Download Chapter 11: Radical Expressions and Triangles

Notes
Introduction
In this unit, students will be introduced to additional nonlinear
functions such as radical and
rational equations. Students learn
how to simplify radical and
rational expressions, and how to
solve equations involving these
expressions.
Students also explore triangles
through the Pythagorean Theorem
and trigonometric ratios.
Nonlinear functions
such as radical and
rational functions
can be used to model
real-world situations
such as the speed of
a roller coaster. In this
unit, you will learn
about radical and
rational functions.
Assessment Options
Unit 4 Test Pages 779–780
of the Chapter 12 Resource Masters
may be used as a test or review
for Unit 4. This assessment contains both multiple-choice and
short answer items.
TestCheck and
Worksheet Builder
This CD-ROM can be used to
create additional unit tests and
review worksheets.
582 Unit 4 Radical and Rational Functions
582
Unit 4 Radical and Rational Functions
Radical and
Rational
Functions
Chapter 11
Radical Expressions and Triangles
Chapter 12
Rational Expressions and
Equations
Teaching
Suggestions
Have students study the
USA TODAY Snapshot.
• Ask students how the data in
the graph differs from that of
a linear function. The data in
the graph does not follow a
straight line, therefore it is not
linear.
• Why might math be important
to a roller coaster designer?
Sample answer: A roller coaster
designer might use math to calculate the speed of a roller coaster
before it is built to make sure it
is safe.
Building the Best
Roller Coaster
USA TODAY Snapshots®
Each year, amusement park owners compete to earn
part of the billions of dollars Americans spend
at amusement parks. Often the parks draw customers
with new taller and faster roller coasters. In this
project, you will explore how radical and rational
functions are related to buying and building a new
roller coaster.
Log on to www.algebra1.com/webquest.
Begin your WebQuest by reading the Task.
Then continue working
on your WebQuest as
you study Unit 4.
A day in the park
What the typical family of four
pays to visit a park1:
$163
$180
11-1
590
12-2
652
$116
$120
$60
0
Lesson
Page
Additional USA TODAY
Snapshots appearing in Unit 4:
Chapter 11 Global spending
on construction
(p. 615)
Chapter 12 Most Americans
have one or two
credit cards (p. 672)
Cost of parenthood
rising (p. 689)
’93
’95
’97
’99
’01
1 — Admission for two adults and two children, parking for one car and
purchase of two hot dogs, two hamburgers, four orders of fries, four small
soft drinks and two children’s T-shirts.
Source: Amusement Business
Unit 4
By Marcy E. Mullins, USA TODAY
Radical and Rational Functions 583
Internet Project
A WebQuest is an online project in which students do research on the Internet,
gather data, and make presentations using word processing, graphing,
page-making, or presentation software. In each chapter, students advance to
the next step in their WebQuest. At the end of Chapter 12, the project
culminates with a presentation of their findings.
Teaching notes and sample answers are available in the WebQuest and
Project Resources.
Unit 4 Radical and Rational Functions 583
Radical Expressions
and Triangles
Chapter Overview and Pacing
PACING (days)
Regular
Block
LESSON OBJECTIVES
Basic/
Average
Advanced
Basic/
Average
Advanced
Simplifying Radical Expressions (pp. 586–592)
• Simplify radical expressions using the Product Property of Square Roots.
• Simplify radical expressions using the Quotient Property of Square Roots.
2
2
1.5
1.5
Operations with Radical Expressions (pp. 593–597)
• Add and subtract radical expressions.
• Multiply radical expressions.
2
2
1
1
Radical Equations (pp. 598–604)
• Solve radical equations.
• Solve radical equations with extraneous solutions.
Follow-Up: Use a graphing calculator to graph radical equations.
2
3
(with 11-3
Follow-Up)
1
2
The Pythagorean Theorem (pp. 605–610)
• Solve problems by using the Pythagorean Theorem.
• Determine whether a triangle is a right triangle.
1
1
0.5
0.5
The Distance Formula (pp. 611–615)
• Find the distance between two points on the coordinate plane.
• Find a point that is a given distance from a second point in a plane.
1
1
0.5
0.5
Similar Triangles (pp. 616–621)
• Determine whether two triangles are similar.
• Find the unknown measures of sides of two similar triangles.
1
1
0.5
0.5
Trigonometric Ratios (pp. 622–630)
Preview: Use paper triangles to investigate trigonometric ratios.
• Define the sine, cosine, and tangent ratios.
• Use trigonometric ratios to solve right triangles.
2
2
1
1
Study Guide and Practice Test (pp. 632–637)
Standardized Test Practice (pp. 638–639)
1
1
0.5
0.5
Chapter Assessment
1
1
0.5
0.5
13
14
7
8
TOTAL
Pacing suggestions for the entire year can be found on pages T20–T21.
584A Chapter 11 Radical Expressions and Triangles
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™
All-In-One Planner
and Resource Center
Chapter Resource Manager
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CHAPTER 11 RESOURCE MASTERS
See pages T12–T13.
SC 21
83
11-1
11-1
84
11-2
11-2
GCS 43
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11-3
11-3
GCS 44,
SC 22
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702–704
90
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699, 701
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Materials
graphing calculator
graphing calculator
(Follow-Up: graphing
calculator)
32
(Follow-Up: ruler, grid
paper, protractor), string,
drinking straw, paper clip,
protractor, tape, meter
sticks or tape measure
*Key to Abbreviations: GCS Graphing Calculator and Speadsheet Masters,
SC School-to-Career Masters,
SM Science and Mathematics Lab Manual
ELL Study Guide and Intervention, Skills Practice, Practice, and Parent
and Student Study Guide Workbooks are also available in Spanish.
Chapter 11 Radical Expressions and Triangles 584B
Mathematical Connections
and Background
Continuity of Instruction
Prior Knowledge
In Chapter 2, students learned to find principal square roots. They solved proportions in
Chapter 3. Students found the factors of the
difference of squares in Chapter 8.
This Chapter
In this chapter, students simplify and perform
operations with radical expressions. They
apply these skills to solving radical equations.
The Pythagorean Theorem is used to solve
problems involving right triangles. The
Distance Formula is introduced as an application of the Pythagorean Theorem. Similar
triangles and trigonometric ratios are used
to find missing measures in triangles.
Future Connections
The basics of trigonometry and the study of
triangles are explored in this chapter. These
basics will be applied to deeper studies of
triangles in later mathematics courses. Many
of these concepts are used in construction
and engineering.
584C
Chapter 11 Radical Expressions and Triangles
Simplifying Radical Expressions
A radical expression contains a square root. The
expression is in simplest form if the expression inside the
radical sign, or radicand, has only 1 as a perfect square
factor. The Product Property of Square Roots states that
the square root of a product equals the product of each
square root. Prime factorizations combined with the
Product Property of Square Roots can be used to simplify
radical expressions. Principal square roots are never
negative, so absolute value symbols must be used to
signify that some results are not negative. The Quotient
Property of Square Roots states that the square root of a
quotient equals the quotient of each square root. The
Quotient Property of Square Roots can be used to derive
the Quadratic Formula.
A fraction does not have a radical in its denominator if it is in simplest form. Since squaring and taking
a square root are inverse functions, you multiply the
numerator and the denominator by the same number
so that the denominator contains a perfect square.
Remember that the numerator must be multiplied by
the same amount so that the whole fraction is being
multiplied by a value of 1. This process is called rationalizing the denominator. After rationalizing the denominator, check for coefficients of the radical in the numerator
that will simplify with the denominator. If the denominator is an expression containing a radical, multiply by
the conjugate. For example, if the denominator is
a b, multiply by a b.
Operations with Radical
Expressions
Use the process of combining like terms to simplify
expressions in which radicals are added or subtracted. For
terms to be combined, their radicands must be the same.
As in combining monomials with variables, only the coefficients of the radicals are combined. Be sure to simplify all
radicals first. When multiplying radical expressions, first
multiply the coefficients, and then use the Product
Property of Square Roots to multiply the radicals. Simplify
each term and then combine like terms as necessary.
Radical Equations
Equations that contain variables in the radicand
are called radical equations. To solve radical equations,
the radical must first be isolated on one side. Then square
each side. This will eliminate the radical. This process
sometimes produces results that are not solutions of the
original equation. These are called extraneous solutions.
All solutions must be substituted back into the original
equation to check their validity.
The Pythagorean Theorem
The two sides of a right triangle that form the
right angle are the legs of the triangle. The third, and
longest, side is the hypotenuse. The hypotenuse is the
longest side because it is always across from the angle
with the greatest measure. The Pythagorean Theorem
states that the sum of the squares of the legs equals the
square of the hypotenuse. The formula is c2 a2 b2,
where a and b are the measures of the legs and c is the
measure of the hypotenuse. This formula can be used
to find the length of any missing side of a right triangle if the lengths of the other two sides are known.
Any three whole numbers that satisfy this equation
are known as a Pythagorean Triple. These triples
represent side lengths that always form right triangles.
It follows that if three numbers do not satisfy the
Pythagorean Theorem, then sides of their length will
not form a right triangle.
Trigonometric Ratios
Trigonometry is the mathematical study of
angles and triangles. Ratios comparing the measures of
two sides of a right triangle are called trigonometric
ratios. The three most common trigonometric ratios
are sine, cosine, and tangent. These ratios can be used
to find the measures of missing sides or the measures
of the acute angles. The relationship of the two sides
necessary for the problem to a specific acute angle
determines which ratio is used. If the measures of just
two sides of a triangle or the measures of one side
and one acute angle are known, then the measures of
all of the rest of the sides and angles can be found.
This is called solving the triangle. Trigonometric ratios
are used to find distances in problems involving
angles of elevation and angles of depression.
The Distance Formula
If the Pythagorean Theorem is solved for c, the
result is the Distance Formula. The variable a is
expressed as the difference of the x-coordinates of the
endpoints of the hypotenuse and b is expressed as the
difference of the y-coordinates. This formula is used to
find the distance between any two points. You can also
find one missing coordinate of an endpoint if you know
the other coordinate, the coordinates of the other endpoint, and the distance between the two points.
Similar Triangles
Similar triangles have the same shape, but are
not necessarily the same size. All of the corresponding
angles will have equal measures, and the corresponding sides will all be proportional. If the sides have a
1 to 1 ratio of proportionality, then the similar triangles
are the same size. When determining whether triangles are similar, all you need to check is if the corresponding angles have the same measure. If all the
angle measures cannot be determined, then check the
corresponding sides to see if they are proportional.
Proportions can be used to find the lengths of
missing sides of similar triangles. You must know the
lengths of at least one pair of corresponding sides
and the length of the side that corresponds to the
missing side’s length. Set up the proportion, and then
cross multiply. Solve the resulting equation.
Chapter 11 Radical Expressions and Triangles 584D
and Assessment
ASSESSMENT
INTERVENTION
Type
Student Edition
Teacher Resources
Ongoing
Prerequisite Skills, pp. 585, 592,
597, 603, 610, 615, 621
Practice Quiz 1, p. 603
Practice Quiz 2, p. 621
5-Minute Check Transparencies
Prerequisite Skills Workbook, pp. 37–38, 61–62
Quizzes, CRM pp. 699–700
Mid-Chapter Test, CRM p. 701
Study Guide and Intervention, CRM pp. 643–644,
649–650, 655–656, 661–662, 667–668, 673–674,
679–680
Mixed
Review
pp. 592, 597, 603, 610, 615,
621, 630
Cumulative Review, CRM p. 702
Error
Analysis
Find the Error, pp. 600, 618
Find the Error, TWE pp. 600, 618
Unlocking Misconceptions, TWE p. 612
Tips for New Teachers, TWE pp. 587, 624
Standardized
Test Practice
pp. 591, 597, 602, 606, 608,
610, 615, 620, 630, 637,
638–639
TWE pp. 638–639
Standardized Test Practice, CRM pp. 703–704
Open-Ended
Assessment
Writing in Math, pp. 591, 597,
602, 610, 614, 620, 630
Open Ended, pp. 589, 595, 600,
607, 612, 618, 627
Standardized Test, p. 639
Modeling: TWE pp. 603, 610, 621
Speaking: TWE pp. 597, 615
Writing: TWE pp. 592, 630
Open-Ended Assessment, CRM p. 697
Chapter
Assessment
Study Guide, pp. 632–636
Practice Test, p. 637
Multiple-Choice Tests (Forms 1, 2A, 2B),
CRM pp. 685–690
Free-Response Tests (Forms 2C, 2D, 3),
CRM pp. 691–696
Vocabulary Test/Review, CRM p. 698
Technology/Internet
AlgePASS: Tutorial Plus
www.algebra1.com/self_check_quiz
www.algebra1.com/extra_examples
Standardized Test Practice
CD-ROM
www.algebra1.com/
standardized_test
TestCheck and Worksheet Builder
(see below)
MindJogger Videoquizzes
www.algebra1.com/
vocabulary_review
www.algebra1.com/chapter_test
Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters
Additional Intervention Resources
The Princeton Review’s Cracking the SAT & PSAT
The Princeton Review’s Cracking the ACT
ALEKS
TestCheck and Worksheet Builder
This networkable software has three modules for intervention
and assessment flexibility:
• Worksheet Builder to make worksheet and tests
• Student Module to take tests on screen (optional)
• Management System to keep student records (optional)
Special banks are included for SAT, ACT, TIMSS, NAEP, and
End-of-Course tests.
584E
Chapter 11 Radical Expressions and Triangles
Reading and Writing
in Mathematics
Intervention Technology
AlgePASS: Tutorial Plus CD-ROM offers a
complete, self-paced algebra curriculum.
Algebra 1
Lesson
11-4
AlgePASS Lesson
32 Solving Problems Using the Pythagorean
Theorem
ALEKS is an online mathematics learning system that
adapts assessment and tutoring to the student’s needs.
Subscribe at www.k12aleks.com.
Glencoe Algebra 1 provides numerous opportunities to
incorporate reading and writing into the mathematics
classroom.
Student Edition
• Foldables Study Organizer, p. 585
• Concept Check questions require students to verbalize
and write about what they have learned in the lesson.
(pp. 589, 595, 600, 607, 612, 618, 627)
• Reading Mathematics, p. 631
• Writing in Math questions in every lesson, pp. 591, 597,
602, 610, 614, 620, 630
• Reading Study Tip, pp. 586, 611, 616, 623
• WebQuest, p. 590
Intervention at Home
Parent and Student Study Guide
Parents and students may work together to reinforce the
concepts and skills of this chapter. (Workbook, pp. 83–90
or log on to www.algebra1.com/parent_student )
Log on for student study help.
• For each lesson in the Student Edition, there are Extra
Examples and Self-Check Quizzes.
www.algebra1.com/extra_examples
www.algebra1.com/self_check_quiz
• For chapter review, there is vocabulary review, test
practice, and standardized test practice.
www.algebra1.com/vocabulary_review
www.algebra1.com/chapter_test
www.algebra1.com/standardized_test
For more information on Intervention and
Assessment, see pp. T8–T11.
Teacher Wraparound Edition
• Foldables Study Organizer, pp. 585, 632
• Study Notebook suggestions, pp. 589, 595, 600, 608,
613, 618, 627, 631
• Modeling activities, pp. 603, 610, 621
• Speaking activities, pp. 597, 615
• Writing activities, pp. 592, 630
• Differentiated Instruction, (Verbal/Linguistic), p. 599
• ELL Resources, pp. 584, 591, 596, 599, 602, 609,
614, 620, 629, 631, 632
Additional Resources
• Vocabulary Builder worksheets require students to
define and give examples for key vocabulary terms as
they progress through the chapter. (Chapter 11 Resource
Masters, pp. vii-viii)
• Reading to Learn Mathematics master for each lesson
(Chapter 11 Resource Masters, pp. 647, 653, 659, 665,
671, 677, 683)
• Vocabulary PuzzleMaker software creates crossword,
jumble, and word search puzzles using vocabulary lists
that you can customize.
• Teaching Mathematics with Foldables provides
suggestions for promoting cognition and language.
• Reading and Writing in the Mathematics Classroom
• WebQuest and Project Resources
• Hot Words/Hot Topics Sections 3.2, 7.9, 7.10, 8.6
For more information on Reading and Writing in
Mathematics, see pp. T6–T7.
Chapter 11 Radical Expressions and Triangles 584F
Radical
Expressions
and Triangles
Notes
Have students read over the list
of objectives and make a list of
any words with which they are
not familiar.
• Lessons 11-1 and 11-2 Simplify and perform
operations with radical expressions.
• Lesson 11-3 Solve radical equations.
• Lessons 11-4 and 11-5 Use the Pythagorean
Theorem and Distance Formula.
• Lessons 11-6 and 11-7 Use similar triangles and
trigonometric ratios.
Point out to students that this is
only one of many reasons why
each objective is important.
Others are provided in the
introduction to each lesson.
Key Vocabulary
•
•
•
•
•
radical expression (p. 586)
radical equation (p. 598)
Pythagorean Theorem (p. 605)
Distance Formula (p. 611)
trigonometric ratios (p. 623)
Physics problems are among the many applications of radical equations.
Formulas that contain the value for the acceleration due to gravity, such
as free-fall times, escape velocities, and the speeds of roller coasters,
can all be written as radical equations. You will learn how to calculate the
time it takes for a skydiver to fall a given distance in Lesson 11-3.
Lesson
11-1
11-2
11-3
11-3
Follow-Up
11-4
11-5
11-6
11-7
Preview
11-7
NCTM
Standards
Local
Objectives
1, 2, 6, 8, 9,
10
1, 6, 8, 9, 10
2, 6, 8, 9, 10
2, 6, 8
1, 2, 3, 6, 8, 9,
10
1, 2, 3, 6, 8, 9,
10
1, 2, 3, 6, 8, 9,
10
1, 3, 7
3, 6, 8, 9, 10
Key to NCTM Standards:
1=Number & Operations, 2=Algebra,
3=Geometry, 4=Measurement,
5=Data Analysis & Probability, 6=Problem
Solving, 7=Reasoning & Proof,
8=Communication, 9=Connections,
10=Representation
584
584 Chapter 11 Radical Expressions and Triangles
Vocabulary Builder
ELL
The Key Vocabulary list introduces students to some of the main vocabulary terms
included in this chapter. For a more thorough vocabulary list with pronunciations of
new words, give students the Vocabulary Builder worksheets found on pages vii and
viii of the Chapter 11 Resource Masters. Encourage them to complete the definition
of each term as they progress through the chapter. You may suggest that they add
these sheets to their study notebooks for future reference when studying for the
Chapter 11 test.
Chapter 11 Radical Expressions and Triangles
Prerequisite Skills To be successful in this chapter, you’ll need to master
these skills and be able to apply them in problem-solving situations. Review
these skills before beginning Chapter 11.
For Lessons 11-1 and 11-4
This section provides a review of
the basic concepts needed before
beginning Chapter 11. Page
references are included for
additional student help.
Additional review is provided in
the Prerequisite Skills Workbook,
pp. 37–38 and 61–62.
Find Square Roots
Find each square root. If necessary, round to the nearest hundredth.
(For review, see Lesson 2-7.)
