Download Chapter 5: Geometry Math@Work

Chapter 5: Geometry
Study Skills
5.1Angles
5.2Perimeter
5.3Area
5.5
Volume and Surface Area
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5.6Triangles
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5.4Circles
5.7
Square Roots and the Pythagorean Theorem
Chapter 5 Projects
Math@Work
Foundations Skill Check for Chapter 6
Math@Work
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Introduction
If you plan to go into architecture, there are several different areas to choose from which have a varying range of
daily tasks. You can be an architect that creates the broad sketches to present to clients, you could design the detailed
construction documents, or you could be on site during construction to ensure everything goes smoothly according to
the plans. Every career path in architecture requires many math skills, from measuring walls and doorframes to working
with geometric structures to converting units, and the ability to effectively communicate with members of your team.
Suppose you decide to pursue a career as a project architect at a large firm. While creating detailed construction
drawings, you will need to know how to answer several questions that will be asked during the creation of the project.
What is the final square footage of the building and individual rooms? Is the cost of materials needed to construct the
project within the budget? Does the design meet fire safety guidelines? Determining the answers to these questions (and
many more) require several of the skills covered in this chapter and the previous chapter. At the end of the chapter, we’ll
come back to this topic and explore how math is used as an architect
Chapter 5: Geometry 173
Study Skills
Study Skills for Success in a Math Course
1. Reading Your Textbook/Workbook­ One of the most important skills when taking a math class is learning how
to read a math textbook. This was explained in the study skills for Chapter 4. Reading a section before the instructor
teaches the content and then reading it again afterwards are important strategies for success in a math course. Even if
you don’t have time to read the entire assigned section, you should get an overview by reading the introduction and
summary and looking at section objectives, headings, and vocabulary terms.
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2. Taking Notes Take notes in class using a method that works for you. There are many different note-taking
strategies like the Cornell method and Concept Mapping. You can try researching these methods and others on the
Internet to see if it they might work better than your current note-taking system. Be sure to date your class notes and
write the topic or section heading at the top of the page so that you can organize your notes later.
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3. Review Always go back and read through your notes as soon as possible after class to make sure they are readable,
write down any questions you had, or fill in any gaps that you missed during class. Mark any information that is
incomplete so that you can get it from the textbook or your instructor later. It is important for you to review your notes
as soon as possible after class so that you can make any changes while the information is fresh in your mind.
4. Organize As you review your notes each day, be sure to label them using categories such as definitions, theorems,
formulas, examples, and procedures. You could also highlight each category using a different colored highlighter as
long as you are consistent through your notes.
5. Study Aids Use index or note cards to help you remember important definitions, theorems, formulas, or
procedures. Use the front of the card for the vocabulary term, theorem name, formula name, or procedure description.
Write the definition, the theorem, the formula, or the procedure on the back of the index card, along with a description
in your own words. You might also try using the Frayer model presented in Chapter 2.
6. Practice, practice, practice! Math is like playing a sport. You don’t get good at basketball if you don’t
practice—the same is true of math. Math can’t be learned by just listening and watching your instructor work through
problems. You have to be actively involved in doing the math yourself. Work through the examples in the book, do
some practice exercises at the end of the section or chapter, and keep up with homework assignments on a daily basis.
8. Understand; Don’t memorize Don’t try to memorize formulas or theorems without understanding them. Try
describing or explaining them in your own words or look for patterns in formulas so that you don’t have to memorize
them. In this chapter, you will learn several formulas to find the perimeter of different shapes. You don’t need to
memorize every perimeter formula if you understand that perimeter is equal to the sum of the lengths of the sides of
the figure.
9. Study Plan to study 2-3 hours outside of class for every hour spent in math class. If your math class meets 4 days
a week for an hour then you should spend 8-12 hours outside of class, reviewing, studying, and practicing. If math is
your most difficult subject then study it while you are alert and fresh. Also, pick a study time when you will have the
least interruptions or distractions so that you can concentrate.
10. Manage Your Time Don’t spend more than 10-15 minutes working on a single problem. If you can’t get the
answer, put it aside and go on to another one. You may learn something from the next problem that will help you with
the one you couldn’t do. Mark the problem so that you can ask your instructor about it at the next class. It may also help
to work a similar but perhaps easier problem that appears near that problem in the exercises. Most textbooks include the
answers to the even or odd-numbered exercises, so if you are assigned an odd-numbered problem for homework, work
the even-numbered problem right before or after it for practice.
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7. Homework When doing homework, always allow plenty of time to get it done before it is due. Work some
practice problems before starting the assigned problems to make sure you know what you are doing and to build up
your confidence. Check your answers when possible to make sure they are correct. With word or application problems,
always review your answer to see if it appears reasonable. Use the estimation techniques that you have learned to
determine if your answer makes sense. Try working the problem a different way to see if you come up with the same
answer.
Name:Date:
5.1 Angles
Objectives
Success Strategy
Understand the terms point, line, and plane.
Know the definition of an angle and how to
measure an angle.
Be able to classify an angle by its measure.
There are a lot of terms in this section, so be sure
to devote a section in your notebook to writing
down all of the terms and their definitions. You
could also use index cards and the Frayer model
from Chapter 2.
Recognize congruent angles, vertical angles, and
adjacent angles.
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Know when lines are parallel and perpendicular.
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Recognize complementary angles and
supplementary angles.
Understand
Concepts
Go to Software First, read through Learn in Lesson 5.1 of the software. Then, work through the
problems in this section to expand your understanding of the concepts related to angles.
1. Geometry has a lot of important terminology. Knowing what these terms mean and being able
to give an example of each is important when learning geometry. Fill in the following table by
drawing an example and writing a definition or description of each term.
Quick Tip
Pay close attention to the
notation used to denote
each geometric form. There
may be more than one way
to refer to some of these.
Term
Example
Definition or Description
Point
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Line
Plane
Line Segment
Ray
Angle
5.1 Angles 175
Quick Tip
It is important to use the
correct mathematical
notation when talking
about angles or degrees
in math. Using the wrong
symbol can lead to
confusion when someone
else reads your work.
2. This problem explores the mathematical notation related to angles and how to translate the
symbols into English words.
a. The symbol for the word “angle” is ∠. Translate the symbols “∠A” into words.
b. The symbol for the phrase “measure of” is m. Translate the symbols “m∠A” into words.
