 11 Maintaining Mathematical Proficiency Chapter

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11 Maintaining Mathematical Proficiency Chapter
```Name _________________________________________________________ Date _________
Chapter
11
Maintaining Mathematical Proficiency
Find the area of the figure.
1.
2.
6.8 in.
4.3 in.
12.8 ft
8.9 ft
16.2 ft
Find the missing dimension.
3. A rectangle has an area of 25 square inches and a length of 10 inches. What is the
width of the rectangle?
4. A triangle has an area of 32 square centimeters and a base of 8 centimeters. What
is the height of the triangle?
318 Geometry
Student Journal
Name_________________________________________________________
Date __________
and Arc Length
11.1 Circumference
For use with Exploration 11.1
Essential Question How can you find the length of a circular arc?
1
EXPLORATION: Finding the Length of a Circular Arc
Work with a partner. Find the length of each gray circular arc.
a. entire circle
b. one-fourth of a circle
5
y
5
3
1
−5
−3
−1
3
1
A
1
3
5x
−5
−3
−5
−5
4
2
A
−4
5x
y
4
−2
B
3
d. five-eighths of a circle
y
−2
A
1
−3
2
−4
−1
−3
c. one-third of a circle
C
y
C
2
B
4
x
−4
A
−2
C
2
B
4
x
−2
−4
Geometry
Student Journal
319
Name _________________________________________________________ Date _________
11.1
2
Circumference and Arc Length (continued)
EXPLORATION: Writing a Conjecture
Work with a partner. The rider is attempting to stop with the front tire of the motorcycle
in the painted rectangular box for a skills test. The front tire makes exactly one-half additional
revolution before stopping. The diameter of the tire is 25 inches. Is the front tire still in contact
with the painted box? Explain.
3 ft
3. How can you find the length of a circular arc?
4. A motorcycle tire has a diameter of 24 inches. Approximately how many inches
does the motorcycle travel when its front tire makes three-fourths of a revolution?
320 Geometry
Student Journal
Name_________________________________________________________
Date __________
with Vocabulary
11.1 Notetaking
For use after Lesson 11.1
In your own words, write the meaning of each vocabulary term.
circumference
arc length
Core Concepts
Circumference of a Circle
The circumference C of a circle is C = π d or C = 2π r , where
d is the diameter of the circle and r is the radius of the circle.
Notes:
r
d
C
C = π d = 2π r
Geometry
Student Journal
321
Name _________________________________________________________ Date _________
11.1
Notetaking with Vocabulary (continued)
Arc Length
In a circle, the ratio of the length of a given arc to the circumference
is equal to the ratio of the measure of the arc to 360°.
A
P
Arc length of 
AB
m
AB
=
, or
2π r
360°
m
AB
 2π r
Arc length of 
AB =
360°
r
B
Notes:
Multiply degree measure by
, or
.
360°
180°
360°
180°
, or
.
Notes:
322 Geometry
Student Journal
Name_________________________________________________________
11.1
Date __________
Notetaking with Vocabulary (continued)
Extra Practice
In Exercises 1–5, find the indicated measure.
1. diameter of a circle with a circumference of 10 inches
2. circumference of a circle with a radius of 3 centimeters
3. radius of a circle with a circumference of 8 feet
4. circumference of a circle with a diameter of 2.4 meters
AC
5. arc length of 
A
3 in.
B
70°
C
In Exercises 6 and 7, convert the angle measure.
7. Convert
5π
6
Geometry
Student Journal
323
Name _________________________________________________________ Date _________
of Circles and Sectors
11.2 Areas
For use with Exploration 11.2
Essential Question How can you find the area of a sector of a circle?
1
EXPLORATION: Finding the Area of a Sector of a Circle
Work with a partner. A sector of a circle is the region bounded by two radii of the
circle and their intercepted arc. Find the area of each shaded circle or sector of a circle.
a. entire circle
8
b. one-fourth of a circle
y
8
4
−8
4
−4
4
8x
−8
−4
4
−4
−4
−8
−8
c. seven-eighths of a circle
4
y
8x
d. two-thirds of a circle
y
y
4
−4
4x
−4
4
x
−4
324 Geometry
Student Journal
Name_________________________________________________________
11.2
2
Date __________
Areas of Circles and Sectors (continued)
EXPLORATION: Finding the Area of a Circular Sector
Work with a partner. A center pivot irrigation system consists of 400 meters of
sprinkler equipment that rotates around a central pivot point at a rate of once every 3
days to irrigate a circular region with a diameter of 800 meters. Find the area of the
sector that is irrigated by this system in one day.