1. 25
5
2. 80
8.94
3. 56
7.48
4. 324
18
For Lesson 11-2
Combine Like Terms
Prerequisite Skills in the Getting
Ready for the Next Lesson section
at the end of each exercise set
review a skill needed in the next
lesson.
Simplify each expression. (For review, see Lesson 1-6.)
5. 3a 7b 2a a 7b
6. 14x 6y 2y 14x 4y
7. (10c 5d) (6c 5d) 16c
8. (21m 15n) (9n 4m) 25m 6n
For Lesson 11-3
Solve Quadratic Equations
Solve each equation. (For review, see Lesson 9-3.)
10. x2 10x 24 0 {6, 4}
1
12. 2x2 x 1 2 , 1
2
9. x(x 5) 0 {0, 5}
11. x2 6x 27 0 {3, 9}
For Lesson 11-6
Proportions
Use cross products to determine whether each pair of ratios forms a proportion.
Write yes or no. (For review, see Lesson 3-6.)
2 8
13. , 3 12
yes
4 16
14. , 5 25
no
8 12
15. , 10 16
6 3
16. , 30 15
no
For
Lesson
Prerequisite
Skill
11-2
11-3
Multiplying Binomials (p. 592)
Finding Special Products
(p. 597)
Evaluating Radical Expressions
(p. 603)
Simplifying Radical Expressions
(p. 610)
Solving Proportions (p. 615)
Evaluating Expressions
(p. 621)
11-4
yes
11-5
11-6
11-7
Make this Foldable to help you organize information about
radical expressions and equations. Begin with a sheet of
1
plain 8" by 11" paper.
2
Fold in Half
Fold Again
Fold the
top to the
bottom.
Fold in half
lengthwise.
Cut
Label
Open. Cut along the
second fold to make
two tabs.
Label each
tab as shown.
Radical
Expression
s
Radical
Equations
As you read and study the chapter, write notes and examples
for each lesson under each tab.
Reading and Writing
Chapter 11 Radical Expressions and Triangles 585
TM
For more information
about Foldables, see
Teaching Mathematics
with Foldables.
Organization of Data: Annotating As students read and work
through the chapter, have them make annotations under the tabs
of their Foldable. Explain to them that annotations are usually
notes taken in the margins of books we own to organize the text
for review or studying. Annotations often include questions that
arise as we read the chapter, reader comments and reactions,
sentence length summaries, steps or data numbered by the
reader, and key points highlighted or underlined.
Chapter 11 Radical Expressions and Triangles 585
Simplifying
Radical Expressions
Lesson
Notes
1 Focus
5-Minute Check
Transparency 11-1 Use as
a quiz or review of Chapter 10.
Mathematical Background notes
are available for this lesson on
p. 584C.
• Simplify radical expressions using the Product Property of Square Roots.
• Simplify radical expressions using the Quotient Property of Square Roots.
• radical expression
• radicand
• rationalizing the
denominator
• conjugate
A spacecraft leaving Earth must have a velocity
of at least 11.2 kilometers per second (25,000
miles per hour) to enter into orbit. This velocity
is called the escape velocity. The escape velocity
of an object is given by the radical expression
2GM
, where G is the gravitational constant,
R
M is the mass of the planet or star, and R is the
radius of the planet or star. Once values are
substituted for the variables, the formula can
be simplified.
Building on Prior
Knowledge
Students were introduced to the
Quadratic Formula in Lesson 10-4
and they learned how to use it to
solve quadratic equations. In this
lesson, students learn how to derive the Quadratic Formula from
the standard form of a quadratic
equation using the Quotient
Property of Square Roots.
are radical expressions
used in space
exploration?
Ask students:
• In the formula for escape
velocity, what does the radical
sign mean? The radical sign
means that you must find the
square root of the value under the
radical sign.
• Based on what you know
about the order of operations,
how do you think you should
simplify the radical expression
in the escape velocity formula?
You should simplify the expression
under the radical sign before
finding the square root.
are radical expressions used in space exploration?
Vocabulary
PRODUCT PROPERTY OF SQUARE ROOTS A radical expression is an
expression that contains a square root. A radicand, the expression under the radical
sign, is in simplest form if it contains no perfect square factors other than 1. The
following property can be used to simplify square roots.
Product Property of Square Roots
• Words
For any numbers a and b, where a 0 and b 0, the square root
of a product is equal to the product of each square root.
• Symbols ab
a b
• Example 4 25 4
25
The Product Property of Square Roots and prime factorization can be used to
simplify radical expressions in which the radicand is not a perfect square.
Example 1 Simplify Square Roots
Simplify.
a. 12
Study Tip
Reading Math
23 is read two times
the square root of 3 or
two radical three.
12 2 2 3
22 3
23
Prime factorization of 12
Product Property of Square Roots
Simplify.
b. 90
90 23
35
32 25
310
Prime factorization of 90
Product Property of Square Roots
Simplify.
586 Chapter 11 Radical Expressions and Triangles
Resource Manager
Workbook and Reproducible Masters
Chapter 11 Resource Masters
• Study Guide and Intervention, pp. 643–644
• Skills Practice, p. 645
• Practice, p. 646
• Reading to Learn Mathematics, p. 647
• Enrichment, p. 648
Parent and Student Study Guide
Workbook, p. 83
School-to-Career Masters, p. 21
Transparencies
5-Minute Check Transparency 11-1
Answer Key Transparencies
Technology
Interactive Chalkboard
The Product Property can also be used to multiply square roots.
Study Tip
Alternative Method
To find 3 15
, you
could multiply first and then
use the prime factorization.
45
3 15
32 5
35
2 Teach
Example 2 Multiply Square Roots
Find 3 15
.
3 3 5
3 15
32 5
35
PRODUCT PROPERTY OF
SQUARE ROOTS
Product Property of Square Roots
Product Property
Simplify.
Intervention
In order to
simplify square
roots with the
Product Property of Square Roots, students
need to be able to find the
prime factorization of the
radicand. Take a few minutes
to review finding prime
factorizations. After a quick
refresher, students can focus
on learning the new concept,
rather than trying to recall
earlier material.
New
When finding the principal square root of an expression containing variables, be
x2. It may seem that
sure that the result is not negative. Consider the expression x2 x. Let’s look at x 2.
x2 x
2 2
(2)
4 2
2 2
Replace x with 2.
(2)2 4
4 2
For radical expressions where the exponent of the variable inside the radical is even
and the resulting simplified exponent is odd, you must use absolute value to ensure
nonnegative results.
x2 x
x3 xx
x4 x2
x5 x2x
x6 x3
Example 3 Simplify a Square Root with Variables
Simplify 40x4y5
z3.
40x4y5
z3 23 5 x4 y5
z3
In-Class Examples
Prime factorization
22 2 5 x4 y4 y z2 z Product Property
1 Simplify.
2 2 5 x2 y2 y z z
a. 52
213
Simplify.
The absolute value of z ensures
a nonnegative result.
2x2y2z10yz
Power
Point®
b. 72
62
Teaching Tip
Remind students
that since 3 is a prime number,
it cannot be factored further
using prime factorization.
QUOTIENT PROPERTY OF SQUARE ROOTS You can divide square
roots and simplify radical expressions that involve division by using the Quotient
Property of Square Roots.
2 Find 2 24
.
Quotient Property of Square Roots
• Words
For any numbers a and b, where a 0 and b 0, the square root of
a quotient is equal to the quotient of each square root.
• Symbols
a
ba b
• Example
43
3 Simplify 45a4b5
c6.
3a 2b 2 |c 3 | 5b
49
49
4
4
Study Tip
Look Back
To review the Quadratic
Formula, see Lesson 10-4.
You can use the Quotient Property of Square Roots to derive the Quadratic
Formula by solving the quadratic equation ax2 bx c 0.
Interactive
ax2 bx c 0 Original equation
b
a
Chalkboard
c
a
x2 x 0 Divide each side by a, a 0.
(continued on the next page)
www.algebra1.com/extra_examples
PowerPoint®
Presentations
Lesson 11-1 Simplifying Radical Expressions 587
This CD-ROM is a customizable
Microsoft® PowerPoint®
presentation that includes:
• Step-by-step, dynamic solutions of
each In-Class Example from the
Teacher Wraparound Edition
• Additional, Your Turn exercises for
each example
• The 5-Minute Check Transparencies
• Hot links to Glencoe Online
Study Tools
Lesson 11-1 Simplifying Radical Expressions 587
In-Class Example
11y
33y
b. 9
27
3 6
8
c. 4
c
Subtract from each side.
b2
b
a
a
b2
c
a
4a
2
x 2ba 2
2a
4a
4ac b2
4a
b
a
2
b2 4ac
2
Remove the absolute value symbols and insert .
2
Plus or Minus
Symbol
b
2a
2 4ac
b
4a
x 2
The symbol is used with the
radical expression since both
square roots lead to solutions.
Quotient Property of Square Roots
2 4ac
b
x b
2a
b2
4a
Take the square root of each side.
2
Study Tip
4a
Factor x2 x 2.
x 2a  4a b
b 4ac
x 2a
4a b
2
b
b
Complete the square; 2.
x2 x 2 2
Power
Point®
4 Simplify.
12
215
a. 5
5
c
a
b
a
x2 x QUOTIENT PROPERTY OF
SQUARE ROOTS
4a2 2a
2a
b b 4ac
x 2
2a
b
Subtract from each side.
2a
Thus, we have derived the Quadratic Formula.
A fraction containing radicals is in simplest form if no prime factors appear under
the radical sign with an exponent greater than 1 and if no radicals are left in the
denominator. Rationalizing the denominator of a radical expression is a method
used to eliminate radicals from the denominator of a fraction.
Example 4 Rationalizing the Denominator
Simplify.
7x
b. 10
a. 8
3
10
3
10
3
3 3
3 .
Multiply by 30
Product Property
of Square Roots
3
3
7x
7x
2
2
8
2
2
7x
2 .
Multiply by 14x
Product Property
of Square Roots
22
4
2
c. Prime
factorization
2
2
6
2 6
2
6 6
6
12
6
Product Property of Square Roots
2 2 3
6
Prime factorization
23
22 2
3
Divide the numerator and denominator by 2.
6
6
3
588
6 .
Multiply by Chapter 11 Radical Expressions and Triangles
Differentiated Instruction
Logical Have students use the same method that is shown for the
derivation of the Quadratic Formula to solve actual quadratic equations.
Have students turn back to Lesson 10-4 and solve an example problem
using this method.
588
Chapter 11 Radical Expressions and Triangles
Binomials of the form pq rs and pq rs are called conjugates.
For example, 3 2 and 3 2 are conjugates. Conjugates are useful when
simplifying radical expressions because if p, q, r, and s are rational numbers,
their product is always a rational number with no radicals. Use the pattern
(a b)(a b) a2 b2 to find their product.
3 2 3 2 32 2 2
9 2 or 7
In-Class Example
Power
Point®
3
5 Simplify .
15 32
23
5 2
a 3, b 2
3 Practice/Apply
2
2 2 2 or 2
Example 5 Use Conjugates to Rationalize a Denominator
2
Simplify .
6 3
6 3
2
2
6 3
6 3
6 3
26 3
2
62 3
(a b)(a b) 12 23
36 3
3
2 3
12 23
Simplify.
33
Study Notebook
6 3
1
6 3
a2
Have students—
• add the definitions/examples of
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 11.
• include examples of how to use the
Product and Quotient Properties of
Square Roots to simplify radical
expressions.
• include examples of how to
rationalize denominators, with and
without conjugates.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
b2
When simplifying radical expressions, check the following conditions to
determine if the expression is in simplest form.
Simplest Radical Form
A radical expression is in simplest form when the following three
conditions have been met.
1. No radicands have perfect square factors other than 1.
2. No radicands contain fractions.
3. No radicals appear in the denominator of a fraction.
Answer
Concept Check
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
4
5, 6, 13
7, 8
9, 10, 14
11, 12
1
2
3
4
5
1. Both x 4 and x 2 are positive even if
x is a negative number.
x4 x2. See margin.
1. Explain why absolute value is not necessary for 1
1
a for a 0. a
a 2. Show that a
a
a
a a
3. OPEN ENDED Give an example of a binomial in the form ab cd and its
conjugate. Then find their product. Sample answer: 22
33 and
Simplify. 8. 2x2y315x
22
33; 19
4. 20
25
5. 2 8 4
7. 54a2b2 3ab6
8. 60x5y6
10.
3
10
30
10
6. 310
410
120
8
83 2 11. 7
3 2
4
6
2
9. 6
3
25 10
25
12. 2
4 8
Lesson 11-1 Simplifying Radical Expressions 589
Lesson 11-1 Simplifying Radical Expressions 589
Applications
About the Exercises …
Organization by Objective
• Product Property of Square
Roots: 15–26, 39, 41
• Quotient Property of
Square Roots: 27–38, 40
Odd/Even Assignments
Exercises 15–40 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Assignment Guide
Basic: 15–41 odd, 42, 43, 50–53,
62–89
Average: 15–41 odd, 45–47,
50–53, 62–89 (optional: 54–61)
Advanced: 16–40 even, 44–83
(optional: 84–89)
14. PHYSICS The period of a pendulum is the time
required for it to make one complete swing back
and forth. The formula of the period P of a pendulum
32
ᐉ
is P 2 , where ᐉ is the length of the pendulum
in feet. If a pendulum in a clock tower is 8 feet long,
find the period. Use 3.14 for . 3.14 s
★ indicates increased difficulty
Practice and Apply
Homework Help
51. A lot of formulas and calculations
that are used in space exploration
contain radical expressions.
Answers should include the
following.
• To determine the escape
velocity of a planet, you would
need to know its mass and the
radius. It would be very
important to know the escape
velocity of a planet before you
landed on it so you would know
if you had enough fuel and
velocity to launch from it to get
back into space.
• The astronomical body with the
smaller radius would have a
greater escape velocity. As the
radius decreases, the escape
velocity increases.
Simplify. 25. 7x3y33y
26. 6xy2z22xz
For
Exercises
See
Examples
15. 18
32
16. 24
26
15–18, 41,
44–46
19–22, 39,
40, 48, 49
23–26
27–32, 42,
43, 47
33–38
1
18. 75
53
19. 5 6
30
20. 3
8 26
2
21. 730
26 845
22. 23 527
90
23. 40a4 2a210
3
4
24. 50m3n5 5mn 22mn
147x6y7
25. 5
27.
Extra Practice
See page 844.
x3xy
32. 3
Answer
13. GEOMETRY A square has sides each measuring 27
feet. Determine the area
of the square. 28 ft2
2y 
5 10
2
34. 2
35. 27 22
26 23
36. 3
The speed of a roller
coaster can be determined
by evaluating a radical
expression. Visit
www.algebra1.com/
webquest to continue
work on your WebQuest
project.
16 123
37. 11
521
335
38. 10
6
27 73 3
27 33
30. p p
28.
31.
2
26. 72x3y4
z5
310
35 64 10
5c
c 5cd
4d 2d
5
2
5
3
18
54 92
33. 17
6 2
25
34. 2
36. 3 6
4
37. 4 3
3
17. 80
45
4 8
29.
8t
2t
4
5y
9x
12x
y
32. 2 6
10
35. 7 2
37
★ 38. 53 35
39. GEOMETRY A rectangle has width 35 centimeters and length
4
10 centimeters. Find the area of the rectangle. 602
or 84.9 cm2
a
a
40. GEOMETRY A rectangle has length meters and width meters.
8
2
What is the area of the rectangle? a 2
m
4
41. GEOMETRY The formula for the area A of a square with side length s is A s2.
Solve this equation for s, and find the side length of a square having an area of
72 square inches. s A
; 62 in.
PHYSICS For Exercises 42 and 43, use the following information.
1
2
The formula for the kinetic energy of a moving object is E mv2, where E is
the kinetic energy in joules, m is the mass in kilograms, and v is the velocity in
meters per second.
2E
42. Solve the equation for v. v m
43. Find the velocity of an object whose mass is 0.6 kilogram and whose kinetic
energy is 54 joules. 6 5
or 13.4 m/s
44. SPACE EXPLORATION Refer to the application at the beginning of the
lesson. Find the escape velocity for the Moon in kilometers per second if
6.7 1020 km
s kg
22 kg, and R 1.7 103 km. How does this
G
2 , M 7.4 10
compare to the escape velocity for Earth? 2.4 km/s; The Moon has a much lower
590
escape velocity than Earth.
Chapter 11 Radical Expressions and Triangles
Teacher to Teacher
Judy Buchholtz
Dublin Scioto H.S., Dublin, OH
“To make algebra more meaningful to my students, I bring in the school
police officer to discuss the formula used in Exercises 45–47.”
590
Chapter 11 Radical Expressions and Triangles
INVESTIGATION For Exercises 45–47, use the following information.
NAME ______________________________________________ DATE
2d
45. Write a simplified expression for the speed if f 0.6 for a wet asphalt road. 3
46. What is a simplified expression for the speed if f 0.8 for a dry asphalt road?
Example 1
Simplify 180
.
Prime factorization of 180
Product Property of Square Roots
Simplify.
Simplify.
Simplify 120a2
b5 c4.
120a2
b5 c4
3 3 2 b5 2
5 a
c4
2
2 2 3 5 a2 b4 b c4
2 2
3
5
| a | b2 b
c2
2 | a | b2c230b
GEOMETRY For Exercises 48 and 49, use the following information.
Hero’s Formula can be used to calculate the area A of a triangle given the three side
lengths a, b, and c.
Exercises
Simplify.
1. 28
1
2
2. 68
27
s(s a
)(s b
)(s c), where s (a b c)
A 5. 162
48. Find the value of s if the side lengths of a triangle are 13, 10, and 7 feet. 15
49. Determine the area of the triangle. 203
or 34.6 ft2
50. CRITICAL THINKING
For any numbers a and b, where a 0 and b 0, ab
a
b
.
Product Property of Square Roots
Example 2
51.4 mph
46. 26d
Insurance investigators
decide whether claims are
covered by the customer’s
policy, assess the amount
of loss, and investigate the
circumstances of a claim.
The Product Property of Square Roots and
prime factorization can be used to simplify expressions involving irrational square roots.
When you simplify radical expressions with variables, use absolute value to ensure
nonnegative results.
180
2 2 33
5
22 32 5
2 3 5
65
47. An officer measures skid marks that are 110 feet long. Determine the speed of
the car for both wet road conditions and for dry road conditions. 44.5 mph,
Insurance
Investigator
p. 643
(shown)
and p. 644
Simplifying
Radical Expressions
Product Property of Square Roots
Lesson 11-1
Police officers can use the formula s 30fd
to determine the speed s that a car
was traveling in miles per hour by measuring the distance d in feet of its skid marks.
In this formula, f is the coefficient of friction for the type and condition of the road.
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
11-1
Study
Guide and
6. 3
6
7. 2
5
92
32
10. 9x4
2 |a |
4. 75
215
9. 4a2
10
52
12. 128c6
8 |c 3 | 2
10a 23
3x 2
15. 20a2b4
9 |x3|
2415
53
8. 5
10
11. 300a4
14. 3x2 3
3x4
13. 410
36
a 1 a
1
Simplify . a2 3a 1
a 1 a
3. 60
217
16. 100x3y
2 |a |b 25
10 | x |xy
17. 24a4b2
18. 81x4y2
19. 150a2
b2c
3c2
20. 72a6b
5z8
21. 45x2y
6z2
22. 98x4y
2a 2 | b | 6
9x 2 | y |
6 |a 3bc | 2b
5 |ab | 6c
3 |x |y 2z 45y
7x 2 |y 3z | 2
NAME ______________________________________________ DATE
Online Research
For more information
about a career as an
insurance investigator,
visit:
www.algebra1.com/
careers
Source: U.S. Department
of Labor
51. WRITING IN MATH
Skills
Practice,
11-1
Practice
(Average)
Simplify.