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c. The symbol for the word “degree(s)” is °. Translate the symbols “72°” into words.
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d. Putting these all together, translate the symbols “m∠A = 72°” into words.
3. There are several different types of angles and relationships between angles. Knowing what
these types of angles are and being able to give an example of each is important when learning
geometry. Fill in the following table by drawing an example and writing a definition or
description of each term.
Term
Example
Definition
Acute
Right
Obtuse
Adjacent Angles
Congruent Angles
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Straight
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4
2
4
A
4. You need to be careful when naming adjacent angles, which are
angles that have a common side. If you don’t properly name the
angles, it will be unclear which angle you are referring to.
a. How many angles are in the figure?
B
C
D
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b. If someone writes “∠D”, is it clear which angle they are referring to? Explain why
or why not.
c. Name all of the angles by referring to the three points associated with each angle. Remember
that the vertex point needs to be the center point listed in the angle name.
Quick Tip
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An easy way to remember
or distinguish between
the terms complementary
and supplementary is that
complementary comes
before supplementary when
written in alphabetical order
and 90° is less than 180°,
so complementary goes
with 90° and supplementary
goes with 180°.
Definitions
Complementary angles are angles whose measures add to 90°.
Supplementary angles are angles whose measures add to 180°. When two lines intersect,
they form two pairs of vertical angles. The vertical angles are opposite of each other. Vertical
angles are congruent, which means they have the same measure.
Intuition and observation, along with logic, can be used to show that mathematical properties or
theorems are true. If you understand why a property or theorem is true, then remembering and using
it properly will be easier. The following problem will help you understand the above theorem about
vertical angles.
2 4
5. According to the vertical angles property, if ∠1 and ∠3 in the figure
are congruent, then m∠1 = m∠3.
a. Which angles are supplementary with ∠1?
p
1
2
3
4
q
b. Write an equation of the form m∠X + m∠Y = 180° for each pair of supplementary angles
from part a. by replacing X and Y with the angle names for each supplementary pair.
c. Which angles are supplementary with ∠3?
5.1 Angles 177
d. Write an equation of the form m∠X + m∠Y = 180° for each pair of supplementary angles
from part c. by replacing X and Y with the angle names for each supplementary pair.
e. What must be true about m∠1 and m∠3 for both pairs of equations from parts b. and d.
to be true?
Definitions
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Two lines intersect if they cross at any point.
Two lines are parallel if they never cross.
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Two lines are perpendicular if they intersect at a 90° angle.
A transversal is a line in a plane that intersects two or more lines at different points.
6. Consider the figure to the right to answer the
following questions.
a
b
c
d
e
a. Are any lines parallel? If yes, list them.
Quick Tip
In geometric figures, a 90°
angle is often represented
by a small square.
b. Are any lines perpendicular? If yes,
list them.
c. Is line c a transversal of lines a and b? If no, why not?
d. Is line d a transversal of lines c and e? If no, why not?
When two parallel lines are intersected by a transversal, the following statements are
true.
1. When two parallel lines are intersected by a transversal, four angles are created on
each parallel line. The angles in matching corners are called corresponding angles.
Corresponding angles are congruent.
2. The pairs of angles on opposite sides of the transversal but inside the two parallel lines are
called alternate interior angles. Alternate interior angles are congruent.
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Angles Created by Transversals
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7. Consider the figure to the right and answer the following
questions.
a. Angles 1 and 5 are corresponding angles. List any
other pairs of corresponding angles.
p
1
4
2
q
3
5
6
8 7
r
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c. List any pairs of vertical angles.
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b. Angles 4 and 6 are alternate interior angles. List any
other pairs of alternate interior angles.
d. List any pairs of supplementary angles.
e. If m∠1 = 45°, use the properties of angles find the measures of the other angles.
Skill Check
Go to Software Work through Practice in Lesson 5.1 of the software before attempting the following
exercises.
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8. Assume ∠1 and ∠2 are complimentary.
a. If m∠1 = 15°, what is m∠2?
b.If m∠2 = 43°, what is m∠1?
9. Assume ∠3 and ∠4 are supplementary.
a. If m∠3 = 115°, what is m∠4?
b.If m∠4 = 74°, what is m∠3?
5.1 Angles 179
Apply Skills
Work through the problems in this section to apply the skills you have learned related to angles.
2 4
10. Consider the proposed road plan shown here.
A
B
a. Are the right-hand lane and left-hand lane of a roadway
parallel or perpendicular?
110°
140°
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C
140°
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b. A city building code prohibits the construction of
roadway intersections that result in an angle of less than
45°. Does the proposed road plan violate this building
code? Why or why not?
c. Roads B and C are parallel roads. Label the corresponding angles formed by the roads on
the figure.
d. What properties allowed you to determine the answers to part c.?
11. The navigator of a submarine sees that
there are two unknown ships located at
points A and B.
C
B
40˚
18˚
Submarine
b. What is the measure of the angle formed by the two unknown ships and the submarine?
c. In order to remain undetected, the navigator wants to keep as much distance as possible
between the sub and the two unknown ships. In order to do this, he sets a course at an angle
which bisects the angle between the unknown ships. The submarines northeasterly course
is towards point C in the figure. At what angle from horizontal, indicated by the arrow, is
the submarine traveling?
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a. Is the angle formed by the unknown
ships and the submarine acute,
obtuse, right, or straight?
A
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5.2 Perimeter
Objectives
Success Strategy
Know what types of geometric figures are
polygons.
Formulas aren’t necessary for finding perimeter.
Just add the lengths of all sides of the figure.
Find the perimeters of polygons.
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Go to Software First, read through Learn in Lesson 5.2 of the software. Then, work through the
problems in this section to expand your understanding of the concepts related to perimeter.
While having formulas to find the perimeter of a shape isn’t necessary, they can be used to practice
substituting values into equations and simplifying. You can confirm that you substituted the correct
values into the perimeter formula by finding the sum of all of the side lengths of the figure and then
comparing the solutions.
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Understand
Concepts
Quick Tip
1. Fill in the table with the name of each shape and the formulas to find the perimeter.
Substitute is a verb
which means “to put or
use in place of another.”
So, when substituting a
value for a variable in a
formula, put that value in
place of the variable.