3. How can you find the area of a sector of a circle?
4. In Exploration 2, find the area of the sector that is irrigated in 2 hours.
Geometry
Student Journal
325
Name _________________________________________________________ Date _________
with Vocabulary
11.2 Notetaking
For use after Lesson 11.2
In your own words, write the meaning of each vocabulary term.
population density
sector of a circle
Core Concepts
Area of a Circle
The area of a circle is
r
A = π r2
where r is the radius of the circle.
Notes:
326 Geometry
Student Journal
Name_________________________________________________________
11.2
Date __________
Notetaking with Vocabulary (continued)
Area of a Sector
The ratio of the area of a sector of a circle to the area of the whole
circle (π r 2 ) is equal to the ratio of the measure of the intercepted arc
to 360°.
A
P
r
B
Area of sector APB
m
AB
, or
=
2
πr
360°
m
AB
 π r2
Area of sector APB =
360°
Notes:
Extra Practice
In Exercises 1–2, find the indicated measure.
1. area of  M
5 cm
2. area of  R
7m
M
R
Geometry
Student Journal
327
Name _________________________________________________________ Date _________
11.2
Notetaking with Vocabulary (continued)
In Exercises 3–8, find the indicated measure.
3. area of a circle with a diameter of 1.8 inches
4. diameter of a circle with an area of 10 square feet
5. radius of a circle with an area of 65 square centimeters
6. area of a circle with a radius of 6.1 yards
7. areas of the sectors formed by ∠ PQR
8. area of Y
P
4 cm 140°
Q
R
Y
25°
X
Z
A = 4.1 square feet
9. About 70,000 people live in a region with a 30-mile radius. Find the population density in
people per square mile.
328 Geometry
Student Journal
Name_________________________________________________________
Date __________
of Polygons and Composite Figures
11.3 Areas
For use with Exploration 11.3
Essential Question How can you find the area of a regular polygon?
The center of a regular polygon is the center of its
circumscribed circle.
The distance from the center to any side of a regular
polygon is called the apothem of a regular
polygon.
1
apothem CP
P
C
center
EXPLORATION: Finding the Area of a Regular Polygon
Go to BigIdeasMath.com for an interactive tool to investigate this exploration.
Work with a partner. Use dynamic geometry software to construct each regular
polygon with side lengths of 4, as shown. Find the apothem and use it to find the area of
the polygon. Describe the steps that you used.
a.
b.
7
4
6
C
5
3
E
3
2
1
−3
−2
1
B
0
−1
C
4
2
A
D
0
1
2
3
A
−5
−4
−3
−2
B
0
−1
0
1
2
3
Geometry
Student Journal
4
5
329
Name _________________________________________________________ Date _________
11.3
1
Areas of Polygons and Composite Figures (continued)
EXPLORATION: Finding the Area of a Regular Polygon (continued)
c.
d.
8
E
7
8
G
5
5
C
3
4
H
2
2
C
3
2
1
−5 −4 −3 −2 −1
D
7
6
4
A
E
10
9
6
F
F
D
A
B
0
0
1
2
3
4
5
−6 −5 −4 −3 −2 −1
1
0
0 1
2
B
3
4
5
6
EXPLORATION: Writing a Formula for Area
Work with a partner. Generalize the steps you used in Exploration 1 to develop a
formula for the area of a regular polygon.
3. How can you find the area of a regular polygon?
4. Regular pentagon ABCDE has side lengths of 6 meters and an apothem of
approximately 4.13 meters. Find the area of ABCDE.
330 Geometry
Student Journal
Name_________________________________________________________
Date __________
with Vocabulary
11.3 Notetaking
For use after Lesson 11.3
In your own words, write the meaning of each vocabulary term.
center of a regular polygon
apothem of a regular polygon
central angle of a regular polygon
Core Concepts
Area of a Rhombus or Kite
The area of a rhombus or kite with diagonals d1 and d 2 is
d2
1
d1d 2 .