How are radical expressions used in space exploration?
Include the following in your answer:
• an explanation of how you could determine the escape velocity for a planet
and why you would need this information before you landed on the planet,
and
• a comparison of the escape velocity for two astronomical bodies with the
same mass, but different radii.
1. 24
26
2. 60
215
3. 108
63
4. 8
6
43
5. 7
14
72
6. 312
56
902
32a3
B
48a3
C
64a3
D
96a3
10. 108x6
y4z5 6| x 3 | y 2z 2 3z
9. 50p5 5p 22p
23
12. 8
6
4o5 2| m | n 2o 214o
11. 56m2n
15
15. 5
13.
19.
1
7
4
4y
3y2
2
9ab
4ab4
15 32
23
3
21. 2
12 43
3
8
22. 3 3
53
57
4
921
37
37
1 27
26
5
23. 7
3
7
11
18
x3
23y
3|y|
5 2
5
32
14.
4
5
3k
8
3
10
8
11
16. 11
32x
18. x
3b
20. 2b
2 5
10
5
6k
17. 52. If the cube has a surface area of 96a2,
what is its volume? C
A
8. 27su3 3| u | 3su
7. 43
318
366
3
4
Standardized
Test Practice
____________ PERIOD _____
p. 645 and
Practice,
646Expressions
(shown)
Simplifyingp.
Radical
Answer the question that was posed at the beginning of
the lesson. See margin.
24. 25. SKY DIVING When a skydiver jumps from an airplane, the time t it takes to free fall a
given distance can be estimated by the formula t 53. If x 81b2 and b 0, then x
B
A
9b.
B
9b.
C
3b27
.
D
27b3.
, where t is in seconds and s is in
2s
9.8
meters. If Julie jumps from an airplane, how long will it take her to free fall 750 meters?
Surface area
of a cube 6s2
about 12.4 s
METEOROLOGY For Exercises 26 and 27, use the following information.
To estimate how long a thunderstorm will last, meteorologists can use the formula
t
, where t is the time in hours and d is the diameter of the storm in miles.
d3
216
26. A thunderstorm is 8 miles in diameter. Estimate how long the storm will last. Give your
answer in simplified form and as a decimal. 83
h 1.5 h
9
27. Will a thunderstorm twice this diameter last twice as long? Explain.
No; it will last about 4.4 h, or nearly 3 times as long.
NAME ______________________________________________ DATE
WEATHER For Exercises 54 and 55, use the following information.
The formula y 91.4 (91.4 t)
0.478 0.301x 0.02 can be used to find
the windchill factor. In this formula, y represents the windchill factor, t represents
the air temperature in degrees Fahrenheit, and x represents the wind speed in miles
per hour. Suppose the air temperature is 12°.
Pre-Activity
54. Use a graphing calculator to find the wind speed to the nearest mile per hour
if it feels like 9° with the windchill factor. 7 mph
Reading the Lesson
____________ PERIOD _____
Reading
11-1
Readingto
to Learn
Learn Mathematics
Mathematics,
p. 647
Simplifying Radical Expressions
ELL
How are radical expressions used in space exploration?
Read the introduction to Lesson 11-1 at the top of page 586 in your textbook.
Suppose you want to calculate the escape velocity for a spacecraft taking off
from the planet Mars. When you substitute numbers in the formula, which
number is sure to be the same as in the calculation for the escape velocity
for a spacecraft taking off from Earth? the value of G
1. a. How can you tell that the radical expression 28x2y4 is not in simplest form?
The radicand contains perfect square factors other than 1.
55. What does it feel like to the nearest degree if the wind speed is 4 miles per hour? 6°F
www.algebra1.com/self_check_quiz
b. To simplify 28x2y4, you first find the
You then apply the
Lesson 11-1 Simplifying Radical Expressions 591
prime factorization
Product Property of Square Roots
4 7 x2 y4 is equal to the product 4
again to get a final answer of 2 | x | y27
.
of 28x2y4.
. In this case,
7
x 2 y 4 . You can simplify
2. Why is it correct to write y4 y2, with no absolute value sign, but not correct to write
x2 x?
NAME ______________________________________________ DATE
11-1
Enrichment
Enrichment,
____________ PERIOD _____
p. 648
Squares and Square Roots From a Graph
The graph of y x2 can be used to find the squares and square
roots of numbers.
rationalizing the denominator
4. What should you do to write the conjugate of a binomial of the form ab
cd
? To
write the conjugate of a binomial of the form ab
cd
?
The arrows show that 32 9.
A small part of the graph at y x2 is shown below. A 1:10 ratio
for unit length on the y-axis to unit length on the x-axis is used.
Example
Find 11
.
3.3
The arrows show that 11
to the nearest tenth.
y
12t
15
3. What method would you use to simplify ?
y
To find the square of 3, locate 3 on the x-axis. Then find its
corresponding value on the y-axis.
To find the square root of 4, first locate 4 on the y-axis. Then find
its corresponding value on the x-axis. Following the arrows on the
graph, you can see that 4
2.
Sample answer: The square of y 2 is y 4 and the expressions y 4 and y 2
both represent positive numbers for all values of y. Although it is true
2
2
that the square of x is x , when x is less than 0, x represents a
positive quantity and x represents a negative quantity.
Change the plus sign to a minus sign; change the minus sign to a
plus sign.
O
x
Helping You Remember
5. What should you remember to check for when you want to determine if a radical
expression is in simplest form?
Sample answer: Check radicands for perfect squares and fractions, and
check fractions for radicals in the denominator.
20
Lesson 11-1 Simplifying Radical Expressions 591
Lesson 11-1
Graphing
Calculator
Extending
the Lesson
4 Assess
1
x 2 x. Using the properties of exponents, simplify each expression.
1
5
57. x 2 x2
1
1
56. x 2 x 2 x
Open-Ended Assessment
Writing Have students copy the
Concept Summary on p. 589 into
their Study Notebooks. For each
of the three conditions listed,
have students provide an
example of an expression that
does not satisfy the condition
and then show how they
simplified the expression to
satisfy the condition.
Radical expressions can be represented with fractional exponents. For example,
4
x2
x
3
58. x 2 or xx
a
5
a
. a 6 or ★ 59. Simplify the expression 3
6
aa
a
1
for y.
★ 60. Solve the equation y3 3
3
★ 61. Write 1
in simplest form.
s2t 2
8
s5t4
3
3
s18t 6
s
Maintain Your Skills
Mixed Review
Find the next three terms in each geometric sequence.
62. 2, 6, 18, 54 162, 486, 1458
64. 384, 192, 96, 48 24, 12, 6
3
3
3 3 3
3
66. 3, , , , , 4 16 64 256 1024 4096
Getting Ready for
Lesson 11-2
(Lesson 10-7)
63. 1, 2, 4, 8 16, 32, 64
1 2
65. , , 4, 24 144, 864, 5184
9 3
67. 50, 10, 2, 0.4 0.08, 0.016, 0.0032
68. BIOLOGY A certain type of bacteria, if left alone, doubles its number every
2 hours. If there are 1000 bacteria at a certain point in time, how many bacteria
will there be 24 hours later? (Lesson 10-6) 4,096,000
PREREQUISITE SKILL Students
will learn how to perform operations with radical expressions in
Lesson 11-2. In order to perform
operations with radical expressions, students must be able to
use the Distributive Property to
multiply binomials. Use
Exercises 84–89 to determine
your students’ familiarity with
multiplying binomials.
69. PHYSICS According to Newton’s Law of Cooling, the difference between the
temperature of an object and its surroundings decreases in time exponentially.
Suppose a cup of coffee is 95°C and it is in a room that is 20°C. The cooling
of the coffee can be modeled by the equation y 75(0.875)t, where y is the
temperature difference and t is the time in minutes. Find the temperature
of the coffee after 15 minutes. (Lesson 10-6) 84.9°C
Factor each trinomial, if possible. If the trinomial cannot be factored using
integers, write prime. (Lesson 9-4)
70. 6x2 7x 5 (3x 5)(2x 1)
71. 35x2 43x 12 (5x 4)(7x 3)
72. 5x2 3x 31 prime
73. 3x2 6x 105 3(x 7)(x 5)
74.
4x2
12x 15 prime
75. 8x2 10x 3 (4x 3)(2x 1)
Find the solution set for each equation, given the replacement set. (Lesson 4-4)
76. y 3x 2; {(1, 5), (2, 6), (2, 2), (4, 10)} {(1, 5), (4, 10)}
77. 5x 2y 10; {(3, 5), (2, 0), (4, 2), (1, 2.5)} {(2, 0), (1, 2.5)}
78. 3a 2b 11; {(3, 10), (4, 1), (2, 2.5), (3, 2)} {(3, 10), (2, 2.5)}
3
1
79. 5 x 2y; (0, 1), (8, 2), 4, , (2, 1) 4, 1 , (2, 1)
2
2
2
Solve each equation. Then check your solution. (Lesson 3-3)
80. 40 5d 8
81. 20.4 3.4y 6
h
82. 25 275
11
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Find each product.
(To review multiplying binomials, see Lesson 8-7.)
84. (x 3)(x 2) x2 x 6
85.
11t 6
88. (5x 3y)(3x y) 15x2 4xy 3y2
86. (2t 1)(t 6)
592
592
Chapter 11 Radical Expressions and Triangles
r
29
83. 65 1885
Chapter 11 Radical Expressions and Triangles
2t 2
(a 2)(a 5) a2 7a 10
87. (4x 3)(x 1) 4x2 x 3
89. (3a 2b)(4a 7b) 12a2 13ab 14b2
Operations with
Radical Expressions
Lesson
Notes
• Add and subtract radical expressions.
1 Focus
• Multiply radical expressions.
can you use radical expressions to
determine how far a person can see?
The formula d 3h
represents
2
World’s Tall Structures
the distance d in miles that a
person h feet high can see. To
determine how much farther a
person can see from atop the
Sears Tower than from atop the
Empire State Building, we can
substitute the heights of both
buildings into the equation.
5-Minute Check
Transparency 11-2 Use as
a quiz or review of Lesson 11-1.
Mathematical Background notes
are available for this lesson on
p. 584C.
984 feet 1250 feet
Eiffel Empire State
Tower,
Building,
Paris
New York
1380 feet
Jin Mau
Building,
Shanghai
1450 feet 1483 feet
Sears
Petronas
Tower,
Towers,
Chicago Kuala Lumpur
ADD AND SUBTRACT RADICAL EXPRESSIONS Radical expressions
in which the radicands are alike can be added or subtracted in the same way that
monomials are added or subtracted.
Monomials
Radical Expressions
2x 7x (2 7)x
211
711
(2 7)11
911
9x
152 32 (15 3)2
122
15y 3y (15 3)y
12y
Notice that the Distributive Property was used to simplify each radical expression.
Example 1 Expressions with Like Radicands
Simplify each expression.
a. 43 63 53
43 63 53 (4 6 5)3
53
Distributive Property
Simplify.
b. 125 37 67 85
Commutative
125 37 67 85 125 85 37 67 Property
(12 8)5 (3 6)7
Distributive Property
45 97
can you use radical expressions to determine
how far a person can see?
Ask students:
• What is an assumption that the
formula makes? Sample answer:
That there are no obstructions that
would block a person from seeing
the entire distance.
• Assuming that nothing blocks
the view, how much farther
can a person atop the Sears
Tower see than a person atop
the Empire State Building?
about 3.3 miles farther
• Mountain Climbing Mount
Everest, the tallest mountain in
the world, is about 12,000 feet
above the plateau out of which
it rises. How far would a
climber be able to see from the
peak of Mount Everest? about
134 miles
Simplify.
In Example 1b, 45 97 cannot be simplified further because the radicands
are different. There are no common factors, and each radicand is in simplest form.
If the radicals in a radical expression are not in simplest form, simplify them first.
Lesson 11-2 Operations with Radical Expressions 593
Resource Manager
Workbook and Reproducible Masters
Chapter 11 Resource Masters
• Study Guide and Intervention, pp. 649–650
• Skills Practice, p. 651
• Practice, p. 652
• Reading to Learn Mathematics, p. 653
• Enrichment, p. 654
• Assessment, p. 699
Parent and Student Study Guide
Workbook, p. 84
Prerequisite Skills Workbook,
pp. 37–38
Transparencies
5-Minute Check Transparency 11-2
Answer Key Transparencies
Technology
Interactive Chalkboard
Lesson x-x Lesson Title 593
Example 2 Expressions with Unlike Radicands
2 Teach
Simplify 220
345
180
.
ADD AND SUBTRACT
RADICAL EXPRESSIONS
In-Class Examples
220
22 5 3
32 5 62 5
345
180
2
2
22 5 3
32 5
62 5
Power
Point®
225 335 65
1 Simplify each expression.
45 95 65
a. 65 25 55 35
195
b. 72 811
411
62
2 411
The simplified form is 195.
2 627
812
275
You can use a calculator to verify that a simplified radical expression is equivalent
to the original expression. Consider Example 2. First, find a decimal approximation
for the original expression.
443
KEYSTROKES:
MULTIPLY RADICAL
EXPRESSIONS
In-Class Example
2 2nd [2 ] 20 )
180
)
ENTER
3 2nd [2 ] 45 )
[2 ]
2nd
42.48529157
Next, find a decimal approximation for the simplified expression.
Power
Point®
KEYSTROKES:
3 Find the area of a rectangle
19 2nd [2 ] 5 ENTER
42.48529157
Since the approximations are equal, the expressions are equivalent.
with a width of 46 210
and a length of 53 75.
1830
102
MULTIPLY RADICAL EXPRESSIONS Multiplying two radical expressions
with different radicands is similar to multiplying binomials.
Example 3 Multiply Radical Expressions
Find the area of the rectangle in simplest form.
To find the area of the rectangle multiply the
measures of the length and width.
45 23
36 10
45 23 36 10
First
terms
Inner
terms
Last
terms
Study Tip
Outer
terms
45 36 45 10
23 36 23 10
Look Back
To review the FOIL
method, see Lesson 8-7.
1230
450
618
230
Multiply.
226
222
1230
45
3
30
Prime factorization
1230
202 182 230
Simplify.
1430
382
Combine like terms.
The area of the rectangle is 1430
382 square units.
594 Chapter 11 Radical Expressions and Triangles
Differentiated Instruction
Visual/Spatial Have students write all the perfect squares from 1 to
100 on an index card using a colored pen or pencil. Then have them
rework Example 3. After they multiply the binomials, have students use
their colored pen or pencil to circle the terms inside radicals that have
perfect square factors. Using the same color, have students write the
factorization in the line below each radical expression that can be
factored.
594 Chapter 11 Radical Expressions and Triangles
Concept Check
1. Explain why you should simplify each radical in a radical expression before
adding or subtracting. to determine if there are any like radicands
2. Explain how you use the Distributive Property to simplify like radicands that
are added or subtracted. See margin.
3. OPEN ENDED Choose values for x and y. Then find x y .
2
GUIDED PRACTICE KEY
Exercises
Examples
4, 5
6–9
10–13
1
2
3
Sample answer: 2 3 2 26
3 or 5 26
Simplify each expression.
4. 43 73 113
5. 26 76 56
6. 55 320
5
7. 23 12
43
8. 35 56 320
95 56
9. 83 3 9 93
3
Find each product.
10. 28 43 4 46
Applications
11. 4 5 3 5 17 75
12. GEOMETRY Find the perimeter and the area of a square whose sides measure
4 36 feet. P 16 126
ft; A 70 246 ft2
13. ELECTRICITY The voltage V required for a circuit is given by V PR
, where
P is the power in watts and R is the resistance in ohms. How many more volts
are needed to light a 100-watt bulb than a 75-watt bulb if the resistance for both
is 110 ohms? 10110
5330
14.05 volts
★ indicates increased difficulty
Practice and Apply
Homework Help
For
Exercises
See
Examples
14–21
22–29
30–48
1
2
3
Simplify each expression.
14. 85 35 115
16. 215
615
315
715
15. 36 106 136
17. 519
619
1119
0
18. 16x 2x 18x
19. 35b
45b
115b
105b
Extra Practice
20. 83 22 32 53
21. 46 17
62 417
20. 133 2
22. 18
12
8 52 23
23. 6 23 12
24. 37 228
7
See page 844.
25.
6 43
250
332
22
32
410
21. 46
62 ★ 26. 2 1 ★ 27. 10
25 2
2
5
5
17
537
★ 28. 33 45
3 13 43 35 ★ 29. 6 74 328
10 17 About the Exercises …
Organization by Objective
• Add and Subtract Radical
Expressions: 14–29
• Multiply Radical
Expressions: 30–37
Odd/Even Assignments
Exercises 14–39 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Alert! Exercise 42 requires the
use of the Internet or other
research materials.
Assignment Guide
7
Find each product.
34. 103 16
Study Notebook
Have students—
• include examples of how to add,
subtract, and multiply radical
expressions.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
2
Guided Practice
3 Practice/Apply
30. 63 52 32
103
31. 5210
32 102 310
32. 3 5 3 5 4
33. 7 10
2 59 1410
★ 34. 6 8 24
2 ★ 35. 5 2 14
35
37
★ 36. 210
315
33 22 195
★ 37. 52 35 210
3
152
115
Basic: 15–25 odd, 31, 33, 39,
41, 42, 49–76
Average: 15–39 odd, 41–44,
49–76
Advanced: 14–40 even, 45–70
(optional: 71–76)
38. GEOMETRY Find the perimeter of a rectangle whose length is
87 45 inches and whose width is 27 35 inches. 20
7 25 in.
www.algebra1.com/extra_examples
Lesson 11-2 Operations with Radical Expressions 595
Answer
2. The Distributive Property allows
you to add like terms. Radicals
with like radicands can be added
or subtracted.
Lesson 11-2 Operations with Radical Expressions 595
NAME ______________________________________________ DATE
39. GEOMETRY The perimeter of a rectangle is 23 411
6 centimeters,
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
11-2
Study
Guide and
p. 649
(shown)
and
p. 650
Operations
with Radical
Expressions
and its length is 211
1 centimeters. Find the width.
Add and Subtract Radical Expressions
When adding or subtracting radical
expressions, use the Associative and Distributive Properties to simplify the expressions. If
radical expressions are not in simplest form, simplify them.
Example 1
Simplify 106
53
63
46
.
53
63
46
(10 4)6
(5 6)3
106
6
6 3
Example 2
Simplify.
1
2
Simplify.
the diagonals of the rhombus. What is the
area of the rhombus at the right? 156
cm2
Simplify.