Formulas for Perimeter
Shape
Shape Name
Perimeter Formula
s
a
c
b
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l
d
a
c
b
d
a
c
b
5.2 Perimeter 181
Lesson Link Perimeter was first
introduced in relation
to whole numbers
in Section 1.2.
2. Consider the completed table from Problem 1 and answer the following questions.
a. Are any of the formulas the same? If so, for which shapes?
b. Why do you think that the shapes from part a. have the same perimeter formula?
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c. Instead of using a formula, what do you need to remember to find the perimeter of any
geometric shape?
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When using a formula to solve a problem, it is important to recognize what the variables in the
formula represent. Some are easy to identify, such as the variable for the side length of a square,
which is usually referred to as s. Other formulas are flexible with which letter can be used for the
variables, such as the side lengths of a triangle.
3. For each description, determine which formula to use to find the perimeter and which
measurement will be substituted for each variable in the formulas.
a. A square has a side length of 5 inches.
b. A rectangle has width 4 inches and length 7 inches.
c. A parallelogram has side lengths 5 inches and 4 inches.
d. A triangle has sides 3 inches, 4 inches, and 5 inches.
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e. A trapezoid has top length 5 cm, bottom length 15 cm, and side lengths 9 cm and 12 cm.
Name:Date:
Skill Check
Go to Software Work through Practice in Lesson 5.2 of the software before attempting the following
exercises.
Find the perimeter of each figure.
4.
6 ft
5.
4 ft
6 ft
30 ft
25 ft
18 ft
5 ft
5 ft
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24 ft
6. 4 m
8 in.
4m
4m
7m
8m
Apply Skills
12 in.
7.
4m
5 in.
2 in. 5 in.
12 in.
5m
4 in.
2 in.
Work through the problems in this section to apply the skills you have learned related to perimeter. Circle or
underline any key words that indicate which perimeter formula should be used.
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8. A police officer needs to tape off a crime scene with caution tape. The smallest area he can tape
off is outlined by trees and road signs, which he can wrap the tape around. The trees and road
signs mark the vertices of the figure.
110°
2 4
180 ft
78 ft
107 ft
82 ft
a. What is the perimeter of the crime scene?
b. The officer needs 6 feet of caution tape in addition to the perimeter to properly tape off the
crime scene. What is the total amount of caution tape needed?
5.2 Perimeter 183
Quick Tip
Remember that drawing a
figure can be helpful when
solving a word problem.
9. Jessica wants to redecorate her living room by updating items she already owns.
a. Jessica wants to add a decorative fringe to a throw rug. The rug is a rectangle with length
8 feet and width 5 feet. If Jessica wants to buy 1 foot more than the perimeter of the rug,
how many feet of fringe must she buy?
b. Jessica wants to outline her mirror with tube lighting. The mirror is in the shape of a regular
octagon (all 8 sides have equal length). One side length of the mirror is 5 inches. How
many inches of tube lighting must Jessica buy?
c. Jessica wants to put new trim around the windows in her living room. She has two
windows of the same size. The windows measure 4.5 feet tall and 6 feet wide. She needs
an additional 0.25 feet of trim for each corner of the window. How many feet of trim will
she need to buy?
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Trim is a type of material
that is used for decorating
something especially
around its edges. Window
trim can be made of wood,
vinyl, or other materials.
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Quick Tip
Quick Tip
Neoprene is a synthetic
rubber made for use in
variety of applications such
as laptop sleeves, wet suits,
and automotive fan belts.
10. An engineer designing a new smartphone decides to add a soft neoprene edging to the phone.
2
1
The phone itself is 4 inches tall and 2 inches wide.
5
2
a. How much neoprene edging is needed to go along the outside edge of each smartphone?
b. The neoprene edging will cost $0.12 per inch. How much will the edging cost per phone?
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5.3 Area
Objectives
Success Strategy
Understand the concept of area.
It is important to understand the difference
between the perimeter and the area of a figure.
Also note that perimeter is measured in standard
units and area is measured in square units.
Know the formulas for finding the area of five
polygons.
Go to Software First, read through Learn in Lesson 5.3 of the software. Then, work through the
problems in this section to expand your understanding of the concepts related to area.
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Understand
Concepts
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1. Fill in the table by sketching a figure of each shape with the variables labeled and the formulas
to find the area.
Formulas for Area
Shape Name
Figure
Area Formula
Square
Triangle
Rectangle
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Trapezoid
Parallelogram
Lesson Link The concept of area was
introduced in Section 1.3.
2. In Problem 3 of Section 1.3, you wrote down some area formulas that you remembered from
previous courses. Compare those formulas to the formulas presented in this section.
a. Does this section cover any formulas you did not write down?
b. Did you write down any formulas not covered in this section?
5.3 Area 185
c. Many formulas are presented in different situations with different notation or variables.
Did any of the formulas you remembered use different notation than the notation used in
this section
To Find the Area of a Figure with a Section Cut Out
1. Find the area of the full figure (ignoring the cut out).
2. Find the area of the cut out.
Quick Tip
3. This problem will guide you through the steps to
determine the area of a figure with sections cut out.
4
2
4
110°
8 in.
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Height is the perpendicular
distance from the base of
the figure to the highest
point. The height of
a figure is not always
equal to a side length.
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3. Subtract the area of the cut out from the area of the full figure.
a. List the shapes in the figure. Indicate whether the
shape is a cut out.
5 in.
2 in.
2 in.
2 in.
2 in.
b. What formula is needed to find the area of each shape in the figure?
c. What is the area of the full figure (that is, the largest shape)?
d. What is the area of the cut outs?
e. Subtract the area in part d. from the area in part c.
To find the area of more
complicated shapes, it
may be necessary to first
breakdown the figure
into smaller pieces that
you recognize. Then,
find the sum of the
areas of the pieces.
5 ft
4. Learning how to break complicated geometric figures
into easy-to-work-with parts is a skill that you can
develop with practice.
a. What shapes are marked off by the dashed lines in
the figure?
5 ft
5 ft
4 ft
12 ft
b. Which area formulas do you need to determine the area of the entire figure?
186
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Quick Tip
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c. Find the area of each shape in the figure.
d. Find the sum of the areas of each shape to determine the area of the entire figure.
Go to Software Work through Practice in Lesson 5.3 of the software before attempting the following
exercises.
Find the area of each figure.
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Skill Check
15 yd
5.
6.
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35 cm
12 yd
55 cm
12 yd
7.
