc in terms of π. SMP 1
b. FIND A PATTERN Enlarge the circle so the measure of the central angle remains 45.
Complete the table by finding the length of the arc (in terms of π) for each of the given
radii. SMP 7
Radius of circle, r
3
5
11
15
r
Length of arc, c. COMMUNICATE PRECISELY Look for a pattern. When the measure of the central angle
is 45, what type of relationship do you notice between the arc length and the radius?
Explain. SMP 6
d. FIND A PATTERN Repeat the above process and complete the following table,
but this time consider arcs with a central angle of 60°. SMP 7
Radius of circle, r
2
4
5
10
r
60°
r
Length of arc, f. DESCRIBE A METHOD Now generalize your findings. Suppose the measure of the
central angle is x. What type of relationship will exist between the arc length and the
radius? How can you find the length of the arc if you know the radius? SMP 8
296 CHAPTER 9 Circles
Copyright © McGraw-Hill Education
e. COMMUNICATE PRECISELY Look for a pattern in your table. When the measure
of the central angle is 60, what type of relationship do you notice between the arc
length and the radius? Explain. SMP 6
You can use your findings from Example 3 to define the radian measure of an angle. Much like a
foot and an inch are two different units for measuring the length of a line, a radian and a degree
are two different units for measuring the size of an angle. An angle that measures 1 radian will
mark off an arc on a circle that is equal in length to the radius of the circle. Because degrees and
radians are two units of measurement, there is a conversion factor between them.
KEY CONCEPT
Radian Measure
Complete the formula.
The length of the arc intercepted by an angle is proportional
to the radius. The constant of proportionality for this
relationship is the radian measure of the angle.
x°
Formula: The radian measure of an angle of x°=
EXAMPLE 4
Apply Degrees and Radians
A carpenter is ordering metal brackets from a specialty
company. The brackets are available in a variety of angles,
as shown. The carpenter wants to know which of the
brackets has the greatest angle measure and which of
the brackets, if any, are a right angle.
a. CALCULATE ACCURATELY Explain how to express the
measure of Bracket A in radians. SMP 6
r
G.C.5
Bracket
Angle
A
70°
B
3π
_____
radians
4
C
1 radian
D
π
___
radians
2
70°
Bracket A
Copyright © McGraw-Hill Education
b. REASON QUANTITATIVELY Without doing any further calculations, can the carpenter
determine which of the brackets has the greatest angle measure? Explain. SMP 2
c. DESCRIBE A METHOD Suppose you know the radian measure of an angle. Explain how
you can convert the measure to degrees. SMP 8
d. CALCULATE ACCURATELY Are any of the brackets a right angle? Explain. SMP 6
e. CRITIQUE REASONING The carpenter orders a bracket that can be adjusted to any
angle measure from 0 radians to π radians. The carpenter claims that this bracket can
be adjusted to form a straight angle. Do you agree? Explain. SMP 6
9.2 Angles and Arcs 297
PRACTICE
1. The figure
⏜ shows a circular flower bed. Part of the fence surrounding the flower
bed, AB, is damaged and needs to be replaced. The fencing costs $8.75 per
linear foot. G.C.2
A
72°
B
a. USE A MODEL What will it cost to replace the damaged portion of the
fence? SMP 4
18 ft
b. REASON QUANTITATIVELY What is the length of the undamaged portion
of the fence to the nearest tenth of a foot? SMP 2
2. COMMUNICATE PRECISELY The clock at the Palace of Westminster in London is
best known by the name Big Ben. The minute hand of the clock is 14 feet long.
Explain how you can determine how far the tip of the minute hand moves between
3 pm and 3:10 pm. Round to the nearest tenth. G.C.2, SMP 6
Q
3. At an amusement park, go-karts travel around a circular track with a radius of
90 feet. As the go-karts travel from point P to point Q, they cover a distance
of 165 feet. G.C.2
a. USE A MODEL What is the measure of ∠QRP to the nearest degree? SMP 4
165 ft
R
90 ft
P
b. DESCRIBE A METHOD Suppose you know the distance in feet, d, that a go-kart
travels. How can you find the measure x (in degrees) of the arc that describes the
go-kart’s path? Include a written and algebraic description. SMP 8
CALCULATE ACCURATELY Convert each degree measure to the equivalent measure in
radians. G.C.5, SMP 6
5. 135°
298 CHAPTER 9 Circles
6. 18°
7. 65°
8. 120°
Copyright © McGraw-Hill Education
⏜
⏜
4. CRITIQUE REASONING In C, AB has a central angle⏜
of 25°. In D, EF has a central
angle of 25°.
⏜ A student claimed that the arc length of AB must be equal to the arc
length of EF. Do you agree? If so, explain why. If not, explain whether there is ever a
situation in which you can conclude that the arc lengths are equal. G.C.2, SMP 3
CALCULATE ACCURATELY Convert each radian measure to the equivalent measure in
degrees. G.C.5, SMP 6
π
9. ___ radians
6
5π
10. _____ radians
π
11. ____ radians
6
12
12. 3 radians
REASON ABSTRACTLY Determine whether each statement is always, sometimes, or
never true. Explain. G.C.5, SMP 2
13. If a central angle measures 2 radians, then the length of the arc it intercepts is twice
the length of the circle’s radius.
π
2
14. The radian measure of an obtuse angle is less than ___.
15. In a circle, the length of the arc intercepted by a central angle is equal to the radius of
the circle.
16. An angle’s measure is 1 in radians and 45 in degrees.
17. USE A MODEL As a pendulum swings from point A to point B, it sweeps out an angle
5π
of _____ radians. What is the distance the weight at the end of the pendulum travels as
18
it swings from A to B and back to A? Round to the nearest tenth. G.C.5, SMP 4
10 cm
A
18. CRTIQUE REASONING An archery target consists
⏜ of three
concentric⏜
circles, as shown. Latanya says that ST must be 3 times
as long as QR. Melanie says that it is not possible to determine this
relationship without knowing the measure of ∠SPT. Who is correct?
Explain. G.C.2, G.C.5, SMP 3
B
S
Q
P
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1 ft
R
1 ft
1 ft
T
19. REASON ABSTRACTLY A bicycle wheel has a diameter of 27.5 inches. On a trip
through town, the wheel rotated 2143 times around plus an additional 62°. How far
G.C.5, SMP 2
did the bicycle travel? 9.2 Angles and Arcs 299