Lesson 11-2
Distributive Property
Exercises
1. 25
45
65
2. 6
46
36
2
3. 8
2
5
6. 23
6
53
33
7. 12
23
53
3
6
41. How much farther can a person see from atop the Sears Tower than from atop
the Empire State Building? 587
253 3.34 mi
8. 3
6 3
2 50 24 5
6 2
2
9. 8a
2a
52a
62a
10. 54
24
56
43
3
1
3
12. 12
176
6
1
6
42. A person atop the Empire State Building can see approximately 4.57 miles
farther than a person atop the Texas Commerce Tower in Houston. Explain
how you could find the height of the Texas Commerce Tower. See margin.
73
3
1
3
14. 80
20
180
85
83
125
3
1
3
15. 50 18 75 27 8
2 2
3 16. 23 445
2
17. 125
2
1
5
1 235
3
5
3
3
18.
1 6
12
3
2
33
4
3
283
12
1. 830
430
430
2. 25
75
55
45
3. 713x
1413x
213x
513x
4. 245
420
145
5. 40
10
90
410
6. 232
350
318
142
7. 27
18
300
32
133
8. 58
320
32
62
65
Source: www.the-skydeck.com
NAME ______________________________________________ DATE
____________ PERIOD _____
p. 651 and
Practice,
(shown)
Operationsp.
with652
Radical
Expressions
Simplify each expression.
9. 14
614
7
2
7
10. 50
32
11. 519
428
819
63
Online Research Data Update What are the tallest buildings and towers
in the world today? Visit www.algebra1.com/data_update to learn more.
Distance
The Sears Tower was
the tallest building in the
world from 1974 to 1996.
You can see four states
from its roof: Michigan,
Indiana, Illinois, and
Wisconsin.
Skills
Practice,
11-2
Practice
(Average)
ENGINEERING For Exercises 43 and 44, use the following information.
rate F of water passing through it in gallons per minute.
43. Find the radius of a pipe that can carry 500 gallons of water per minute.
Round to the nearest whole number. 6 in.
172
2
1
2
10
33
44. An engineer determines that a drainpipe must be able to carry 1000 gallons
of water per minute and instructs the builder to use an 8-inch radius pipe.
Can the builder use two 4-inch radius pipes instead? Justify your answer.
Find each product.
(10
15
) 215
310
13. 6
14. 5
(52
48
) 310
15. 27
(312
58
) 1221
2014
16. (5 15
) 40 1015
2
17. (10
6
)(30
18
) 43
18. (8
12
)(48
18
) 36 146
19. (2
28
)(36
5
)
20. (43
25
)(310
56
)
303
510
F
relates the radius r of a drainpipe in inches to the flow
5
The equation r 12. 310
75
240
412
319
117
54 cm
DISTANCE For Exercises 41 and 42, refer to the application at the beginning
of the lesson.
4. 375
25
153
25
5. 20
25
35
36 cm
A d1d2, where d1 and d2 are the lengths of
Simplify.
Simplify each expression.
13. 54
40. GEOMETRY A formula for the area A of a
rhombus can be found using the formula
Associative and Distributive Properties
Simplify 312
575
.
575
3
22 3 5
52 3
312
3 23
5 53
6
3 253
31
3
11. 3
3 2 cm
No, each pipe would need to carry 500 gallons per minute, so the pipes would
need at least a 6-inch radius.
230
302
MOTION For Exercises 45–47, use the following information.
The velocity of an object dropped from a certain height can be found using the
SOUND For Exercises 21 and 22, use the following information.
t 27
3,
The speed of sound V in meters per second near Earth’s surface is given by V 20
where t is the surface temperature in degrees Celsius.
formula v 2gd
, where v is the velocity in feet per second, g is the acceleration
due to gravity, and d is the distance in feet the object drops.
21. What is the speed of sound near Earth’s surface at 15°C and at 2°C in simplest form?
2402
m/s, 10011
m/s
22. How much faster is the speed of sound at 15°C than at 2°C?
45. Find the speed of an object that has fallen 25 feet and the speed of an object
that has fallen 100 feet. Use 32 feet per second squared for g. 40 ft/s; 80 ft/s
2402
10011
7.75 m/s
GEOMETRY For Exercises 23 and 24, use the following information.
A rectangle is 57
23
centimeters long and 67
33
centimeters wide.
23
cm
23. Find the perimeter of the rectangle in simplest form. 227
46. When you increased the distance by 4 times, what happened to the velocity?
24. Find the area of the rectangle in simplest form. 192 321
cm2
NAME ______________________________________________ DATE
The velocity doubled.
47. MAKE A CONJECTURE Estimate the velocity of an object that has fallen
225 feet. Then use the formula to verify your answer. The velocity should be
____________ PERIOD _____
Reading
11-2
Readingto
to Learn
Learn MathematicsELL
Mathematics,
p. 653
Operations with Radical Expressions
Pre-Activity
9 or 3 times the velocity of an object falling 25 feet; 3 40 120 ft/s,
25)
120 ft/s.
2(32)(2
How can you use radical expressions to determine how far a person
can see?
Read the introduction to Lesson 11-2 at the top of page 593 in your textbook.
48. WATER SUPPLY The relationship between a city’s size and its capacity to supply
Suppose you substitute the heights of the Sears Tower and the Empire State
Building into the formula to find how far you can see from atop each building.
What operation should you then use to determine how much farther you can
see from the Sears Tower than from the Empire State Building?
water to its citizens can be described by the expression 1020P
1 0.01P
,
where P is the population in thousands and the result is the number of gallons
per minute required. If a city has a population of 55,000 people, how many
gallons per minute must the city’s pumping station be able to supply?
subtraction
Reading the Lesson
about 7003.5 gal/min
1. Indicate whether the following expressions are in simplest form. Explain your answer.
596 Chapter 11 Radical Expressions and Triangles
a. 63
12
No; 12 can be simplified to 22 3
or 23
.
b. 126
710
Yes; both the addends are radical expressions in simplest form, the
radicands are different, and there are no common factors.
2. Below the words First terms, Outer terms, Inner terms, and Last terms, write the
products you would use to simplify the expression (215
315
)(63
52
).
First terms
Outer terms
Inner terms
(215
)(63
)
(215
)(52
) (35 )(63)
Last terms
(35 )(52 )
Helping You Remember
3. How can you use what you know about adding and subtracting monomials to help you
remember how to add and subtract radical expressions?
Sample answer: Check that the addends have been simplified. Next,
group addends that involve like radicals. Then use the Distributive
Property to combine the addends that involve like radicals.
596
NAME ______________________________________________ DATE
11-2
Enrichment
Enrichment,
____________ PERIOD _____
p. 654
42. Approximately 1000 feet;
determine h so that
The Wheel of Theodorus
The Greek mathematicians were intrigued by problems of
representing different numbers and expressions using
geometric constructions.
1
D
C
1
Theodorus, a Greek philosopher who lived about 425 B.C., is
said to have discovered a way to construct the sequence
1, 2, 3, 4, … .
B
1
The beginning of his construction is shown. You start with an
isosceles right triangle with sides 1 unit long.
O
1
Use the figure above. Write each length as a radical expression in simplest form.
1. line segment AO
1
2. line segment BO
2
3. line segment CO
3
4. line segment DO
4
Chapter 11 Radical Expressions and Triangles
Answer
A
3(1250)
3h
4.57;
2
2
may use guess and test,
graphical, or analytical methods.
49. CRITICAL THINKING Find a counterexample to disprove the following
statement. Sample answer: a 4, b 9: 4 9 4
9
4 Assess
For any numbers a and b, where a 0 and b 0, a b a b.
a b a b 50. CRITICAL THINKING Under what conditions is true? a 0 or b 0 or both
2
51. WRITING IN MATH
2
2
Answer the question that was posed at the beginning of
the lesson. See margin.
How can you use radical expressions to determine how far a person can see?
Include the following in your answer:
• an explanation of how this information could help determine how far apart
lifeguard towers should be on a beach, and
• an example of a real-life situation where a lookout position is placed at a high
point above the ground.
Standardized
Test Practice
C
7
57
B
47
D
77
PREREQUISITE SKILL Students
will learn how to solve radical
equations in Lesson 11-3. In
order to solve radical equations,
students will need to be able to
recognize and find special
products. Use Exercises 71–76 to
determine your students’
familiarity with finding special
products.
53. Simplify 34 12
. D
2
A
43 6
B
283
C
28 163
D
48 283
Maintain Your Skills
Mixed Review
56. 14xy y
Simplify.
(Lesson 11-1)
54. 40
210
55. 128
82
5
50
57. 8 2
58.
Find the nth term of each geometric sequence.
60. a1 4, n 6, r 4
2 y3
56. 196x
225c4d 5c2d
18c2
2
59.
5103
4096
Assessment Options
314
63a
128a3b2 16ab
Quiz (Lessons 11-1 and 11-2) is
available on p. 699 of the Chapter
11 Resource Masters.
(Lesson 10-7)
61. a1 7, n 4, r 9
62. a1 2, n 8, r 0.8
0.4194304
Solve each equation by factoring. Check your solutions. (Lesson 9-5)
6
9
36
63. 81 49y2 64. q2 0 121
11
7
5
5
65. 48n3 75n 0 , 0, 66. 5x3 80x 240 15x2 {4, 3, 4}
4
4
Solve each inequality. Then check your solution. (Lesson 6-2)
5
3
w
7k
21
67. 8n 5 n 68. 14 w 126
69. k 9
2
10
8
5
70. PROBABILITY A student rolls a die three times. What is the probability that
each roll is a 1? (Lesson 2-6) 1
216
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Find each product. (To review special products, see Lesson 8-8.)
71. (x 2)2 x2 4x 4
72. (x 5)2 x2 10x 25 73. (x 6)2 x2 12x 36
74. (3x 1)2 9x2 6x 1 75. (2x 3)2
www.algebra1.com/self_check_quiz
4x2 12x 9
Speaking Ask students to compare and contrast adding, subtracting, and multiplying radical
expressions and variable expressions. What are the similarities?
What are the differences? Allow
students to give examples on the
chalkboard if they wish.
Getting Ready for
Lesson 11-3
52. Find the difference of 97 and 228
. C
A
Open-Ended Assessment
76. (4x 7)2
16x2 56x 49
Lesson 11-2 Operations with Radical Expressions 597
Answer
51. The distance a person can see is
related to the height of the person
using d 3h2 . Answers should
include the following.
• You can find how far each
lifeguard can see from the
height of the lifeguard tower.
Each tower should have some
overlap to cover the entire
beach area.
• On early ships, a lookout
position (Crow’s nest) was
situated high on the foremast.
Sailors could see farther from
this position than from the
ship’s deck.
Lesson 11-2 Operations with Radical Expressions 597
Lesson
Notes
Radical Equations
• Solve radical equations.
1 Focus
• Solve radical equations with extraneous solutions.
5-Minute Check
Transparency 11-3 Use as
a quiz or review of Lesson 11-2.
Vocabulary
• radical equation
• extraneous solution
Mathematical Background notes
are available for this lesson on
p. 584D.
are radical equations
used to find freefall times?
Ask students:
h
4
are radical equations used
to find free-fall times?
Skydivers fall 1050 to 1480 feet every 5 seconds,
reaching speeds of 120 to 150 miles per hour at
terminal velocity. It is the highest speed they can
reach and occurs when the air resistance equals
the force of gravity. With no air resistance, the
time t in seconds that it takes an object to fall
h
h feet can be determined by the equation t .
4
How would you find the value of h if you are
given the value of t?
h
that contain radicals with
RADICAL EQUATIONS Equations like t 4
• Is in simplest form?
variables in the radicand are called radical equations. To solve these equations,
first isolate the radical on one side of the equation. Then square each side of the
equation to eliminate the radical.
Explain. Yes, since there is no
radical in the denominator, the
expression is in simplest form.
x
4
• How would you solve t Example 1 Radical Equation with a Variable
for x? Multiply each side by 4.
• What do you think might be a
way to remove the radical sign
from 4t h? Square each side
of the equation.
FREE-FALL HEIGHT Two objects are dropped simultaneously. The first object
reaches the ground in 2.5 seconds, and the second object reaches the ground
1.5 seconds later. From what heights were the two objects dropped?
Find the height of the first object. Replace t with 2.5 seconds.
h
t Original equation
4
h
2.5 Replace t with 2.5.
4
10 h
Multiply each side by 4.
102 h 100 h
2
Square each side.
Simplify.
h
CHECK t 4
100
t 4
10
t 4
t 2.5
Original equation
h 100
100 10
Simplify.
The first object was dropped from 100 feet.
598
Chapter 11 Radical Expressions and Triangles
Resource Manager
Workbook and Reproducible Masters
Chapter 11 Resource Masters
• Study Guide and Intervention, pp. 655–656
• Skills Practice, p. 657
• Practice, p. 658
• Reading to Learn Mathematics, p. 659
• Enrichment, p. 660
Graphing Calculator and
Spreadsheet Masters, p. 43
Parent and Student Study Guide
Workbook, p. 85
Transparencies
5-Minute Check Transparency 11-3
Answer Key Transparencies
Technology
Interactive Chalkboard
The time it took the second object to fall was 2.5 1.5 seconds or 4 seconds.
h
t Original equation
h
4 Replace t with 4.
4
4
16 h
2 Teach
RADICAL EQUATIONS
In-Class Examples
Multiply each side by 4.
162 h 256 h
2
Power
Point®
Square each side.
1 FREE-FALL HEIGHT An object
Simplify.
is dropped from an unknown
height and reaches the ground
in 5 seconds. From what
height is it dropped? The
object is dropped from 400 ft.
The second object was dropped from 256 feet. Check this solution.
Example 2 Radical Equation with an Expression
Solve x 1 7 10.
1 7 10
x
13
x
2
x 1 32
x19
x8
Teaching Tip
Remind students
that they must isolate the
radical part of the expression
before squaring each side.
Original equation
Subtract 7 from each side.
Square each side.
2 Solve x 3 8 15. 52
2
x 1 x 1
Subtract 1 from each side.
The solution is 8. Check this result.
EXTRANEOUS SOLUTIONS
EXTRANEOUS SOLUTIONS Squaring each side of an equation sometimes
produces extraneous solutions. An extraneous solution is a solution derived from
an equation that is not a solution of the original equation. Therefore, you must check
all solutions in the original equation when you solve radical equations.
Example 3 Variable on Each Side
Solve x 2 x 4.
x2x4
2
x
2 (x 4)2
Study Tip
Look Back
To review Zero Product
Property, see Lesson 9-2.
Original equation
Square each side.
x2
x2
8x 16
Simplify.
0
x2
9x 14
Subtract x and 2 from each side.
0 (x 7)(x 2) Factor.
x70
or x 2 0
x7
x2
Zero Product Property
33 ⻫
In Chapter 10, students learned
that there are two solutions to
most quadratic equations. In this
lesson, students will learn that
when solving radical equations,
it is possible to introduce a
second solution when squaring
both sides of an equation.
However, since the original
equation was not quadratic, one
of the solutions will be
extraneous.
Solve.
CHECK x2x4
274
7
9 3
Building on Prior
Knowledge
x7
x2x4
2224
4 2
2 2 ⫻
In-Class Example
Power
Point®
3 Solve 2 y y. 1
x2
Since 2 does not satisfy the original equation, 7 is the only solution.
www.algebra1.com/extra_examples
Lesson 11-3 Radical Equations
Differentiated Instruction
599
ELL
Verbal/Linguistic Have students write a short paragraph in their own
words explaining why checking solutions is important when solving
radical equations. Students should include an example of a radical
equation that has an extraneous solution.
Lesson 11-3 Radical Equations 599
3 Practice/Apply
Study Notebook
Have students—
• include examples of how to solve
radical equations.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
Solving Radical Equations
2. (10, 5)
3. x 10; the
solution is the same
as the solution from
the graph. However,
when solving
algebraically, you
have to check that
x 3 is an extraneous
solution.
Concept Check
FIND THE ERROR
This problem
may require close
inspection for students to find the
error. Tell students to focus on
checking the given solutions. One
solution produces an impossible
radicand. Also point out that Alex
and Victor could have begun by
multiplying each side by 1 to
eliminate the negative signs.
1. Isolate the radical
on one side of the
equation. Square each
side of the equation
and simplify. Then
check for extraneous
solutions.
4. Alex; the square of
x 5 is x 5.
You can use a TI-83 Plus graphing calculator to solve radical equations
3x 5 x 5. Clear the Y list. Enter the left side of the
such as 3x 5. Enter the right side of the equation as
equation as Y1 Y2 x 5. Press GRAPH .
Think and Discuss
1. Sketch what is shown on the screen. See margin.
2. Use the intersect feature on the CALC menu, to find the point of intersection.
3. Solve the radical equation algebraically. How does your solution compare to
the solution from the graph?
1. Describe the steps needed to solve a radical equation.
2. Explain why it is necessary to check for extraneous solutions in radical
equations. The solution may not satisfy the original equation.
3. OPEN ENDED Give an example of a radical equation. Then solve the equation
for the variable. Sample answer: x 1 8; 63
4. FIND THE ERROR Alex and Victor are solving x 5 2.
Alex
Victor
x – 5 = –2
–∫
–∂
x – 5 = –2
x – 5)2 = (–2)2
(– ∫
(–∂
x – 5) 2 = (–2) 2
x–5=4
–(x – 5) = 4
x=9
–x + 5 = 4
x = 1
Answer
Who is correct? Explain your reasoning.
1.
Guided Practice
Solve each equation. Check your solution.
5. x 5 25
6. 2b
8 no solution 7. 7x
7 7
8. 3a
6 12
9. 8s
15 2
11. 5x 1 2 6 3
Application
GUIDED PRACTICE KEY
Exercises
Examples
5–8,
14–16
9–11
12, 13
1
600
2
3
12. 3x 5 x 5 10
10. 7x 18
9 9
13. 4 x2x 6
OCEANS For Exercises 14–16, use the following information.
Tsunamis, or large tidal waves, are generated by undersea earthquakes in the Pacific
d, where d is the
Ocean. The speed of the tsunami in meters per second is s 3.1
depth of the ocean in meters. 15. 5994 m
14. Find the speed of the tsunami if the depth of the water is 10 meters. 9.8 m/s
15. Find the depth of the water if a tsunami’s speed is 240 meters per second.
16. A tsunami may begin as a 2-foot high wave traveling 450–500 miles per hour. It
can approach a coastline as a 50-foot wave. How much speed does the wave
lose if it travels from a depth of 10,000 meters to a depth of 20 meters?
Chapter 11 Radical Expressions and Triangles
approximately 296 m/s
Solving Radical Equations An alternative to graphing each side of the
equation separately is to graph the single equation as it is. Move the
x 5 to the left side of the equation and then enter the equation as
Y1 3x
5 x 5. Then press 2nd [CALC] 2 to calculate the zero point
of the graph. The zero point is the solution, which for this equation is 10.
600
Chapter 11 Radical Expressions and Triangles
★ indicates increased difficulty
Practice and Apply
Homework Help
For
Exercises
See
Examples
17–34
35–47
48–59
1, 2
3
1–3
Extra Practice
See page 844.
About the Exercises …
Solve each equation. Check your solution.
17. a 10 100
18. k
4 16
19. 52 x 50
20. 37 y
63
21. 34a
2 10 4
22. 3 5n
18 9
23. x 3 5 no solution
24. x 5 26 29
25. 3x 12 33 5
26. 2c 4 8 34
27. 4b 1 3 0 2
28. 3r 5 7 3 no solution
3
4t
30. 5 2 0 3
25
32. y 2 y2 5
y4 0
29.