10 in.
3 in.
7 in.
4 in.
8.
11 km
3 in.
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12 in.
Apply Skills
8 km
9 km
17 km
Work through the problems in this section to apply the skills you have learned related to area. Circle or underline
any key words that indicate which area formula should be used.
9. A parking space is in the shape of a rectangle that is 2
is the area of the parking space?
1
meters wide and 5 meters long. What
2
5.3 Area 187
15 feet
10. The main stage at a theater is in the shape of a trapezoid. The
owner of the theater is planning to install a new specially
designed flooring system on the stage. The stage is 12 feet
wide in the front and 15 feet wide in the back. The stage is
10 feet deep.
10 feet
a. What is the area of the stage?
12 feet
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b. If the wooden flooring system costs $35.50 per square foot for purchase and installation.
How much will it cost to replace the stage floor?
11. A warehouse has several different rooms, each in the shape of a rectangle. The floor of one room
in the warehouse is 25 feet by 40 feet.
a. What is the area of the floor for this room of the warehouse?
b. A pallet for storage measures 4 feet by 3.5 feet. What is the area of a pallet?
c. The warehouse room is empty except for 38 pallets on the floor. What is the area of the
empty floor space in the room?
12. Lee is making a box. He starts with a piece of cardboard that is 14 inches by 20 inches.
a. What is the area of the piece of cardboard?
20 in.
14 in.
3 in.
c. When the sides are folded up, what will be the area of the bottom of the box?
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b. Lee cuts a square with a side length of 3 inches from
each corner of the cardboard. What is the area of the
cardboard with the corners removed?
Name:Date:
5.4 Circles
Objectives
Success Strategy
Know the definition of a circle and its related terms.
Be able to find the circumference (perimeter) and
area of a circle.
Go to Software First, read through Learn in Lesson 5.4 of the software. Then, work through the
problems in this section to expand your understanding of the concepts related to circles.
When working with circles, the value π is something you should be familiar with. This
mathematical value has a long and interesting history. Use the key words “history of pi” to find
answers to the following questions.
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1.
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Understand
Concepts
Most calculators have a button for π, a special
constant that you will be working with in this
section. You should find where this value is on your
calculator and learn how to use it.
Quick Tip
a. What ratio does π represent?
Pi Day is celebrated on
March 14 every year. Pi
Approximation Day is
celebrated on July 22
(22/7 in the day/month
format) since 22/7 is
a common fractional
approximation of pi.
b. Who was the first civilization to approximate the value of π to find the area of a circle?
c. Who was the first mathematician to approximate the value of π?
d. Who was the first person to use the symbol π to stand for this value?
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Quick Tip
2. Fill in the definition of each term in the table.
A
“N/A” means “not
applicable” or “no answer”.
This notation is used when
a question doesn’t apply
to a certain case or the
answer is not available.
C
D
B
Term
Part of Figure
Circle
N/A
Circumference
A
Center
D
Radius
C
Diameter
B
Definition
5.4 Circles 189
Circle Formulas
Circumference of a Circle C = 2πr or C = πd
Area Enclosed by a Circle A = πr 2
A common approximation
of π is 3.14. If a problem
in this workbook requires
the use of π, use 3.14
unless otherwise directed.
When working with circles, you will need to determine the values of the variables r and d.
Occasionally you need to determine the value of one of these variables given the value of the other.
It is important to know if a problem statement is giving you the value of the radius or of the diameter.
For the next two problems, answer the questions based on the information given.
3. Suppose you need to find the circumference of this circle.
7 ft
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a. The value of which measurement is given?
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Quick Tip
b. Which variable do you need the value of?
c. Find the circumference of the circle.
4. Suppose you need to find the area of the circle.
6 in.
a. The value of which measurement is given?
b. Which variable do you need the value of?
c. Find the area of the circle.
Half of a circle is called a
semicircle. The area of
a semicircle is one half of
the area of a circle. The
perimeter of a semicircle is
half the circumference of a
circle plus the diameter.
5. Some complicated geometric figures contain circles or parts of a
circle. Knowing how to identify these figures is a skill that you can
develop with practice.
a. What two shapes can you identify in the figure?
4 in.
8 in.
b. Which area formulas do you need to determine the area of the entire figure?
c. Find the area of each shape in the figure.
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Quick Tip
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d. Find the sum of the areas of each shape to determine the area of the entire figure.
Skill Check
Go to Software Work through Practice in Lesson 5.4 of the software before attempting the following
exercises.
Find a. the perimeter and b. the area of each figure.
6.
7.
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3.5 m
3 cm
8.
12 yd
9.
2m
12 yd
4m
12 yd
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Apply Skills
Work through the problems in this section to apply the skills you have learned related to circles. Use π = 3.14
and round your answers to the nearest hundredth.
10. The prices for three different sizes of one‑topping pizzas at Romito’s pizza are shown in
the table.
Price for 1-Topping Pizza
Quick Tip
9-inch
12-inch
16-inch
$7.25
$10.25
$13.50
a. Find the area of each pizza. (Note: units will be in square inches.)
The size of the pizza
indicates the diameter
of the pizza.
b. Use ratios to find the price per square inch for each pizza size.
c. Based on your answer to part b., which pizza is the best value?
5.4 Circles 191
11. The city is planning to put a fountain in the middle of the public park. The park is a rectangle
with length of 70 feet and width of 45 feet. The base of the fountain will be a circle with
diameter 10 feet. What area of the public park will not be taken up by the fountain? (Hint: draw
a picture to help you solve this problem.)
85 ft
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14 ft
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12. The parking lot of the emergency room at a hospital
is in the shape of a rectangle with length 100 feet and
width 85 feet. There is also a semicircle with radius 14
feet near the entrance for ambulance drop off. What is
the area of the parking lot including the drop off area?
100 ft
Finding the area of a
washer is similar to finding
the area of a shape
with cut outs. This was
covered in Section 5.3.
13. A machine shop receives an order for 80 millimeter wide
washers with an area of 2198 mm2. They have machines set up
to make the following washers.
Machine
Inner Radius
Outer Radius
A
10 mm
40 mm
B
20 mm
40 mm
C
30 mm
40 mm
Area of Washer
Outer
Radius
Inner
Radius
Quick Tip
a. Are all of the machines set up to produce a washer with the correct outer radius?
c. Calculate the area of each washer and place the areas in the fourth column of the table.
d. Are any of the machines set up to create the washer size that was ordered? If yes,
which machine?