2
d2
d1
d1
Notes:
Geometry
Student Journal
331
Name _________________________________________________________ Date _________
11.3
Notetaking with Vocabulary (continued)
Area of a Regular Polygon
The area of a regular n-gon with side length s is one-half the
product of the apothem a and the perimeter P.
A =
a
1
1
aP, or A = a  ns
2
2
s
Notes:
Extra Practice
In Exercises 1 and 2, find the area of the kite or rhombus.
1.
2.
5 cm
10 in.
7 in.
2 cm
3 cm
2 cm
7 in.
10 in.
332 Geometry
Student Journal
Name_________________________________________________________
11.3
Date __________
Notetaking with Vocabulary (continued)
3. Find the measure of a central angle of a regular polygon with 8 sides.
4. The central angles of a regular polygon are 40°. How many sides does the polygon have?
5. A regular pentagon has a radius of 4 inches and a side length of 3 inches.
a. Find the apothem of the pentagon.
b. Find the area of the pentagon.
6. A regular hexagon has an apothem of 10 units.
a. Find the radius of the hexagon and the length of one side.
b. Find the area of the hexagon.
Geometry
Student Journal
333
Name _________________________________________________________ Date _________
of Changing Dimensions
11.4 Effects
For use with Exploration 11.4
Essential Question How does changing one or more dimensions of a rectangle
affect its perimeter and area?
1
EXPLORATION: Changing One Dimension
Work with a partner.
a. Fold a piece of paper in half twice so that there are four
layers.
b. Draw a rectangle on the paper. Then use scissors to cut
through the four layers so that you cut out four
congruent rectangles.
c. Place two rectangles side-by-side along either the
length or the width so that you form a figure with
double the length or double the width of a single
rectangle.
d. Compare the perimeter and the area of the figure
formed by the two rectangles to the perimeter and
the area of a single rectangle.
e. Make a conjecture about how doubling the length or the
width of a rectangle affects the perimeter and the area.
f. Make a conjecture about how multiplying the length or
the width of a rectangle by a positive number k affects
the perimeter and the area.
334 Geometry
Student Journal
Name_________________________________________________________
11.4
2
Date __________
Effects of Changing Dimensions (continued)
EXPLORATION: Changing Dimensions Proportionally
Work with a partner. Use the rectangles from Exploration 1.
a. Arrange the four rectangles so that you form a rectangle
with double the length and double the width of a single
rectangle.
b. Compare the perimeter and the area of the figure
formed by the four rectangles to the perimeter and the
area of a single rectangle.
c. Make a conjecture about how doubling the length and
the width of a rectangle affects the perimeter and the
area.
d. Make a conjecture about how multiplying the length
and the width of a rectangle by a positive number k
affects the perimeter and the area.
3. How does changing one or more dimensions of a rectangle affect its perimeter and
area?
Geometry
Student Journal
335
Name _________________________________________________________ Date _________
with Vocabulary
11.4 Notetaking
For use after Lesson 11.4
In your own words, write the meaning of each vocabulary term.
perimeter
area
similar figures
Core Concepts
Changing Dimensions Proportionally
When you multiply all the linear dimensions of a figure by a positive number k, the perimeter and the area
change as shown.
Before multiplying all
dimensions by k
After multiplying all
dimensions by k
Perimeter
P
kP
Area
A
k2A
Notes:
336 Geometry
Student Journal
Name_________________________________________________________
11.4
Date __________
Notetaking with Vocabulary (continued)
Extra Practice
In Exercises 1–6, describe how the change affects the perimeter and the area of
the figure.
2. multiplying the base by 2
3
1. tripling the length
5 cm
20 m
8 cm
15 m
3. multiplying all the linear dimensions by 6
4. doubling the height and multiplying all side
lengths by 4
4 ft
5 ft
9 ft
3 in.
Geometry
Student Journal
337
Name _________________________________________________________ Date _________
11.4
Notetaking with Vocabulary (continued)
5. multiplying the length by 4
3
6. multiplying all the linear dimensions by 1
6
and tripling the width
5 ft
12 ft
6 mm
7. Describe how the change affects the circumference and the area of the circle.