4x
9 3
5
180
31. x2 9
x 14 x 4 2
Organization by Objective
• Radical Equations: 17–34
• Extraneous Solutions: 35–47
Odd/Even Assignments
Exercises 17–46 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Assignment Guide
33. The square root of the sum of a number and 7 is 8. Find the number. 57
Basic: 17–47 odd, 48, 49,
60–63, 70–89
Average: 17–47 odd, 48–53,
60–63, 70–89 (optional: 64–69)
Advanced: 18–46 even, 50–85
(optional: 86–89)
All: Practice Quiz 1 (1–10)
34. The square root of the quotient of a number and 6 is 9. Find the number. 486
Solve each equation. Check your solution.
35. x 6x 2
36. x x 20 5
37. 5x 6 x 2, 3
38. 28 3x x 4
39. x1x1 3
40. 1 2b 1 b 0
41. 4 m2m 6
42. 3d 8 d 2 3, 4
43. x 6x4 2
44. 6 3x x 16 10
45. 121
r 11
46. 5p2 7 2p
2r2
7
47. State whether the following equation is sometimes, always, or never true.
5
)2 x 5 sometimes
(x
AVIATION For Exercises 48 and 49, use the following information.
The formula L kP
represents the relationship between a plane’s length L and
the pounds P its wings can lift, where k is a constant of proportionality calculated
for a plane.
Aviation
Piloted by A. Scott
Crossfield on November
20, 1953, the Douglas
D-558-2 Skyrocket became
the first aircraft to fly
faster than Mach 2, twice
the speed of sound.
Source: National Air and Space
Museum
48. The length of the Douglas D-558-II, called the Skyrocket, was approximately
42 feet, and its constant of proportionality was k 0.1669. Calculate the
maximum takeoff weight of the Skyrocket. 10,569 lb
49. A Boeing 747 is 232 feet long and has a takeoff weight of 870,000 pounds.
Determine the value of k for this plane. 0.0619
GEOMETRY For Exercises 50–53, use the figure below. The area A of a circle is
equal to r2 where r is the radius of the circle.
A
50. Write an equation for r in terms of A. r 51. The area of the larger circle is 96 square meters.
Find the radius. 46
or 9.8 m
52. The area of the smaller circle is 48 square
meters. Find the radius. 43
or 6.9 m
★ 53. If the area of a circle is doubled, what is the
change in the radius? It increases by a factor of 2
.
Lesson 11-3 Radical Equations
601
Lesson 11-3 Radical Equations 601
NAME ______________________________________________ DATE
PHYSICAL SCIENCE For Exercises 54–56, use the following information.
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
11-3
Study
Guide and
32
ᐉ
The formula P 2 gives the period of a pendulum of length ᐉ feet. The period
p. 655
and p. 656
Radical(shown)
Equations
Radical Equations Equations containing radicals with variables in the radicand are
called radical equations. These can be solved by first using the following steps.
P is the number of seconds it takes for the pendulum to swing back and forth once.
Isolate the radical on one side of the equation.
Square each side of the equation to eliminate the radical.
Step 1
Step 2
Example 1
x
16 2
x
2(16) 2 2
32 x
(32)2 (x
)2
1024 x
x
2
Example 2
Solve 16 for x.
Original equation
Multiply each side by 2.
Simplify.
Square each side.
Simplify.
The solution is 1024, which checks in the
original equation.
54. Suppose we want a pendulum to complete three periods in 2 seconds. How long
should the pendulum be? 0.36 ft
Solve 4x 7
2 7.
4x 7 2 7
4x 7 2 2 7 2
4x 7 5
2
(
4x 7 ) 52
4x 7 25
4x 7 7 25 7
4x 32
x8
Original equation
Subtract 2 from each side.
Simplify.
Square each side.
Simplify.
Add 7 to each side.
Simplify.
Divide each side by 4.
The solution is 8, which checks in the original
equation.
Exercises
1. a
8 64
2. a
6 32 676
3. 2x
8 16
4. 7 26 n
23
5. a
6 36
6. 3r2 3 3
7. 23
y 12
8. 23a
2 7 6
3
4
10. 2c 3 5 11
9. x 4 6 40
11. 3b 2
19 24 9
10
3
3r 2 23
13. 1
2
4
3
16. 6x2 5x 2 , 14.
17.
6 8 12
x
2
1 1
2 2
5
2
12. 4x 1 3 15.
4 128
x
8
3 11 2623
x
3
18. 2
NAME ______________________________________________ DATE
Skills
Practice,
11-3
Practice
(Average)
3x
5
____________ PERIOD _____
p. 657 and
Practice,
p. 658 (shown)
Radical Equations
Lesson 11-3
Solve each equation. Check your solution.
Physical Science
The Foucault pendulum
was invented in 1851. It
demonstrates the rotation
of Earth. The pendulum
appears to change its path
during the day, but it
moves in a straight line.
The path under the
pendulum changes
because Earth is rotating
beneath it.
Source: California Academy
of Sciences
55. Two clocks have pendulums of different lengths. The first clock requires
1 second for its pendulum to complete one period. The second clock requires
2 seconds for its pendulum to complete one period. How much longer is one
pendulum than the other? 2.43 ft
24t 2
★ 56. Repeat Exercise 55 if the pendulum periods are t and 2t seconds. 2
SOUND For Exercises 57–59, use the following information.
The speed of sound V near Earth’s surface can be found using the equation
t 273, where t is the surface temperature in degrees Celsius.
V 20
57. Find the temperature if the speed of sound V is 356 meters per second. 43.8°C
58. The speed of sound at Earth’s surface is often given at 340 meters per second,
but that is only accurate at a certain temperature. On what temperature is this
figure based? 16°C
★ 59. What is the speed of sound when the surface temperature is below 0°C?
V 330.45 m/s
60. CRITICAL THINKING Solve h 9 h 3
. 3
Solve each equation. Check your solution.
1. b
8 64
2. 43
x 48
3. 24c
3 11 4
4. 6 2y
2 32
5. k 2 3 7 98
6. m 5 43
53
7. 6t 12
86
62
8. 3j 11
2 9 20
9. 2x 15
5 18 77
3 0 14
11. 6
61. WRITING IN MATH
How are radical equations used to find free-fall times?
4 2 60
12. 6 2 no solution
10.
3s
5
Include the following in your answer:
• the time it would take a skydiver to fall 10,000 feet if he falls 1200 feet every
h
5 seconds and the time using the equation t , with an explanation of
4
why the two methods find different times, and
5r
6
3x
3
13. y y6 3
14. 15 2x
x 3
15. w 4 w 4 4, 3
16. 17 k
k5 8
17. 5m 16 m 2 4, 5
18. 24 8q
q 3 3, 5
19. 4s 17
s30 2
20. 4 3m 28 m 1
21. 10p 61 7 p 6, 2
22. 2x2 9x 3
• ways that a skydiver can increase or decrease his speed.
ELECTRICITY For Exercises 23 and 24, use the following information.
The voltage V in a circuit is given by V PR
, where P is the power in watts and R is the
resistance in ohms.
Standardized
Test Practice
23. If the voltage in a circuit is 120 volts and the circuit produces 1500 watts of power, what
is the resistance in the circuit? 9.6 ohms
24. Suppose an electrician designs a circuit with 110 volts and a resistance of 10 ohms. How
much power will the circuit produce? 1210 watts
QUANTITATIVE COMPARISON In Exercises 62 and 63, compare the quantity in
Column A and the quantity in Column B. Then determine whether:
FREE FALL For Exercises 25 and 26, use the following information.
Assuming no air resistance, the time t in seconds that it takes an object to fall h feet can be
h
determined by the equation t .
4
25. If a skydiver jumps from an airplane and free falls for 10 seconds before opening the
parachute, how many feet does the skydiver fall? 1600 ft
26. Suppose a second skydiver jumps and free falls for 6 seconds. How many feet does the
second skydiver fall? 576 ft
NAME ______________________________________________ DATE
Answer the question that was posed at the beginning of
the lesson. See margin.
A
the quantity in Column A is greater,
B
the quantity in Column B is greater,
C
both quantities are equal, or
D
the relationship cannot be determined from the given information.
____________ PERIOD _____
Reading
11-3
Readingto
to Learn
Learn Mathematics
Mathematics,
p. 659
Radical Equations
Pre-Activity
ELL
Column A
Column B
the solution of
the solution of
x36
y 3 6
2
a 1
1
)2
(a
How are radical equations used to find free-fall times?
Read the introduction to Lesson 11-3 at the top of page 598 in your textbook.
on one side of the equation?
How can you isolate h
62.
Multiply each side of the equation by 4.
Reading the Lesson
A
1. To solve a radical equation, you first isolate the radical on one side of the equation. Why
do you then square each side of the equation?
to eliminate the radical
63.
2. a. Provide the reason for each step in the solution of the given radical equation.
5x 1 4 x 3
5x 1 x 1
2
(
5x 1)
(x 1)2
5x 1 x2 2x 1
0 x2 3x 2
0 (x 1)(x 2)
x 1 0 or
x1
x20
x2
Original equation
Add 4 to each side.
Square each side.
Simplify.
Subtract 5x and add 1 to each side.
Factor.
Zero Product Property
Solve.
b. To be sure that 1 and 2 are the correct solutions, into which equation should you
substitute to check? the original equation
3. a. How do you determine whether an equation has an extraneous solution?
Substitute the solution(s) into the original equation. If a solution does
not satisfy the original equation, then it is an extraneous solution.
b. Is it necessary to check all solutions to eliminate extraneous solutions? Explain.
Yes; since you square each side of a radical equation, and squaring
each side can sometimes produce an extraneous solution, you need
to check all solutions. The only way to be sure that a solution is not
extraneous is to check it in the original equation.
Helping You Remember
4. How can you use the letters ISC to remember the three steps in solving a radical
equation?
Sample answer: Isolate the radical on one side of the equation, Square
each side to eliminate the radical, and Check for extraneous solutions.
602
602
C
Chapter 11 Radical Expressions and Triangles
NAME ______________________________________________ DATE
11-3
Enrichment
Enrichment,
____________ PERIOD _____
p. 660
Special Polynomial Products
Sometimes the product of two polynomials can be found readily with
the use of one of the special products of binomials.
For example, you can find the square of a trinomial by recalling the
square of a binomial.
Example 1
Find (x y z)2.
(a b)2 a2 2 a b b2
[(x y) z]2 (x y)2 2(x y)z z2
x2 2xy y2 2xz 2yz z2
Example 2
Find (3t x 1)(3t x 1).
(Hint: (3t x 1)(3t x 1) is the product of a sum 3t (x 1)
and a difference 3t (x 1).)
(3t x 1)(3t x 1) [3t (x 1)][3t (x 1)]
9t2 (x 1)2
9t2 x2 2x 1
Chapter 11 Radical Expressions and Triangles
Graphing
Calculator
RADICAL EQUATIONS Use a graphing calculator to solve each radical equation.
Round to the nearest hundredth.
65. 3x 8 5 11
66. x 6 4 x 2
67. 4x 5 x 7 15.08
Open-Ended Assessment
68. x 7 x 4 1.70
69. 3x 9 2x 6 no solution
Modeling Write radical equations
on the overhead projector, using
a cut-out overlay for the radical.
Have students explain what
must be done to the equation in
order to “remove” the radical.
When students explain the
correct procedure, remove the
overlay and solve the equation.
Maintain Your Skills
Mixed Review
Simplify each expression. (Lesson 11-2)
70. 56 126 176
Simplify.
71. 12
627
203 72. 18
52 332
42
(Lesson 11-1)
73. 192
83
74. 6 10
215
21
75. 10 3
310
3 Getting Ready for
Lesson 11-4
Determine whether each trinomial is a perfect square trinomial. If so, factor it.
(Lesson 9-6)
76. d2 50d 225 no
77. 4n2 28n 49
78. 16b2 56bc 49c2
yes; (2n 7)2
yes; (4b 7c)2
Find each product. (Lesson 8-7)
79. (r 3)(r 4)
r 2 r 12
80. (3z 7)(2z 10)
6z2 44z 70
81. (2p 5)(3p2 4p 9)
6p3 7p 2 2p 45
82. PHYSICAL SCIENCE A European-made hot tub is advertised to have a
temperature of 35°C to 40°C, inclusive. What is the temperature range for the
9
5
hot tub in degrees Fahrenheit? Use F C 32. (Lesson 6-4) 95° F 104°
Write each equation in standard form. (Lesson 5-5)
3
7
83. y 2x 14x 7y 3
Getting Ready for
the Next Lesson
84. y 3 2(x 6)
2x y 15
15x 2y 49
86. a 3, b 4 5
2
87. a 24, b 7 25
89. a 8, b 12 413
Lessons 11-1 through 11-3
(Lesson 11-1)
1. 48
43
2. 3 6 32
2 10
3
3. 2
2 10
5. 23 912
203
2
6. 3 6 15 66
115 311
7. GEOMETRY Find the area of a square whose side measure is 2 7 centimeters.
11 47 or 21.6 cm2
(Lesson 11-2)
Solve each equation. Check your solution. (Lesson 11-3)
8. 15 x 4 1
9. 3x2 32
x 4
www.algebra1.com/self_check_quiz
10. 2x 1 2x 7 5
Lesson 11-3 Radical Equations
Practice Quiz 1 The quiz
provides students with a brief
review of the concepts and skills
in Lessons 11-1 through 11-3.
Lesson numbers are given to the
right of exercises or instruction
lines so students can review
concepts not yet mastered.
Answer
(Lesson 11-2)
4. 65
311
55
PREREQUISITE SKILL Students
will learn about the Pythagorean
Theorem in Lesson 11-4. In order
to use the Pythagorean Theorem,
students must be able to evaluate
radical expressions for given
values. Use Exercises 86–89 to
determine your students’
familiarity with evaluating
radical expressions.
Assessment Options
(To review evaluating expressions, see Lesson 1-2.)
P ractice Quiz 1
Simplify.
85. y 2 7.5(x 3)
PREREQUISITE SKILL Evaluate a2 b2 for each value of a and b.
88. a 1, b 1
Simplify.
4 Assess
64. 3 2x
7 8
603
61. You can determine the time it takes
an object to fall from a given height
using a radical equation. Answers
should include the following.
• It would take a skydiver
approximately 42 seconds to fall
10,000 feet. Using the equation,
it would take 25 seconds. The
time is different in the two
calculations because air
resistance slows the skydiver.
• A skydiver can increase the
speed of his fall by lowering air
resistance. This can be done by
pulling his arms and legs close
to his body. A skydiver can
decrease his speed by holding
his arms and legs out, which
increases the air resistance.
Lesson 11-3 Radical Equations 603
Graphing
Calculator
Investigation
A Follow-Up of Lesson 11-3
A Follow-Up of Lesson 11-3
Graphs of Radical Equations
Getting Started
In order for a square root to be a real number, the radicand cannot be negative.
When graphing a radical equation, determine when the radicand would be negative
and exclude those values from the domain.
Know Your Calculator The
ZoomFit option in the ZOOM menu
will let the calculator automatically zoom the viewing window
to fit the graph. Suggest that
students use this option with the
examples in this investigation to
get a better view of the shape of
the graph of a radical equation.
Example 1
Graph y x
. State the domain of the graph.
Enter the equation in the Y= list.
KEYSTROKES:
2nd
[2 ] X,T,␪,n
)
GRAPH
From the graph, you can see that the domain of x is {xx 0}.
Teach
[10, 10] scl: 1 by [10, 10] scl: 1
• Have students use the CALC
Value function to find the value
of the function at different
x-values. Press 2nd [CALC] 1
and then enter an x value.
Students should quickly see
that as the cursor moves along
the curve, the value of the
function is the square root of
the radicand of the function.
Example 2
Graph y x 4. State the domain of the graph.
Enter the equation in the Y= list.
KEYSTROKES:
2nd
[2 ] X,T,␪,n
4 )
GRAPH
The value of the radicand will be positive when x 4 0,
or when x 4. So the domain of x is {xx 4}.
This graph looks like the graph of y x shifted left 4 units.
[10, 10] scl: 1 by [10, 10] scl: 1
Assess
Exercises 1–9. See pp. 639A–639B.
Graph each equation and sketch the graph on your paper. State the domain of
the graph. Then describe how the graph differs from the parent function y x
.
Ask: The graph of y x looks
somewhat flat, as though it has a
horizontal asymptote. Does it?
Explain why or why not. As the
value of x increases, the value of y,
which is the square root of x, should
continue to increase as well. There is
no horizontal asymptote.
1. y x
1
4. y x5
7. y x
2. y x
3
5. y x
8. y 1x6
3. y x2
6. y 3x
9. y 2x 5 4
10. Is the graph of x y2 a function? Explain your reasoning. 10–11. See margin.
11. Does the equation x2 y2 1 determine y as a function of x? Explain.
12. Graph y x
1 x2 in the window defined by [2, 2] scl: 1 by [2, 2] scl: 1.
Describe the graph. heart
www.algebra1.com/other_calculator_keystrokes
604 Investigating Slope-Intercept Form
604 Chapter 11 Radical Expressions and Triangles
Answers
10. No; you must consider the graph of y x and the graph of y x. This graph fails the
vertical line test. For every value of x 0, there are two values for y.
11. No; the equation y 1
x 2 is not a function since there are both positive and
negative values for y for each value of x.
604
Chapter 11 Radical Expressions and Triangles
Lesson
Notes
The Pythagorean Theorem
place chapter
sphere art for
each lesson at
coordinates
x= -2p
y= –1p2
• Solve problems by using the Pythagorean Theorem.
is the Pythagorean Theorem
used in roller coaster design?
Vocabulary
•
•
•
•
hypotenuse
legs
Pythagorean triple
corollary
1 Focus
• Determine whether a triangle is a right triangle.
The roller coaster Superman: Ride of
Steel in Agawam, Massachusetts, is one
of the world’s tallest roller coasters at
208 feet. It also boasts one of the world’s
steepest drops, measured at 78 degrees,
and it reaches a maximum speed of
77 miles per hour. You can use the
Pythagorean Theorem to estimate
the length of the first hill.
5-Minute Check
Transparency 11-4 Use as
a quiz or review of Lesson 11-3.
208 feet
Mathematical Background notes
are available for this lesson on
p. 584D.
78˚
THE PYTHAGOREAN THEOREM In a right triangle, the
side opposite the right angle is called the hypotenuse . This side
is always the longest side of a right triangle. The other two sides
are called the legs of the triangle.
hypotenuse
leg
To find the length of any side of a right triangle when the
lengths of the other two are known, you can use a formula
developed by the Greek mathematician Pythagoras.
leg
The Pythagorean Theorem
Study Tip
• Words
Triangles
If a and b are the measures of the legs of a
right triangle and c is the measure of the
hypotenuse, then the square of the length of
the hypotenuse is equal to the sum of the
squares of the lengths of the legs.
Sides of a triangle are
represented by lowercase
letters a, b, and c.
c
a
is the Pythagorean
Theorem used in roller
coaster design?