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b. Without calculating the areas, which machine will produce the washer with the largest area?
Name:Date:
5.5 Volume and Surface Area
Objectives
Success Strategy
Understand the concept of volume.
Know the formulas for finding the volume of five
geometric solids.
Understand the concept of surface area.
To help you determine when to use volume
formulas and when to use surface area
formulas, remember that volume refers to how
much an object can hold and surface area is a
measurement of the outside area of the object.
Go to Software First, read through Learn in Lesson 5.5 of the software. Then, work through the
problems in this section to expand your understanding of the concepts related to volume and surface area.
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Understand
Concepts
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Know the formulas for finding the surface area of
three geometric solids.
1. Fill in the table with the name of each shape and the formulas to find the volume and surface
area. If the surface area formula is not given for that shape, write “N/A” in the box.
Quick Tip
The shape names and
related formulas can be
found in Learn of Lesson
5.5 of the software.
Shape
Shape Name
Volume Formula
Surface Area Formula
h
w
l
h
s
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h
r
h
r
r
5.5 Volume and Surface Area 193
2. Perimeter, area, and volume are all measurements involving standard units of length which
have different meanings and uses. Each of these corresponds with a dimension of space and
units that go with those dimensions. Fill in the table with the missing information if the unit of
measurement is inches.
Dimension
Perimeter
1-Dimensional
Area
2-Dimensional
Units of Measurement
Shape Example
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Measurement
Volume
3-Dimensional
Knowing how formulas were developed can help you understand them and use them correctly. The next two
problems will guide you through the logic behind the volume formula and the surface area formula for circular
cylinders.
3. The volume of a right circular cylinder is given by the formula v = πr2h.
a. One way to think of a right circular cylinder is as a lot of circles stacked on top of each
other. What is the equation for the area of a circle?
b. Suppose a circle has a radius of 2 inches. What is the area of the circle? (Use π = 3.14.)
d. If the formula for the area of a circle is multiplied by this missing measurement, will we
obtain the formula for the volume of a circular cylinder?
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c. Circles are 2‑dimensional shapes, which mean they have a width and a length. A right
circular cylinder is a 3‑dimensional object. Which additional dimension does the circular
cylinder have that the circle does not have?
Name:Date:
Quick Tip
It is helpful to visualize a
soup can when working
with circular cylinders.
4. The surface area of a right circular cylinder is given by the formula SA = 2πr2 + 2πrh.
a. A right circular cylinder can be divided into three pieces. The top and bottom are circles.
The “tube” piece can be cut down one side and flattened into a rectangle. Draw the pieces
of a disassembled right circular cylinder.
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b. What are the area formulas for a circle and a rectangle? Label the variables from these
formulas on your drawing from part a.
c. For the rectangle from part a., two of the side lengths are the same as the circumference
of the circles which form the top and bottom of the cylinder. What is the formula for the
circumference of a circle?
d. The circumference of a circle is equal to which variable on your rectangle from part a.?
e. Which measurement of the rectangle represents the height of the cylinder?
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f. Rewrite the area formula for the rectangle by using the information from parts c., d., and e.
and the area formula from part b.
g. What do you need to do with the area formulas for the rectangle and the circles to create
the surface area formula of a right circular cylinder?
5.5 Volume and Surface Area 195
Skill Check
Go to Software Work through Practice in Lesson 5.5 of the software before attempting the following
exercises.
Find the volume of each figure. Round your answers to the nearest hundredth when necessary.
Quick Tip
5.
Remember that
3.14 is a common
approximation for π.
4 cm
6.
10 in.
4 cm
7 cm
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8 in.
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6 cm
Find the surface area of each figure. Round your answers to the nearest hundredth when necessary.
7.
8.
4 in.
10 cm
7 in.
Apply Skills
Work through the problems in this section to apply the skills you have learned related to volume and surface
area. Use π = 3.14 and round your answers to the nearest hundredth if necessary.
a. Find the surface area of the soup can.
b. Find the volume of the soup can.
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9. A can of soup is 4 inches high and has a diameter of 2.6 inches.
196
3 in.
Name:Date:
10. Barbara’s Bombtastic Bakery sells wedding cakes and sets a price based on the number of
servings. A serving of wedding cake has volume equivalent to a rectangular piece of cake with
measurements 1 inch by 2 inches by 4 inches. Each serving of wedding cake costs $1.25.
a. What is the volume of one slice of wedding cake?
y
b. The bottom tier of a round wedding cake has a diameter of 16
inches. If the cake tier is 4 inches high, what is the volume of
this tier of the cake? Round to the nearest cubic inch.
16 inches
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c. How many equivalent slices of wedding cake are in the 16 inch
diameter wedding cake? Round to the nearest whole slice.
d. How much should Barbara’s Bombtastic Bakery charge for this tier of the wedding cake?
Quick Tip
A cube is a rectangular
solid where the length,
width, and height have
equal measures.
11. A glass ornament in the shape of a sphere is to be packaged in a box along with soft foam pellets.
The ornament has a diameter of 4 inches. The box is a cube whose side length is 5 inches.
a. What is the volume of the glass ornament?
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b. What is the volume of the box?
c. The volume of the box which is not taken up by the glass ornament will be filled with the
foam pellets. What volume of the box will be filled with the foam pellets?
12. Jerry is a tool and die maker, and is creating a specialized solid steel cone for a customer. The
cone needs to be 12.125 cm tall, with a radius of 4.4 cm. How much steel will be used to create
the solid steel cone?
5.5 Volume and Surface Area 197
13. The Louvre Pyramid is a rectangular pyramid made of glass and metal which is located in the
courtyard of the Louvre Palace in Paris, France. The pyramid has a height of 20.6 meters and
each side of the base has a length of 35 meters.
a. What is the volume of the Louvre Pyramid?
20.6 m
y
b. Each triangular piece that makes up a side of the pyramid has a height of approximately
27 meters. What is the surface area of each triangular piece? (Note: The height of the
triangular side is a different measurement than the height of the pyramid.)
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27 m
35 m
35 m
c. The surface area of the pyramid is equal to the area of the base of the pyramid plus the total
area of the four triangular faces. What is the surface area of the pyramid?