Ask students:
• What shape is being used to
estimate the length of the first
hill? a right triangle
• What information in the
problem is most likely useful
in finding the length of the
hill? What information is
extraneous? The height of the hill
and the angle of the drop is probably
useful, and the maximum speed is
probably not useful.
b
• Symbols c2 a2 b2
Example 1 Find the Length of the Hypotenuse
Find the length of the hypotenuse of a right triangle if a 8 and b 15.
c2 a2 b2
c2
82
152
c2 289
Pythagorean Theorem
a 8 and b 15
Simplify.
c 289
Take the square root of each side.
c 17
Disregard 17. Why?
The length of the hypotenuse is 17 units.
Lesson 11-4 The Pythagorean Theorem
605
Resource Manager
Workbook and Reproducible Masters
Chapter 11 Resource Masters
• Study Guide and Intervention, pp. 661–662
• Skills Practice, p. 663
• Practice, p. 664
• Reading to Learn Mathematics, p. 665
• Enrichment, p. 666
• Assessment, pp. 699, 701
Graphing Calculator and
Spreadsheet Masters, p. 44
Parent and Student Study Guide
Workbook, p. 86
School-to-Career Masters, p. 22
Transparencies
5-Minute Check Transparency 11-4
Real-World Transparency 11
Answer Key Transparencies
Technology
AlgePASS: Tutorial Plus, Lesson 32
Interactive Chalkboard
Lesson x-x Lesson Title 605
Example 2 Find the Length of a Side
2 Teach
Find the length of the missing side.
In the triangle, c 25 and b 10 units.
THE PYTHAGOREAN
THEOREM
In-Class Examples
c2 a2 b2
252
Power
Point®
102
Pythagorean Theorem
b 10 and c 25
25
a
625 a2 100 Evaluate squares.
525 a2
1 Find the length of the hypote-
525
a
nuse of a right triangle if
a 18 and b 24. The length
of the hypotenuse is 30 units.
22.91 a
Subtract 100 from each side.
Use a calculator to evaluate
.
525
10
Use the positive value.
To the nearest hundredth, the length of the leg is 22.91 units.
2 Find the length of the
Whole numbers that satisfy the Pythagorean Theorem are called Pythagorean
triples. Multiples of Pythagorean triples also satisfy the Pythagorean Theorem.
Some common triples are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).
missing side.
b
a2
Standardized Example 3 Pythagorean Triples
Test Practice
Multiple-Choice Test Item
16
B
What is the area of triangle ABC?
9
A
The length of the leg is
approximately 13.23 units.
C
96 units2
160 units2
B
D
120 units2
196 units2
20
A
3 What is the area of triangle
12
C
XYZ? C
Read the Test Item
Z
1
The area of a triangle is A bh. In a right triangle, the legs form the base and
2
height of the triangle. Use the measures of the hypotenuse and the base to find
the height of the triangle.
35
X
28
Y
A 94 units2
B 128 units2
C 294 units2
D 588 units2
Concept Check
Solve the Test Item
Step 1
Test-Taking Tip
Memorize the common
Pythagorean triples and
check for multiples such as
(6, 8, 10). This will save you
time when evaluating
square roots.
4 3 12
4 4 16
4 5 20
The height of the triangle is 16 units.
Pythagorean Theorem Ask
students to rewrite the formula
for the Pythagorean Theorem in
a2 b2
terms of c. c Check to see if the measurements of this triangle are a multiple of a
common Pythagorean triple. The hypotenuse is 4 5 units, and the leg
is 4 3 units. This triangle is a multiple of a (3, 4, 5) triangle.
Step 2
Find the area of the triangle.
1
2
1
A 12 16
2
A bh
A 96
Area of a triangle
b 12 and h 16
Simplify.
The area of the triangle is 96 square units. Choice A is correct.
606
Chapter 11 Radical Expressions and Triangles
Example 3 Caution students that this is a two-step problem.
First they must find the missing side length, then they must
calculate the area of the triangle. Also stress to students that it
is important to read each answer choice carefully before
selecting the correct answer. In Example 3, choices B and C look very similar, and one
could mistakenly choose D, 196, when the correct answer is A, 96.
Standardized
Test Practice
606
Chapter 11 Radical Expressions and Triangles
RIGHT TRIANGLES A statement that can be easily proved using a theorem
is often called a corollary. The following corollary, based on the Pythagorean
Theorem, can be used to determine whether a triangle is a right triangle.
RIGHT TRIANGLES
In-Class Example
Teaching Tip
Point out to
students that with Pythagorean
triples, the greatest value is
always the measure of the
hypotenuse, and the two lesser
values are the measures of the
two legs.
Corollary to the Pythagorean Theorem
If a and b are measures of the shorter sides of a triangle, c is the measure
of the longest side, and c2 a2 b2, then the triangle is a right triangle.
If c2 a2 b2, then the triangle is not a right triangle.
Example 4 Check for Right Triangles
4 Determine whether the
following side measures form
right triangles.
Determine whether the following side measures form right triangles.
a. 20, 21, 29
Since the measure of the longest side is 29, let c 29, a 20, and b 21.
Then determine whether c2 a2 b2.
c2 a2 b2
292
202
Pythagorean Theorem
212
841 841
1.
leg
b. 8, 10, 12
Since the measure of the longest side is 12, let c 12, a 8, and b 10.
Then determine whether c2 a2 b2.
Pythagorean Theorem
122 82 102
144 164
Concept Check
GUIDED PRACTICE KEY
Exercises
Examples
4–9
10, 11
12
1, 2
4
3
Guided Practice
Add.
a2
b2,
hypotenuse
leg
2. Compare the lengths of the sides.
The hypotenuse is the longest
side, which is always the side
opposite the right angle.
a 8, b 10, and c 12
144 64 100 Multiply.
Since
b. 27, 36, 45 right triangle
Answers
Add.
Since c2 a2 b2, the triangle is a right triangle.
c2
a. 7, 12, 15 not a right triangle
a 20, b 21, and c 29
841 400 441 Multiply.
c2 a2 b2
Power
Point®
the triangle is not a right triangle.
1. OPEN ENDED Draw a right triangle and label each side and angle. Be sure to
indicate the right angle. See margin.
2. Explain how you can determine which angle is the right angle of a right triangle
if you are given the lengths of the three sides. See margin.
3. Write an equation you could use to find the length of the diagonal d of a square
with side length s. d 2s 2 or d s 2
Find the length of each missing side. If necessary, round to the nearest hundredth.
18.44
4.
c
12
5.
41
a
9
40
14
www.algebra1.com/extra_examples
Lesson 11-4 The Pythagorean Theorem 607
Differentiated Instruction
Kinesthetic Give students blocks, algebra tiles, or some other square or
cube-shaped manipulatives. Have them use the manipulatives and what
they know about the Pythagorean Theorem and Pythagorean triples to
construct a right triangle, placing the manipulatives along the outside of
the triangle. Have students construct the triangle on a piece of paper,
then trace inside the blocks to transfer the triangle to the paper. Then
ask them to use a protractor to confirm that the right angle is 90°.
Lesson 11-4 The Pythagorean Theorem 607
If c is the measure of the hypotenuse of a right triangle, find each missing
measure. If necessary, round to the nearest hundredth.
3 Practice/Apply
7. a 11, c 61, b ? 60
8. b 13, c 233
, a ? 8
9. a 7, b 4, c ?
8.06
65
Determine whether the following side measures form right triangles. Justify your
answer.
Study Notebook
Have students—
• add the definitions/examples of
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 11.
• include an explanation of the
Pythagorean Theorem and examples
of how to use it to find the sides
or hypotenuse of a right triangle.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
6. a 10, b 24, c ? 26
10. 4, 6, 9 No; 42 62 92.
Standardized
Test Practice
11. 16, 30, 34 Yes; 162 302 342.
12. In right triangle XYZ, the length of Y
Z
is 6, and the length of the hypotenuse
is 8. Find the area of the triangle. A
67 units2
A
30 units2
B
C
40 units2
48 units2
D
★ indicates increased difficulty
Practice and Apply
Homework Help
For
Exercises
See
Examples
13–30
31–36
37–40
1, 2
4
3
Find the length of each missing side. If necessary, round to the nearest hundredth.
14.14
13.
14.
7
53
28
c
a
15
11.40 15.
9
45
c
Extra Practice
See page 845.
5
16.
About the Exercises …
175
5
Organization by Objective
• The Pythagorean Theorem:
13–30, 37–40
• Right Triangles: 31–36
Odd/Even Assignments
Exercises 13–36 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Alert! Exercise 44 requires the
use of the Internet or another
reference source.
13.08 17.
14
a
180
b
20
99
101
b
If c is the measure of the hypotenuse of a right triangle, find each
missing measure. If necessary, round to the nearest hundredth.
27. 253
15.91
28. 124
11.14
19. a 16, b 63, c ? 65
20. a 16, c 34, b ? 30
21. b 3, a 112
, c ? 11
22. a 15
, b 10
, c ? 5
10.72
115
b 77
, c 12, a ? 67
8.19
6.71
45
a 4, b 11
, c ? 27
5.20
23. c 14, a 9, b ?
24. a 6, b 3, c ?
25.
26.
27. a 225
, b 28
, c ?
28. a 31
, c 155
, b ?
29. a 8x, b 15x, c ? 17x
30. b 3x, c 7x, a ?
40x 6.32x
Determine whether the following side measures form right triangles. Justify your
answer.
Assignment Guide
31. 30, 40, 50 Yes; 302 402 502.
Basic: 13–37 odd, 41–43, 49–68
Average: 13–39 odd, 41–43,
45, 48–68
Advanced: 14–40 even, 44,
46–62 (optional: 63–68)
32. 6, 12, 18 No; 62 122 182.
34. 45, 60, 75 Yes; 452 602 752.
2
2
2
35. 15, 31, 16 Yes; 15 31 16 . 36. 4, 7, 65
Yes; 42 72 65
2.
33. 24, 30, 36 No;
242
302
362.
Use an equation to solve each problem. If necessary, round to the nearest
hundredth.
37. Find the length of a diagonal of a square if its area is 162 square feet. 18 ft
608
Chapter 11 Radical Expressions and Triangles
Answer
c 2
4
4
48. The area of the largest semicircle is c 2.
4
The sum of the other two areas is (a 2 b 2). Using
the Pythagorean Theorem, c 2 a 2 b 2, we can show
that the sum of the two small areas is equal to the area
of the largest semicircle.
608
42.1318.
Chapter 11 Radical Expressions and Triangles
★ 38. A right triangle has one leg that is 5 centimeters longer than the other leg.
NAME ______________________________________________ DATE
p. 661
(shown)
and p. 662
The Pythagorean
Theorem
The Pythagorean Theorem The side opposite the right angle in a right triangle is
called the hypotenuse. This side is always the longest side of a right triangle. The other
two sides are called the legs of the triangle. To find the length of any side of a right
triangle, given the lengths of the other two sides, you can use the Pythagorean Theorem.
★ 39. Find the length of the diagonal of the cube if each side
of the cube is 4 inches long. 4
3 in. or about 6.93 in.
★ 40. The ratio of the length of the hypotenuse to the length
B
If a and b are the measures of the legs of a right triangle
and c is the measure of the hypotenuse, then c2 a2 b2.
Pythagorean Theorem
Example 1 Find the length of the
hypotenuse of a right triangle if a 5
and b 12.
112.41 m
c2
c2
c2
c
c
ROLLER COASTERS For Exercises 41–43, use
the following information and the figure.
Suppose a roller coaster climbs 208 feet higher than
its starting point making a horizontal advance of
360 feet. When it comes down, it makes a horizontal
advance of 44 feet.
a2 b2
52 122
169
169
13
208 ft
b
A
Example 2 Find the length of a leg
of a right triangle if a 8 and c 10.
c2 a2 b2
102 82 b2
100 64 b2
36 b2
b 36
b 6
Pythagorean Theorem
a 5 and b 12
Simplify.
Take the square root of each side.
The length of the hypotenuse is 13.
Pythagorean Theorem
a 8 and c 10
Simplify.
Subtract 64 from each side.
Take the square root of each side.
The length of the leg is 6.
Exercises
Find the length of each missing side. If necessary, round to the nearest
hundredth.
1.
2.
3.
100
25
c
a
110
25
c
Lesson 11-4
30
360 ft
41. How far will it travel to get to the top of the
ride? 415.8 ft
40
50
44 ft
Roller Coasters
42. How far will it travel on the downhill track? 212.6 ft
Roller Coaster Records in
the U.S.
Fastest: Millennium Force
(2000), Sandusky, Ohio;
92 mph
Tallest: Millenium Force
(2000), Sandusky, Ohio;
310 feet
Longest: California
Screamin’ (2001), Anaheim,
California; 6800 feet
Steepest: Hypersonic XLC
(2001), Doswell, Virginia;
90°
43. Compare the total horizontal advance, vertical height, and total track length.
45.83
4. a 10, b 12, c ?
44. RESEARCH Use the Internet or other reference to find the measurements of
your favorite roller coaster or a roller coaster that is at an amusement park close
to you. Draw a model of the first drop. Include the height of the hill, length of
the vertical drop, and steepness of the hill. See students’ work.
5. a 9, b 12, c ?
15.62
6. a 12, b ?, c 16
15
10.58
8. a ?, b 8
, c 18
7. a ?, b 6, c 8
5.29
9. a 5
, b 10
, c ?
3.16
3.87
NAME ______________________________________________ DATE
Skills
Practice,
11-4
Practice
(Average)
____________ PERIOD _____
p. 663 and
Practice,
p. 664
(shown)
The Pythagorean
Theorem
Find the length of each missing side. If necessary, round to the nearest
hundredth.
1.
a
2.
3.
c
32
45. SAILING A sailboat’s mast and boom form a right
angle. The sail itself, called a mainsail, is in the shape
of a right triangle. If the edge of the mainsail that is
attached to the mast is 100 feet long and the edge of
the mainsail that is attached to the boom is 60 feet
long, what is the length of the longest edge of the
mainsail? 116.6 ft
35.36
If c is the measure of the hypotenuse of a right triangle, find each missing
measure. If necessary, round to the nearest hundredth.
The roller coaster makes a total horizontal advance of 404 feet, reaches a vertical
height of 208 feet, and travels a total track length of 628.3 feet.
12
4
11
19
b
60
68
mast
mainsail
15.49
11.31
If c is the measure of the hypotenuse of a right triangle, find each missing
measure. If necessary, round to the nearest hundredth.
5. a 28, b 96, c ? 100
4. a 24, b 45, c ? 51
6. b 48, c 52, a ? 20
7. c 27, a 18, b ?
245
15.65
8. b 14, c 21, a ?
boom
10. a 75
, b 6
, c ? 9
405
20.12
9. a 20
, b 10, c ?
120
10.95
11. b 9x, c 15x, a ? 12x
Determine whether the following side measures form right triangles. Justify your
answer.
12. 11, 18, 21
13. 21, 72, 75
No; 112 182 212.
Yes; 212 722 752.
15. 9, 10, 161
14. 7, 8, 11
ROOFING For Exercises 46 and 47, refer to the figures below.
No;
72
82
2
No; 92 102 (161
) .
112.
16. 9, 210
, 11
5 ft
? ft
c
a
C
of the shorter leg in a right triangle is 8:5. The hypotenuse
measures 144 meters. Find the length of the longer leg.
Source: www.coasters2k.com
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
11-4
Study
Guide and
The hypotenuse is 25 centimeters long. Find the length of each leg of the
triangle. 15 cm, 20 cm
17. 7
, 22
, 15
2
2
Yes; 92 2(10
) 112.
2
2
Yes; (7
) (22
) (15
) .
18. STORAGE The shed in Stephan’s back yard has a door that measures 6 feet high and
3 feet wide. Stephan would like to store a square theater prop that is 7 feet on a side.
Will it fit through the door diagonally? Explain. No; the greatest length that will
fit through the door is 45
6.71 ft.
24 ft (entire span)
SCREEN SIZES For Exercises 19–21, use the following information.
The size of a television is measured by the length of the screen’s diagonal.
19. If a television screen measures 24 inches high and 18 inches wide, what size television
is it? 30-in. television
46. Determine the missing length shown in the rafter. 13 ft
20. Darla told Tri that she has a 35-inch television. The height of the screen is 21 inches.
What is its width? 28 in.
47. If the roof is 30 feet long and it hangs an additional 2 feet over the garage walls,
how many square feet of shingles are needed for the entire garage roof? 900 ft2
21. Tri told Darla that he has a 5-inch handheld television and that the screen measures
2 inches by 3 inches. Is this a reasonable measure for the screen size? Explain. No; if
the screen measures 2 in. by 3 in., then its diagonal is only about 3.61 in.
NAME ______________________________________________ DATE
____________ PERIOD _____
Reading
11-4
Readingto
to Learn
Learn Mathematics
Mathematics,
p. 665
The Pythagorean Theorem
★ 48. CRITICAL THINKING Compare the area of the
Pre-Activity
largest semicircle to the areas of the two smaller
semicircles. Justify your reasoning. See margin.
ELL
How is the Pythagorean Theorem used in roller coaster design?
Read the introduction to Lesson 11-4 at the top of page 605 in your textbook.
The diagram in the introduction shows a right triangle and part of the
roller coaster. Which side of the right triangle has a length approximately
equal to the length of the first hill of the roller coaster?
c
a
the longest side, which is the side opposite the right angle
b
Reading the Lesson
Complete each sentence.
1. The words leg and hypotenuse refer to the sides of a
Lesson 11-4 The Pythagorean Theorem 609
triangle.
3. The longest side of a right triangle is called the
hypotenuse
leg
of the right triangle.
Write an equation that you could solve to find the missing side length of each
right triangle.
NAME ______________________________________________ DATE
____________ PERIOD _____
4.
5.
6.
10
11-4
Enrichment
Enrichment,
9
p. 666
10
6
9
10
c2 92 102
Pythagorean Triples
Recall the Pythagorean theorem:
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 legs.
a
c
b
A
a2 b2 c2
Note that c is the length
of the hypotenuse.
9 16 25
25 25
The integers 3, 4, and 5 satisfy the
Pythagorean theorem and can be the
lengths of the sides of a right triangle.
32
42
52
Furthermore, for any positive integer n,
the numbers 3n, 4n, and 5n satisfy the
Pythagorean theorem.
For n 2: 62 82 102
36 64 100
100 100
102 a2 92
7. Suppose you are given three positive numbers. Explain how you can decide whether
these numbers are the lengths of the sides of a right triangle.
B
C
102 62 b2
Sample answer: Put the numbers in order from least to greatest. Square
the three numbers. See whether the sum of the squares of the two
smaller numbers is equal to the square of the largest number. If it is,
then the numbers are the lengths of the sides of a right triangle.
Helping You Remember
8. Think of a word or phrase that you can associate with the Pythagorean Theorem to help
you remember the equation c2 a2 b2.
Sample answer: Think of the word cab. The letters of this word are the
same as those in the equation c 2 a 2 b 2.
If three numbers satisfy the Pythagorean theorem, they are called a
Lesson 11-4 The Pythagorean Theorem 609
Lesson 11-4
www.algebra1.com/self_check_quiz
right
2. In a right triangle, each of the two sides that form the right angle is a
of the right triangle.