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5.6 Triangles
Objectives
Success Strategy
Be able to classify triangles by sides.
There are a lot of terms in this section, so be sure
to devote a section in your notebook to writing
down all of the terms and their definitions. You
could also use index cards and the Frayer model
from Chapter 2.
Be able to classify triangles by angles.
Understand similar triangles.
Understand
Concepts
y
Understand congruent triangles.
Go to Software First, read through Learn in Lesson 5.6 of the software. Then, work through the
problems in this section to expand your understanding of the concepts related to triangles.
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1. For each type of triangle, describe the properties of the triangle and draw an example.
Classification by Sides
Quick Tip
Triangle properties and
examples can be found
in Learn of Lesson 5.6
of the software.
Name
Properties
Example
Scalene
Isosceles
Equilateral
Classification by Angles
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Name
Quick Tip
Acute
Did you know that in
construction the shape that
has the most structural
strength is the triangle? This
is why you see the triangle
shape in bridge designs.
For more information, go to
www.teachengineering.org.
Right
Properties
Example
Obtuse
5.6 Triangles 199
A
2. In Section 5.1, mathematical notation for angles was introduced.
Geometry also has a specific notation for triangles. This problem
explores the notation for triangles and how to translate that symbol
into English words.
a. The symbol for the word “triangle” is ∆. Translate the symbols
“∆ ABC ” into English words.
b. A triangle is named by listing the angles in order as you move clockwise or counterclockwise
around the figure. How many different ways can you name this triangle?
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To move in a
counterclockwise
direction means to move
in a direction that is the
opposite direction in which
the hands of a clock rotate.
C
y
Quick Tip
B
Similar and Congruent Triangles
Notation: ∆ABC ∼ ∆XYZ
Similar triangles have two properties:
1. Corresponding angles have the same measure.
2. Lengths of corresponding sides are proportional.
Notation: ∆ABC ≅ ∆XYZ
Congruent triangles have two properties:
1. Corresponding angles have the same measure.
2. Lengths of corresponding sides are equal.
Quick Tip
The order that the pairs
of congruent angles are
listed in the notation
∆ ABC ~ ∆ XYZ can
vary. The important part
is to correctly pair the
corresponding angles of
the two similar triangles.
When writing the names of two similar triangles or two congruent triangles, it is important to write
the corresponding vertices in the same order for both triangles. This means that for ∆ABC ∼ ∆XYZ ,
∠A corresponds with ∠X, ∠B corresponds with ∠Y, and ∠C corresponds with ∠Z.
3. A common mistake when writing the notation for similar triangles is to incorrectly match up
the corresponding angles. Determine if any mistakes were made in the notation for each pair of
similar triangles. If any mistakes were made, describe the mistake and then write the notation
correctly.
∆ABC ∼ ∆XYZ
A
X
B
C
D
b. ∆DEF ∼ ∆LNM F
200
Y
Z
L
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N
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a.
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Three Properties of Triangles
1. The sum of the measures of the angles is 180°.
2. The sum of the lengths of any two sides must be greater than the length of the third side.
3. Longer sides are opposite angles with larger measures.
4. The following problem will help you understand why the sum
of the measures of the angles of a triangle is equal to 180°.
1
2
B
a. What is the angle measure of a straight line?
A
y
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b. What do you know about the sum of m∠1, m∠2, and m∠B?
C
Lesson Link c. Which of the labeled angles in this figure are alternate interior angles?
Alternate interior angles
were defined in Section 5.1.
d. What do you know about alternate interior angles?
e. What does this tell you about the sum of the angle measures of a triangle?
Skill Check
Go to Software Work through Practice in Lesson 5.6 of the software before attempting the following
exercises.
Hawkes Learning Systems © 2014
Determine whether or not a triangle with the given dimensions exists using the second property of triangles
listed in the box at the top of the page.
5. 4 in., 5 in., 7 in.
Quick Tip
You may need to visually
rotate the triangles to line
up the corresponding
angles or sides.
6. 9 ft, 32 ft, 41 ft
Determine whether or not the pairs of triangles are similar. If they are similar, use the proper notation to indicate
the similarity.
4 2 4
7.
8.
A
D
A
X
110°
4
4
3
4
C
110°
B
C
E
B
2
Z
Y
2
5.6 Triangles 201
Apply Skills
Work through the problems in this section to apply the skills you have learned related to triangles. Round
answers to the nearest hundredth if necessary.
9. A building has two ramps going up to the entrances on different sides. Both ramps have an
incline of 4.5° and form a right angle with the building. The first ramp has a base length of 8
feet. The second ramp has a base of length of 15.25 feet and a height of 1.2 feet.
8 ft
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1.2 ft
4.5°
a. Do these ramps form similar triangles, congruent triangles, or neither?
b. What is the height of the first ramp?
10. The pieces to assemble a spice rack include two congruent triangles. The triangles need to have
corresponding angles lined up.
110°
4 2 4
C
F
70°
A
60°
50°
50°
B
G
70°
60°
E
a. Match up the corresponding angles on the triangles. Write that the two triangles are
congruent in the proper mathematical notation.
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b. Would it be enough for the manufacturer to label just one corresponding angle on each
triangle? That is, can you determine how the other angles correspond just by knowing how
one angle corresponds?
Name:Date:
11. A billboard advertisement has a right triangle as part of its design. In the scaled version the
graphic designers made in Photoshop, the base of the triangle is 8 inches and the height of the
triangle is 5 inches. On the full sized billboard, the base of the triangle is 96 inches. What is the
height of the triangle on the billboard?
4.76°
x
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In geometric figures, a 90°
angle is often represented
by a small square.
12. Handicap ramps must be at an angle no greater than 4.76° from horizontal.
y
Quick Tip
a. What is the measure of angle x?
Lesson Link b. What is the relationship between the 4.76° angle and angle x?
Relationships between
angles were covered
in Section 9.1.
Quick Tip
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Indirect measurement
can be used to find the
height of an object when
measuring the object
directly is not possible.
13. While performing field research, a historian needs to determine the height of an abandoned
lighthouse. Since he is unable to directly measure the height of the lighthouse, he determines
the height indirectly. He places a 2‑foot long stick in the ground and measures the length of the
shadow it casts. He then measures the length of the shadow cast by the lighthouse. What is the
height of the abandoned lighthouse? (Note: The light house and the stick are both at right angles
from the ground.)
h
2 ft
55.5 ft
0.75 ft
5.6 Triangles 203
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5.7 Square Roots and the Pythagorean
Theorem
Objectives
Success Strategy
Understand and calculate square roots.