49. CRITICAL THINKING A model of a part of a roller coaster is shown. Determine
the total distance traveled from start to finish and the maximum height reached
by the roller coaster. 1081.7 ft, 324.5 ft
4 Assess
Open-Ended Assessment
80 ft
Modeling Have students use
pencils, uncooked spaghetti, or
twigs to model right triangles.
Ask students to explain the
Pythagorean Theorem and relate
it to their triangle.
180 ft
750 ft
50. WRITING IN MATH
Answer the question that was posed at the beginning of
the lesson. See margin.
How is the Pythagorean Theorem used in roller coaster design?
Include the following in your answer:
• an explanation of how the height, speed, and steepness of a roller coaster are
related, and
• a description of any limitations you can think of in the design of a new roller
coaster.
Standardized
Test Practice
51. Find the area of XYZ. C
X
A
65 units2
B
185 units2
C
45 units2
D
90 units2
15
52. Find the perimeter of a square whose diagonal
measures 10 centimeters. B
Assessment Options
A
C
102 cm
B
202 cm
252 cm
D
80 cm
Z
2x
x
Y
Maintain Your Skills
Mixed Review
Solve each equation. Check your solution. (Lesson 11-3)
53. y 12 144
54. 3s 126 1764
55. 4
2v 1 3 17 12
Simplify each expression. (Lesson 11-2)
56. 72
62
57. 7z 10z 3z 58.
821
37 21
7
Simplify. Assume that no denominator is equal to zero. (Lesson 8-2)
1
58
26a4b7c5 2a 2b 3
59. 3 55 or 3125
60. d7 7
61. 5
13a2b4c3
d
c8
Answer
610
start
100 ft 130 ft 50 ft
PREREQUISITE SKILL Students
will learn about the Distance
Formula in Lesson 11-5. In order
to use the Distance Formula,
students must be able to simplify
radical expressions. Use
Exercises 63–68 to determine
your students’ familiarity with
simplifying radical expressions.
50. Engineers can use the Pythagorean Theorem to find the total length
of the track, which determines how
much material and land area they
need to build the attraction.
Answers should include the
following.
• A tall hill requires more track
length both going uphill and
downhill, which will add to the
total length of the tracks. Tall,
steep hills will increase the
speed of the roller coaster. So a
coaster with a tall, steep first
hill will have more speed and a
longer track length.
• The steepness of the hill and
speed are limited for safety and
to keep the cars on the track.
120 ft
finish
Getting Ready for
Lesson 11-5
Quiz (Lessons 11-3 and 11-4) is
available on p. 699 of the Chapter
11 Resource Masters.
Mid-Chapter Test (Lessons 11-1
through 11-4) is available on
p. 701 of the Chapter 11 Resource
Masters.
381.2 ft
62. AVIATION Flying with the wind, a plane travels 300 miles in 40 minutes.
Flying against the wind, it travels 300 miles in 45 minutes. Find the air
speed of the plane. (Lesson 7-4) 425 mph
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Simplify each expression.
(To review simplifying radical expressions, see Lesson 11-1.)
63. (6
3
)2 (8
4)2 5
65. (5 3
)2 (2
9)2
53
67. (4 5)2 (4 3)2
610 Chapter 11
Chapter 11 Radical Expressions and Triangles
Radical Expressions and Triangles
64. (10 4)2 (
13 5
)2 10
130
66. (9 5)2 (7 3
)2 253
68. (20 5)2 (
2 6)2 17
Lesson
Notes
The Distance Formula
• Find the distance between two points on the coordinate plane.
1 Focus
• Find a point that is a given distance from a second point in a plane.
can the distance between two points be determined?
Study Tip
Reading Math
AC is the measure of AC
and BC is the measure
of BC.
Consider two points A and B in the coordinate
plane. Notice that a right triangle can be formed
by drawing lines parallel to the axes through
the points at A and B. These lines intersect at
C forming a right angle. The hypotenuse of this
triangle is the distance between A and B. You can
A
determine the length of the legs of this triangle
and use the Pythagorean Theorem to find the
distance between the two points. Notice that
AC is the difference of the y-coordinates,
and BC is the difference of the x-coordinates.
2
(BC
)2.
So, (AB)2 (AC)2 (BC)2, and AB (AC)
y
B
O
x
C
THE DISTANCE FORMULA You can find the distance between any two
points in the coordinate plane using a similar process. The result is called the
Distance Formula.
The Distance Formula
• Words
• Model
The distance d between any two points with coordinates (x1, y1)
(x2 x1)2 (y2 y1)2.
and (x2, y2) is given by d A (x 1, y 1)
y
B (x 2, y 2)
O
x
5-Minute Check
Transparency 11-5 Use as
a quiz or review of Lesson 11-4.
Mathematical Background notes
are available for this lesson on
p. 584D.
can the distance
between two points be
determined?
Ask students:
• What must you know in order
to use the Pythagorean Theorem
to determine the distance
between the two points? You
must know the lengths of the two
legs in order to find the distance
between the two points, which is
the hypotenuse.
• How do you find the length of
the vertical leg? Find the difference in the y-coordinates of A and B.
• How do you find the length of
the horizontal leg? Find the
difference in the x-coordinates of A
and B.
Example 1 Distance Between Two Points
Find the distance between the points at (2, 3) and (4, 6).
d (x2 x1)2 (y2 y1)2 Distance Formula
(4 2)2 (6 3
)2
(x1, y1) (2, 3) and (x2, y2) (4, 6)
Simplify.
45
Evaluate squares and simplify.
(6)2
32
35 or about 6.71 units
Lesson 11-5 The Distance Formula 611
Resource Manager
Workbook and Reproducible Masters
Chapter 11 Resource Masters
• Study Guide and Intervention, pp. 667–668
• Skills Practice, p. 669
• Practice, p. 670
• Reading to Learn Mathematics, p. 671
• Enrichment, p. 672
Parent and Student Study Guide
Workbook, p. 87
Science and Mathematics Lab Manual,
pp. 85–90
Transparencies
5-Minute Check Transparency 11-5
Answer Key Transparencies
Technology
Interactive Chalkboard
Lesson x-x Lesson Title 611
Example 2 Use the Distance Formula
2 Teach
GOLF Tracy hits a golf ball that lands 20 feet short and 8 feet to the right of the
cup. On her first putt, the ball lands 2 feet to the left and 3 feet beyond the cup.
Assuming that the ball traveled in a straight line, how far did the ball travel on
her first putt?
THE DISTANCE FORMULA
In-Class Examples
Power
Point®
Draw a model of the situation on a coordinate grid. If the cup is at (0, 0), then the
location of the ball after the first hit is (8, 20). The location of the ball after the
first putt is (2, 3). Use the Distance Formula.
Teaching Tip
Point out to
students that it does not matter
which of the two points are
designated (x1, y1) or (x2, y2).
2
(x2 x
(y2 y1)2
d 1) (10)2
232
points at (1, 2) and (3, 0).
The distance is 25, or about
4.47 units.
Simplify.
629
or about 25 feet
2 BIATHLON Julianne is sight-
8642 O
5
10
15
20
25
y
2 4 6 8x
(8, 20)
FIND COORDINATES Suppose you know the coordinates of a point, one
Golf
There are four major
tournaments that make up
the “grand slam” of golf:
Masters, U.S. Open,
British Open, and PGA
Championship. In 2000,
Tiger Woods became the
youngest player to win
the four major events
(called a career grand
slam) at age 24.
Source: PGA
coordinate of another point, and the distance between the two points. You can
use the Distance Formula to find the missing coordinate.
Example 3 Find a Missing Coordinate
Find the value of a if the distance between the points at (7, 5) and (a, 3)
is 10 units.
(x2 x1)2 (y2 y1)2 Distance Formula
d 10 (a 7
)2 (
3 5)2
Let x2 a, x1 7, y2 3, y1 5,
10 (a and d 10.
10 1
4a 4
9 64
Evaluate squares.
10 a2 1
4a 113
Simplify.
7)2
(8)2
a2
FIND COORDINATES
In-Class Example
15
10
(2, 3) 5
(x , y ) (8, 20),
(2 8)2 [3 (
20)]2 (x1, y1) (2, 3)
2 2
1 Find the distance between the
ing in her rifle for an upcoming biathlon competition. Her
first shot is 2 inches to the
right and 7 inches below the
bull’s-eye. What is the distance between the bull’s-eye
and where her first shot hit
the target? The distance is
53
or about 7.28 inches.
Distance Formula
102 a2 1
4a 113
Square each side.
100 a2 14a 113
Simplify.
2
Power
Point®
0
3 Find the value of a if the
distance between the points
at (2, 1) and (a, 4) is
5 units. 2 or 6
a2
14a 13
Subtract 100 from each side.
0 (a 1)(a 13)
a10
Factor.
or a 13 0
a1
a 13
Zero Product Property
The value of a is 1 or 13.
Answers
1. The values that are subtracted are
squared before being added and
the square of a negative number
is always positive. The sum of two
positive numbers is positive, so the
distance will never be negative.
2. See students’ graph; the distance
from A to B equals the distance
from B to A. Using the Distance
Formula, the solution is the same
no matter which ordered pair is
used first.
3. See students’ diagrams; there are
exactly two points that lie on the
line y 3 that are 10 units from
the point (7, 5).
Concept Check
1–3. See margin.
1. Explain why the value calculated under the radical sign in the Distance Formula
will never be negative.
2. OPEN ENDED Plot two ordered pairs and find the distance between their
graphs. Does it matter which ordered pair is first when using the Distance
Formula? Explain.
3. Explain why there are two values for a in Example 3. Draw a diagram to support
your answer.
612
Chapter 11 Radical Expressions and Triangles
Unlocking Misconceptions
Not only does it not matter which point is designated (x1, y1) and
(x2, y2) when using the Distance Formula, it also does not matter in
which order the x- and y-coordinates are subtracted. For example,
(x
x1)2 (y2 y1)2 will produce the same results as
2 (x
x2)2 (y1 y2)2 or (x
x2)2 (y2 y1)2. Have students
1 1 test each method with two points.
612
Chapter 11 Radical Expressions and Triangles
Guided Practice
4. (5, 1), (11, 7) 10
GUIDED PRACTICE KEY
Exercises
Examples
4–7
8, 9
10–12
1
3
2
Find the distance between each pair of points whose coordinates are given.
Express in simplest radical form and as decimal approximations rounded to the
nearest hundredth if necessary.
6. (2, 2), (5, 1) 32
4.24
5. (3, 7), (2, 5) 13
7. (3, 5), (6, 4)
3.16
10
Find the possible values of a if the points with the given coordinates are the
indicated distance apart.
8. (3, 1), (a, 7); d 10 9 or 3
Applications
9. (10, a), (1, 6); d 145
2 or 14
10. GEOMETRY An isosceles triangle has two sides of equal length. Determine
whether triangle ABC with vertices A(3, 4), B(5, 2), and C(1, 5) is an
isosceles triangle. yes; AB 68
, BC 85
, AC 85
FOOTBALL For Exercises 11 and 12, use the
information at the right.
11. 25.5 yd, 25 yd
Quarterback
10 yd
5 yd
Practice and Apply
See
Examples
13–26,
33, 34
27–32,
35, 36
37–42
1
13. (12, 3), (8, 3) 20
14. (0, 0), (5, 12) 13
3
15. (6, 8), (3, 4) 5
16. (4, 2), (4, 17) 17
2
17. (3, 8), (5, 4) 45
8.94
18. (9, 2), (3, 6) 213
7.21
Extra Practice
See page 845.
Find the distance between each pair of points whose coordinates are given.
Express in simplest radical form and as decimal approximations rounded to
the nearest hundredth if necessary.
19. (8, 4), (3, 8) 41
6.40
2 10
21. (4, 2), 6, 3.33
3 3
4
1 13
23. , 1, 2, or 1.30
5
2 10
25. 45, 7, 65, 1 214
7.48
Have students—
• include examples of how to use
the Distance Formula.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
10 20 30 40
★ indicates increased difficulty
For
Exercises
Study Notebook
25 yd
11. A quarterback can throw the football to one
of the two receivers. Find the distance from
the quarterback to each receiver.
12. What is the distance between the two
receivers? 20.6 yd
Homework Help
3 Practice/Apply
20. (2, 7), (10, 4) 185
13.60
17
1
22. 5, , (3, 4) or 4.25
4
4
3
2 74
24. 3, , 4, 1.23
7
7
7
26. 52, 8, 72, 10 23
3.46
Find the possible values of a if the points with the given coordinates are the
indicated distance apart.
27. (4, 7), (a, 3); d 5 1 or 7
28. (4, a), (4, 2); d 17 17 or 13
29. (5, a), (6, 1); d 10
2 or 4
30. (a, 5), (7, 3); d 29
2 or 12
About the Exercises …
Organization by Objective
• The Distance Formula:
13–26, 33, 34, 37–42
• Find Coordinates: 27–32,
35, 36
Odd/Even Assignments
Exercises 13–32 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Assignment Guide
Basic: 13–31 odd, 37–39, 43–68
Average: 13–37 odd, 38, 39,
43–68
Advanced: 14–36 even, 40–62
(optional: 63–68)
31. (6, 3), (3, a); d 130
10 or 4 32. (20, 5), (a, 9); d 340
2 or 38
★ 33. Triangle ABC has vertices at A(7, 4), B(1, 2), and C(5, 6). Determine whether
the triangle has three, two, or no sides that are equal in length. two; AB BC 10
34. 157
101
; ★ 34. If the diagonals of a trapezoid have the same length, then the trapezoid is
The trapezoid is not
isosceles. Find the lengths of the diagonals of trapezoid ABCD with vertices
isosceles.
A(2, 2), B(10, 6), C(9, 8), and D(0, 5) to determine if it is isosceles.
www.algebra1.com/extra_examples
Lesson 11-5 The Distance Formula 613
Differentiated Instruction
Interpersonal Place students in pairs. Have each person in a pair drop
a penny on a coordinate grid, and record the coordinates of the point
closest to where the center of the penny landed. Then have the
students work together to find the distance between the two points,
using the Distance Formula.
Lesson 11-5 The Distance Formula 613
NAME ______________________________________________ DATE
★ 35. Triangle LMN has vertices at L(4, 3), M(2, 5), and N(13, 10). If the distance
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
11-5
Study
Guide and
from point P(x, 2) to L equals the distance from P to M, what is the value of x? 3
p. 667
(shown)
The Distance
Formulaand p. 668
The Distance Formula
The Pythagorean Theorem can be used to derive the Distance
Formula shown below. The Distance Formula can then be used to find the distance between
any two points in the coordinate plane.
★ 36. Plot the points Q(1, 7), R(3, 1), S(9, 3), and T(7, d). Find the value of d that makes
each side of QRST have the same length. 9
The distance between any two points with coordinates (x1, y1) and (x2, y2) is given by
Distance Formula
(x2 x1)2 (y2 y1)2.
d Example 1 Find the distance between the
points at (5, 2) and (4, 5).
d (x2 x1)2 ( y2 y1) 2 Distance Formula
(4 (
5))2 (5 2)2
(x1, y1) (5, 2), (x2, y2) (4, 5)
92 32
Simplify.
81 9
Evaluate squares and simplify.
Example 2 Jill draws a line
segment from point (1, 4) on her
computer screen to point (98, 49).
How long is the segment?
37. FREQUENT FLYERS To determine the mileage between cities for their frequent
flyer programs, some airlines superimpose a coordinate grid over the United
States. An ordered pair on the grid represents the location of each airport.
The units of this grid are approximately equal to 0.316 mile. So, a distance of
3 units on the grid equals an actual distance of 3(0.316) or 0.948 mile. Suppose
the locations of two airports are at (132, 428) and (254, 105). Find the actual
distance between these airports to the nearest mile. 109 mi
d (x2 x1)2 ( y2 y1) 2
(98 1)2 (49 4)2
972 452
90
The distance is 90
, or about 9.49 units.
9409
2025
11,43
4
The segment is about 106.93 units
long.
Exercises
Find the distance between each pair of points whose coordinates are given.
Express answers in simplest radical form and as decimal approximations rounded
to the nearest hundredth if necessary.
3. (2, 8), (7, 3)
2. (0, 0), (6, 8)
25
; 4.47
6. (3, 4), (4, 4)
5. (1, 5), (8, 4)
82
; 9.06
17
7. (1, 4), (3, 2)
7
25
; 4.47
34
; 5.83
130
; 11.40
12. (3, 4), (4, 16)
11. (3, 4), (0, 0)
173
; 13.15
193
; 13.89
5
13. (1, 1), (3, 2)
14. (2, 0), (3, 9)
15. (9, 0), (2, 5)
5
; 2.24
82
; 9.06
74
; 8.60
17. (1, 3), (8, 21)
18. (3, 5), (1, 8)
16. (2, 7), (2, 2)
41
; 6.40
38. Kelly has her first class in Rhodes Hall
and her second class in Fulton Lab.
How far does she have to walk between
her first and second class? 0.53 mi
9. (2, 6), (7, 1)
8. (0, 0), (3, 5)
10. (2, 5), (0, 8)
COLLEGE For Exercises 38 and 39, use the
map of a college campus.
106
; 10.30
10
4. (6, 7), (2, 8)
657
; 25.63
Lesson 11-5
1. (1, 5), (3, 1)
5
NAME ______________________________________________ DATE
Skills
Practice,
11-5
Practice
(Average)
____________ PERIOD _____
Find the distance between each pair of points whose coordinates are given.
Express answers in simplest radical form and as decimal approximations rounded
to the nearest hundredth if necessary.
3. (4, 6), (3, 9)
10
3.16
5. (0, 4), (3, 2) 35
6.71
1
2
6. (13, 9), (1, 5) 410
12.65
5
2
13 1
2
1
2
9. 2, , 1, 100
5
3
8. (1, 7), , 6 1.67
7. (6, 2), 4, or 2.50
2
7
2.65
1
20
M I NNE S OTA
40
Find the possible values of a if the points with the given coordinates are the
indicated distance apart.
60
0
15. (6, 7), (a, 4); d 18
a 3 or 9
16. (4, 1), (a, 8); d 50
a 5 or 3
17. (8, 5), (a, 4); d 85
a 6 or 10
18. (9, 7), (a, 5); d 29
a 14 or 4
Duluth
Three players are warming up for a baseball game. Player B
stands 9 feet to the right and 18 feet in front of Player A.
Player C stands 8 feet to the left and 13 feet in front of Player A.
16
20
41. Find the distance between the following pairs of cities:
Minneapolis and St. Cloud, St. Paul and Rochester,
Minneapolis and Eau Claire, and Duluth and St. Cloud.
Eau Claire
Rochester
42. A radio station in St. Paul has a broadcast range of 75 miles.
Which cities shown on the map can receive the broadcast?
B (9, 18)
80
12
C (8, 13)
Fulton Lab
1
mi
8
Eau Claire, (71, 8); Rochester, (27, 58)
St. Paul
Minneapolis
60
y
3
mi
8
40. Estimate the coordinates for Duluth, St. Cloud, Eau Claire,
and Rochester. Duluth, (44, 116); St. Cloud, (46, 39);
St. Cloud
40
BASEBALL For Exercises 19–21, use the following information.
Rhodes Hall
GEOGRAPHY For Exercises 40–42, use the map at the left that
shows part of Minnesota and Wisconsin.