Locate the square root button on your calculator
and practice using it so that you can correctly
work the problems in this section.
Understand the Pythagorean Theorem.
y
Go to Software First, read through Learn in Lesson 5.7 of the software. Then, work through the
problems in this section to expand your understanding of the concepts related to square roots and the
Pythagorean Theorem.
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Understand
Concepts
1. Label the parts of the radical expression.
15
Definitions
A perfect square is the square of a counting number. The square root of a perfect square is
a whole number. The square root of a number which is not a perfect square is an irrational
number. An irrational number is an infinite nonrepeating decimal.
Lesson Link 2. Fill in the table with the first 16 perfect squares and their square roots.
Perfect squares were first
introduced in Section
1.6 when learning
about exponents.
Perfect Square
Square Root
Perfect Square
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Square Root
3.
Before calculators were commonly used in classrooms, people had to calculate the square
roots of numbers which were not perfect squares by hand. Use the keywords “square roots
without calculator” to find at least two different methods of calculating a square root by hand.
a. Describe one method of finding square roots without a calculator.
b. What benefit do you think there is to learning how to calculate a square root by hand?
5.7 Square Roots and the Pythagorean Theorem 205
Quick Tip
The Pythagorean Theorem
The legs of a right triangle
are represented by the
variables a and b. The
hypotenuse, which is
opposite the right angle
and is always the longest
side, is represented
by the variable c.
In a right triangle, the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of the two legs.
c
b
90°
c 2 = a 2 + b 2
a
4. A visual verification of the Pythagorean Theorem uses three squares to
make a right triangle. We will use squares with side lengths 3, 4, and 5.
5
a. Find the area of each square.
3
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b. Write the area of one of the squares as the sum of the two
other squares.
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4
c. Use the Pythagorean Theorem to verify that the triangle made by the squares is a right
triangle. Show your work.
d. What is the similarity between your answers to part b. and part c.?
Read the following paragraph about Pythagorean triples and work through the problems.
A Pythagorean triple is a set of three whole numbers which satisfy the formula of the Pythagorean
Theorem. One example of a Pythagorean triple, (3, 4, 5), was used in the proof of the Pythagorean
Theorem presented in Problem 4. There are several formulas that can be used to create these triples.
The next problems explore how to create Pythagorean triples.
a. Use the Pythagorean Theorem formula to verify that (6, 8, 10) forms a Pythagorean triple.
b. Use this method to create two more Pythagorean triples.
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5. The easiest method to create more Pythagorean triples is if you already know a Pythagorean
triple. In this case, you multiply each number in the triple by the same whole number. For
example, since we know (3, 4, 5) is a Pythagorean triple, then 2 ∙ 3, 2 ∙ 4, and 2 ∙ 5, which is
(6, 8, 10), is also a Pythagorean Triple.
Name:Date:
Quick Tip
This set of formulas
does not produce all of
the Pythagorean triples.
Different formulas to
produce Pythagorean
triples can be found
on the Internet.
6. One of many sets of formulas to create Pythagorean triples is
a = n2 − m2
b = 2nm
c = n2 + m2
where n and m are integers and n > m.
For example, if we use n = 2 and m = 1 we get the triple (3, 4, 5). We have already verified that
this is a triple.
y
a. Substitute two integers n and m, where n > m, into the formulas to create a Pythagorean triple.
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b. Use the formula of the Pythagorean Theorem to verify that the triple you created in part a.
is a Pythagorean triple.
Skill Check
Go to Software Work through Practice in Lesson 5.7 of the software before attempting the following
exercises.
To calculate a square root using a calculator, press the
button and then enter the number you are finding
the square root of, followed by the
button. Determine the square root of each number to the nearest
thousandth.
Quick Tip 8. 72
9. 24
10. 6724
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Some basic calculators
require the number be
entered before the
button is pressed. Care
must be taken when
using calculators so the
desired answer is given.
7. 14
5.7 Square Roots and the Pythagorean Theorem 207
Work through the problems in this section to apply the skills you have learned related to square roots and the
Pythagorean Theorem.
110°
x
11. A police officer needs to tape off a crime scene. The crime
took place in a park that has a fence along one side and a
shed near the fence.
ft
17
a. The side of the shed is 8 feet long and the fence is
8 ft
Apply Skills
The Pythagorean
Theorem is a useful
tool in mathematics as
shown by the application
problems in this section.
17 feet long. The officer wants to attach the caution
tape from the edge of the shed to the end of the fence.
How much caution tape does he need?
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b. What is the area of the taped off crime scene?
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Quick Tip
12. Aya has a triangular wooden porch attached to the side of her house. The “legs” of the triangle
porch measure 12 feet and 16 feet. She is decorating for a party and has 18 feet of party lights.
a. Does she have enough lighting to put along the entire railing of the longest side of the porch?
b. If yes, how many feet of party light are left over? If no, how many additional feet of party
lights are needed?
13. The maximum walking speed of an animal depends on the length of their legs. To calculate the
maximum speed an animal can walk (in feet per second), one needs to multiply the square root
of the animal’s leg length in feet by 5.66.
b. A man has legs that measure 3 feet in length. What is this man’s maximum walking speed
to the nearest hundredth?
c. Since the giraffe’s legs were twice as long as the man’s legs, did this mean that the giraffe
could walk twice as fast? If not, how did the speeds compare?
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a. A giraffe has legs that measure 6 feet in length. What is this giraffe’s maximum walking
speed to the nearest hundredth?
Name:Date:
Chapter 5 Projects
Project A: Before and After
An activity to demonstrate the use of geometric concepts in real life
Suppose HGTV came to your home one day and said, “Congratulations, you have just won a FREE makeover for any
room in your home! The only catch is that you have to determine the amount of materials needed to do the renovations
and keep the budget under $2000.” Could you pass up a deal like that? Would you be able to calculate the amount of
flooring and paint needed to remodel the room? Remember it’s a FREE makeover if you can!
y
Let’s take an average size room that is rectangular in shape and measures 16 feet 3 inches in width by 18 feet 9 inches
in length. The height of the ceiling is 8 feet. The plan is to repaint all the walls and the ceiling and to replace the carpet
on the floor with hardwood flooring. You are also going to put crown molding, a decorative type of trim used along the
top of a wall where the ceiling and the wall meet, for a more sophisticated look.
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1. Take the length and width measurements that are in feet and inches and convert them to a fractional number of feet
1
feet.)
and reduce to lowest terms. (Remember that there are 12 inches in a foot. For example, 12 feet 1 inch is 12
12
2. Now convert these same measurements to a decimal number.
Hawkes Learning Systems © 2014
3. Determine the number of square feet of flooring needed to redo the floor. (Express your answer in terms of a
decimal and do not round the number.)
4. If the flooring comes in boxes that contain 24 square feet, how many boxes of flooring will be needed? (Remember
that the store only sells whole boxes of flooring.)
5. If the flooring you have chosen costs $74.50 per box, how much will the hardwood flooring for the room cost
(before sales tax)?
Chapter 5 Projects 209
6. Figure out the surface area of the four walls and the ceiling that need to be painted, based on the room’s dimensions.
(We will ignore any windows, doors, or closets since this is an estimate.)
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7. Assume that a gallon of paint covers 350 square feet and you are going to have to paint the walls and the ceilings
twice to cover the current paint color. Determine how many gallons of paint you need to paint the room. (Again
assume that you can only buy whole gallons of paint. Any leftover paint can be used for touch‑ups.)
8. If the paint you have chosen costs $18.95 per gallon, calculate the cost of the paint (before sales tax).
a. Determine how many feet of crown molding will be needed to go around the top of the room.
b. The molding comes in 12 foot lengths only. How many 12 foot lengths will you need to buy?
9. If the molding costs $2.49 per linear foot, determine the cost of the molding (before sales tax).
10. Calculate the cost of all the materials for the room makeover (before sales tax).
a. Were you able to stay within budget for the project?
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b. If so, then what extras could you add? If not, what could you adjust in this renovation to stay within budget?
Name:Date:
Project B: Building a Circular Patio
An activity to demonstrate the use of geometric concepts in real life
Bob has just finished converting the back bedroom of his house to a sunroom and has decided to build a concrete patio
in a circular shape outside the sunroom for grilling out. The sunroom is 11 feet by 11 feet.
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pa
tio
11 ft
su
nr
oo
m
11 ft
1. What would the radius of the circular patio be?
2. Calculate the area of the patio. (Hint: One‑fourth of the area of the entire circle is taken up by the sunroom.) Use
π = 3.14 as an approximation and round your answer to the nearest hundredth.
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3. If the concrete for the patio is to be 6 inches thick, calculate the volume of concrete needed in cubic feet. (Hint:
The patio is a cylinder with one‑fourth of it missing.) Use π = 3.14 as an approximation and round your answer to
the nearest hundredth.
3
 1 yard 
4. Convert the volume result from Problem 3 to cubic yards. (Hint: Use the ratio 
and multiply this times
 3 feet 
the volume in cubic feet from the previous question.) Round your answer to the nearest hundredth.
Chapter 5 Projects 211
5. If concrete costs $75 per cubic yard, then determine the cost of the concrete for the patio (before sales tax). Round
your answer to the nearest cent.
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6. Bob’s wife wants to put a short decorative fence around the patio. Calculate how many feet of fencing will be
needed. Round your answer to the nearest hundredth.
Name:Date:
Math@Work
Architecture
As a project architect, you will be part of a team that creates detailed drawings of the project that will be used during the
construction phase. It will be your job to ensure that the project will meet guidelines given to you by your company, such
as square footage requirements and budget constraints. You will also need to meet the design requirements requested
by the client.
Suppose you are part of a team that is designing an apartment building. You are given the task to create the floor plan
for an apartment unit with two bedrooms and one bathroom. The apartment management company that has contracted
your company to do the project has several requirements for this specific apartment unit.
2. All walls must intersect or touch at 90 degree angles.
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3. The kitchen must have an area of no more than 110 square feet.
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1. One bedroom is the “master bedroom” and must have at least 60 square feet more than the other bedroom.
4. The apartment must be between 1000 square feet and 1050 square feet.
A preliminary sketch of the apartment is shown.
44 feet
Dining Room
Kitchen
9.5 feet x 10.5 feet
Bathroom
Master Bedroom
12 feet x 18 feet
26.5 feet
Living Room
Bedroom 2
10.5 feet x 13 feet
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6 feet
12 feet
1. Does the apartment have the required total square footage that was requested? Is it over or under the total required?
2. Does the apartment blueprint meet the other requirements given by the client? If not, what does not meet the
requirements?
3. For this specific apartment unit, the total construction cost per square foot is estimated to be $95.75. Approximately
how much will it cost to construct each two bedroom apartment based on the floor plan?
Math@Work 213
Foundations Skill Check for Chapter 6
This page lists several skills covered previously in the book and software that are needed to learn new skills in Chapter 6.
To make sure you are prepared to learn these new skills, take the self-test below and determine if any specific skills
need to be reviewed.
1.4 Divide. Round to the nearest hundredth when necessary.
m. 8844 ÷ 33
h. 9936 ÷ 27
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e. 51 ÷ 3
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Each skill includes an easy (e.), medium (m.), and hard (h.) version. You should be able to complete each problem type
at each skill level. If you are unable to complete the problems at the easy or medium level, go back to the given lesson
in the software and review until you feel confident in your ability. If you are unable to complete the hard problem for a
skill, or are able to complete it but with minor errors, a review of the skill may not be necessary. You can wait until the
skill is needed in the chapter to decide whether or not you should work through a quick review.
2.1 Reduce each fraction to lowest terms.
e.
25
100
m.
36
48
h.
51
85
3.4 Divide. Round to the nearest hundredth when necessary.
e. 14.5 ÷ 1000
m. 72 ÷ 1.2
h. 267 ÷ 1.2
e.
1
+ 1.75 2
2 1
m.  2.35 + 1 + 1  ÷ 3 
5
2
2
3
2
h.   + 2.75 ⋅
 10 
5
3.5 Arrange the numbers in order from least to greatest.
e.
214
5 1
0.5, , 1 5
m.
7
, 2.3, 2.33 3
h.
4
9
, 0.15,
25
50
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3.5 Simplify each expression following the order of operations. Round to the nearest hundredth when necessary.