A coordinate grid has been superimposed on the map with the
origin at St. Paul. The grid lines are 20 miles apart. Minneapolis
is at (7, 3).
80
20
14. (2, 5), (a, 7); d 15 a 7 or 11
19. Draw a model of the situation on the coordinate grid.
Assume that Player A is located at (0, 0).
40
60
3
12. (22
, 1), (32
, 3) 32
4.24
13. (4, 1), (a, 5); d 10 a 4 or 12
0
80
42
1.89
10. , 1 , 2, 3
3
2
1.41
11. (3
, 3), (23
, 5)
120 100 80 60 40 20
120
4. (3, 8), (7, 2) 229
10.77
Undergraduate
Library
minutes to walk between the two buildings.
170
13.04
2. (0, 9), (7, 2)
1
mi
4
39. She has 12 minutes between the end of
her first class and the start of her second
class. If she walks an average of 3 miles
per hour, will she make it to her second
class on time? Yes; it will take her 10.6
p. 669 and
Practice,
p.Formula
670 (shown)
The Distance
1. (4, 7), (1, 3) 5
Campus Map
all cities except Duluth
8
4
20. To the nearest tenth, what is the distance between Players A
and B and between Players A and C? 20.1 ft; 15.3 ft
A(0, 0)
4 O
4
8
x
8
21. What is the distance between Players B and C? 17.7 ft
22. MAPS Maria and Jackson live in adjacent neighborhoods. If they superimpose a
coordinate grid on the map of their neighborhoods, Maria lives at (9, 1) and Jackson
lives at (5, 4). If each unit on the grid is equal to approximately 0.132 mile, how far
apart do Maria and Jackson live? about 1.96 mi
NAME ______________________________________________ DATE
____________ PERIOD _____
Reading
11-5
Readingto
to Learn
Learn Mathematics
ELL
Mathematics,
p. 671
The Distance Formula
Pre-Activity
How can the distance between two points be determined?
41. MinneapolisSt. Cloud, 53 mi;
St. Paul-Rochester,
64 mi; MinneapolisRochester, 70 mi;
Duluth-St. Cloud,
118 mi
43. CRITICAL THINKING Plot A(4, 4), B(7, 3), and C(4, 0), and connect them
to form triangle ABC. Demonstrate two different ways to show whether ABC is
a right triangle. See margin.
44. WRITING IN MATH
How can the distance between two points be determined?
Read the introduction to Lesson 11-5 at the top of page 611 in your textbook.
What are the coordinates of points A, B, and C?
Include the following in your answer:
• an explanation how the Distance Formula is derived from the Pythagorean
Theorem, and
• an explanation why the Distance Formula is not needed to find the distance
between points P(24, 18) and Q(24, 10).
A (4, 3), B (3, 2), and C (4, 2)
Reading the Lesson
1. Suppose you want to use the Distance Formula to find the
distance between (6, 4) and (2, 1). Use (x1, y1) (6, 4) and
(x2, y2) (2, 1). Complete the equations by writing the
correct numbers in the blanks.
a. x1 b. d 6
y1 4
x2 2
y2 y
A
1
B
O
( 2 6 )2 ( 1 4 )2
x
614
Chapter 11 Radical Expressions and Triangles
2. Suppose you want to use the Distance Formula to find the distance between (3, 7) and
(9, 2). Use (x1, y1) (3, 7) and (x2, y2) (9, 2). Complete the equations by writing the
correct numbers in the blanks.
a. x1 b. d 3
y1 7
x2 9
Answer the question that was posed at the beginning
of the lesson. See pp. 639A–639B.
y2 2
( 9 3 )2 ( 2 7 )2
3. A classmate is using the Distance Formula to find the distance between two points. She
(2 5)2 (7 11)2.
has done everything correctly so far, and her equation is d This equation will give her the distance between what two points?
(5, 11) and (2, 7)
Helping You Remember
4. Sometimes it is easier to remember a formula if you can state it in words. How can you
state the Distance Formula in easy-to-remember words?
Sample answer: Square the difference between the x-coordinates, then
add that to the square of the difference between the y-coordinates. Take
the square root of the sum to find the distance between the two points.
NAME ______________________________________________ DATE
11-5
Enrichment
Enrichment,
____________ PERIOD _____
p. 672
A Space-Saving Method
Two arrangements for cookies on a 32 cm by 40 cm cookie
sheet are shown at the right. The cookies have 8-cm diameters
after they are baked. The centers of the cookies are on the
vertices of squares in the top arrangement. In the other, the
centers are on the vertices of equilateral triangles. Which
arrangement is more economical? The triangle arrangement
is more economical, because it contains one more cookie.
8 cm
In the square arrangement, rows are placed every 8 cm. At
what intervals are rows placed in the triangle arrangement?
20 cookies
Look at the right triangle labeled a, b, and c. A leg a of the
triangle is the radius of a cookie, or 4 cm. The hypotenuse c is
the sum of two radii, or 8 cm. Use the Pythagorean theorem to
find b, the interval of the rows.
c2 a2 b2
82 42 b2
64 16 b2
614 Chapter 11 Radical Expressions and Triangles
b?
a
c
Standardized
Test Practice
45. Find the distance between points at (6, 11) and (2, 4). B
A
16 units
B
17 units
C
18 units
D
19 units
4 Assess
Open-Ended Assessment
46. Find the perimeter of a square ABCD if two of the vertices are A(3, 7)
and B(3, 4). B
A
12 units
B
125 units
C
95 units
D
45 units
Speaking Have students describe
a real-world situation in which
the Distance Formula could be
used to find the distance between
two points. If students have
trouble thinking of situations,
suggest that they think of
anything that involves measurement such as construction, art,
sports, and so on. Have them
explain how the Distance
Formula is used in the situation.
Maintain Your Skills
Mixed Review
If c is the measure of the hypotenuse of a right triangle, find each missing
measure. If necessary, round to the nearest hundredth. (Lesson 11-4)
47. a 7, b 24, c ? 25
48. b 30, c 34, a ? 16
49. a 7, c 16
, b ? 3
50. a 13
, b 50
, c ?
37
7.94
Solve each equation. Check your solution. (Lesson 11-3)
51. p 2 8 p 11
52. r5r1 4
COST OF DEVELOPMENT For
Exercises 54–56, use the graph that
shows the amount of money being
spent on worldwide construction.
(Lesson 8-3)
54. Asia,
1,113,000,000,000;
Europe,
1,016,000,000,000;
U.S./Canada,
884,000,000,000;
Latin America,
241,000,000,000;
Middle East,
101,200,000,000;
Africa,
56,100,000,000.
55. Asia, 1.113 1012;
Europe, 1.016 1012;
U.S./Canada, 8.84 1011; Latin America,
2.41 1011; Middle
East, 1.012 1011;
Africa, 5.61 1010.
Getting Ready for
the Next Lesson
54. Write the value shown for each
continent or region listed in
standard notation.
53. 5t2 29
2t 3
{2, 10}
USA TODAY Snapshots®
PREREQUISITE SKILL Students will
learn about similar triangles in
Lesson 11-6. In order to determine
whether triangles are similar, they
must be able to solve proportions.
Use Exercises 63–68 to determine
your students’ familiarity with
solving proportions.
Global spending on construction
The worldwide construction industry, valued
at $3.41 trillion in 2000, grew 5.8% since
1998. Breakdown by region:
$1.113
trillion $1.016
trillion
55. Write the value shown for each
continent or region in scientific
notation.
$884
billion
56. How much more money is
being spent in Asia than in
Latin America? $8.72 1011
$241
billion $101.2
$56.1
billion billion
or $872 billion
Asia
Europe
U.S./
Latin Middle
Canada America East
Africa
Source: Engineering News Record
By Shannon Reilly and Frank Pompa, USA TODAY
Solve each inequality. Then check your solution and graph it on a number line.
57–62. See pp. 639A–639B for graphs.
57. 8 m 1 {mm 9}
58. 3 10 k {kk 7}
59. 3x 2x 3 {xx 3}
60. v (4) 6 {vv 2}
(Lesson 6-1)
61. r 5.2 3.9 {rr 9.1}
1
1
2
62. s ss 6
3
2
PREREQUISITE SKILL Solve each proportion. (To review proportions, see Lesson 3-6.)
20
5
64. 8
x
3
63. 6
4
2
6
8
65. 12
9
x
x2
3
67. 7
7
1
www.algebra1.com/self_check_quiz
Getting Ready for
Lesson 11-6
Answer
43. Compare the slopes of the two
potential legs to determine
whether the slopes are negative
reciprocals of each other. You can
also compute the lengths of the
three sides and determine
whether the square of the longest
side length is equal to the sum of
the squares of the other two side
lengths. Neither test holds true in
this case because the triangle is
not a right triangle.
x
2
x
10
66. 15
18
12
6
2
68. 5
x4
3
Lesson 11-5 The Distance Formula 615
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Lesson 11-5 The Distance Formula 615
Lesson
Notes
1 Focus
5-Minute Check
Transparency 11-6 Use as
a quiz or review of Lesson 11-5.
Similar Triangles
• Determine whether two triangles are similar.
• Find the unknown measures of sides of two similar triangles.
Vocabulary
• similar triangles
Mathematical Background notes
are available for this lesson on
p. 584D.
Building on Prior
Knowledge
In Lesson 3-6, students learned
how to solve proportions. In this
lesson, students will use their
knowledge of proportions to
determine whether two triangles
are similar.
are similar triangles
related to photography?
Ask students:
• If you photograph two people,
one of whom is twice the
height of the other, what are
the heights of their images in
the photograph? One image will
be twice the height of the other.
• How would you describe this
relationship using ratios? The
ratio of the heights of the people is
the same as the ratio of the heights
of the images.
• What do you call an equation
containing two ratios? a
proportion
are similar triangles related to photography?
When you take a
picture, the image
of the object being
photographed is
projected by the camera
lens onto the film. The
height of the image on
the film can be related to
the height of the object
using similar triangles.
Film
Lens
Distance from lens to object
Distance
from lens to film
SIMILAR TRIANGLES Similar triangles have the same shape, but not
necessarily the same size. There are two main tests for similarity.
• If the angles of one triangle and the corresponding angles of a second triangle
have equal measures, then the triangles are similar.
• If the measures of the sides of two triangles form equal ratios, or are proportional,
then the triangles are similar.
Study Tip
The triangles below are similar. This is written as ABC DEF. The vertices
of similar triangles are written in order to show the corresponding parts.
Reading Math
E
The symbol is read
is similar to.
B
2 cm
A
4 cm
83˚ 2.5 cm
56˚ 41˚
C
3 cm
D
corresponding angles
83˚
5 cm
56˚
41˚
F
6 cm
corresponding sides
2
1
AB
A
B
and D
E
→ A and D
4
2
DE
BC
2.5
1
B
C
and E
F → EF
5
2
AC
3
1
A
C
and D
F → DF
6
2
B and E
C and F
Similar Triangles
Study Tip
Reading Math
Arcs are used to show
angles that have equal
measures.
• Words
If two triangles are similar, then the measures of their corresponding
sides are proportional, and the measures of their corresponding
angles are equal.
• Symbols
If ABC DEF,
• Model
AB
BC
AC
then .
DE
EF
DF
B
E
C
A
D
616
Chapter 11 Radical Expressions and Triangles
Resource Manager
Workbook and Reproducible Masters
Chapter 11 Resource Masters
• Study Guide and Intervention, pp. 673–674
• Skills Practice, p. 675
• Practice, p. 676
• Reading to Learn Mathematics, p. 677
• Enrichment, p. 678
• Assessment, p. 700
Parent and Student Study Guide
Workbook, p. 88
Prerequisite Skills Workbook,
pp. 61–62
Transparencies
5-Minute Check Transparency 11-6
Answer Key Transparencies
Technology
Interactive Chalkboard
F
Example 1 Determine Whether Two Triangles Are Similar
2 Teach
Determine whether the pair of triangles is similar. Justify your answer.
Remember that the sum of the measures of the
angles in a triangle is 180°.
N
SIMILAR TRIANGLES
The measure of P is 180° (51° 51°) or 78°.
M
In MNO, N and O have the same measure.
Let x the measure of N and O.
x x 78° 180°
2x 102°
x 51°
In-Class Example
78˚
P
Teaching Tip
Tell students to
look closely at all the angles and
sides of the two triangles for
clues about whether the triangles
are similar. In Example 1, the
same angles are not marked on
both triangles, but there is sufficient information to determine
that the angles are congruent.
O
51˚
R
Power
Point®
51˚
Q
So N 51° and O 51°. Since the corresponding angles have equal measures,
MNO PQR.
1 Determine whether the pair
FIND UNKNOWN MEASURES Proportions can be used to find the
of triangles is similar. Justify
your answer.
measures of the sides of similar triangles when some of the measurements
are known.
Y
15
Example 2 Find Missing Measures
Study Tip
Corresponding
Vertices
Always use the
corresponding order
of the vertices to
write proportions
for similar triangles.
Find the missing measures if each pair of triangles below is similar.
X
a. Since the corresponding angles have equal measures, TUV WXY.
The lengths of the corresponding sides are proportional.
10
XY Corresponding sides of similar
WX
UV triangles are proportional.
TU
a
16
WX a, XY 16, TU 3, UV 4
3
4
4a 48
a 12
4b 96
b 24
U
3
98˚
47˚
6
T
Find the cross products.
4
98˚
a
47˚
W
51˚
b
Y
Divide each side by 4.
FIND UNKNOWN
MEASURES
a.
E
B
10
A
E
6m
xm
G
D
y
18
77
26
77
9m
I
D
18
27
77
The missing measure is 15.
www.algebra1.com/extra_examples
C
8
C
BE 10, CD x, AE 6, AD 9
Divide each side by 6.
Power
Point®
each pair of triangles below
is similar.
Corresponding sides of similar
triangles are proportional.
15 x
C
16
2 Find the missing measures if
b. ABE ACD
Find the cross products.
8
In-Class Example
Find the cross products.
90 6x
B
16
The missing measures are 12 and 24.
BE
AE
CD
AD
10
6
x
9
Z
24
The corresponding sides of the
triangles are proportional, so the
triangles are similar.
V
X
Divide each side by 4.
XY Corresponding sides of similar
WY
UV triangles are proportional.
TV
b
16
WY b, XY 16, TV 6, UV 4
6
4
A
12
26
77
H
x
The missing measures are 27
and 12.
Lesson 11-6 Similar Triangles 617
b.
Z
10
Y
4
X
cm
cm
3 cm W
a cm
V
The missing measure is 7.5.
Lesson 11-6 Similar Triangles 617
In-Class Example
Example 3 Use Similar Triangles to Solve a Problem
Power
Point®
SHADOWS Jenelle is standing near the
Washington Monument in Washington, D.C.
The shadow of the monument is 302.5 feet,
and Jenelle’s shadow is 3 feet. If Jenelle is
5.5 feet tall, how tall is the monument?
3 SHADOWS Richard is
standing next to the General
Sherman Giant Sequoia tree in
Sequoia National Park. The
shadow of the tree is
22.5 meters, and Richard’s
shadow is 53.6 centimeters. If
Richard’s height is 2 meters,
how tall is the tree?
The tree is about 84 m tall.
3 Practice/Apply
Note: Not drawn
to scale
x ft
The shadows form similar triangles. Write a
proportion that compares the heights of the
objects and the lengths of their shadows.
5.5 ft
Washington
Monument
Let x the height of the monument.
Jenelle’s shadow →
monument’s shadow →
The monument has a
shape of an Egyptian
obelisk. A pyramid made
of solid aluminum caps
the top of the monument.
3
5.5
302.5
x
3 ft
302.5 ft
← Jenelle’s height
← monument’s height
3x 1663.75
Cross products
x 554.6 feet
Divide each side by 3.
The height of the monument is about 554.6 feet.
Study Notebook
Have students—
• add the definitions/examples of
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 11.
• include examples of how to
determine whether two triangles
are similar, and how to find
unknown measures using triangles.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
Concept Check
1. If the measures of
the angles of one
triangle equal the
measures of the
corresponding angles
of another triangle,
and the lengths of the
sides are proportional,
then the two triangles
are similar.
2. OPEN ENDED Draw a pair of similar triangles. List the corresponding angles
and the corresponding sides. See pp. 639A–639B.
3. FIND THE ERROR Russell and Consuela are comparing the similar triangles
below to determine their corresponding parts. See margin.
Russell
ΩX = ΩT
ΩY = ΩU
ΩZ = ΩV
πXYZ , πTUV
Y
Consuela
}X = }V
}Y = }U
}Z = }T
X
Z
U
πXYZ , πVUT
T
V
Who is correct? Explain your reasoning.
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
4, 5
6–9
10
1
2
3
FIND THE ERROR
Remind students
to pay close attention
to the arcs that denote which
angles are congruent.
Determine whether each pair of triangles is similar. Justify your answer.
4.
No; the angle
measures are
not equal.
84˚
46˚
5.
35˚
55˚
Yes; the angle
measures are
equal.
46˚
For each set of measures given, find the measures of the missing sides if
ABC DEF.
6. c 15, d 7, e 9, f 5 a 21, b 27
7. a 18, c 9, e 10, f 6 b 15, d 12
25
30
8. a 5, d 7, f 6, e 5 b , c 7
7
9. a 17, b 15, c 10, f 6 d 10.2, A
Answers
3. Consuela; the arcs indicate which
angles correspond. The vertices of
the triangles are written in order
to show the corresponding parts.
1. Explain how to determine whether two triangles are similar.
618
Chapter 11 Radical Expressions and Triangles
e9
B
c
E
a
f
b
C D
Differentiated Instruction
Naturalist Have students use the method in Example 3 to find the
heights of trees that are native to your area. Make sure students record
the location and type of tree along with the height. Students will need
tape measures and will need to take measurements on a sunny day.
618
Chapter 11 Radical Expressions and Triangles
d
e
F
Application
10. SHADOWS If a 25-foot flagpole casts a shadow that is 10 feet long and the nearby
school building casts a shadow that is 26 feet long, how high is the building? 65 ft
Organization by Objective
• Similar Triangles: 11–16
• Find Unknown Measures:
17–32
★ indicates increased difficulty
Practice and Apply
Homework Help
For
Exercises
See
Examples
11–16
17–24
25–32
1
2
3
Determine whether each pair of triangles is similar. Justify your answer.
11. yes
21˚
12.
54˚
no
21˚
105˚
13. no
42˚
60˚
50˚
Extra Practice
See page 845.
14. yes
15.
44˚
no
50˚
16.
88˚
yes
55˚ 55˚
x
45˚
56˚
Odd/Even Assignments
Exercises 11–24 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
60˚
50˚
60˚
79˚
45˚
45˚
Assignment Guide
2x
80˚
Basic: 11–27 odd, 33–61
Average: 11–27 odd, 29, 30,
33–61
Advanced: 12–28 even, 29–55
(optional: 56–61)
All: Practice Quiz 2 (1–10)
11–16. See margin for justifications.
For each set of measures given, find the measures of the missing sides if
KLM NOP.
17. k 9, n 6, o 8, p 4 ᐉ 12, m 6
L
18. k 24, ᐉ 30, m 15, n 16 o 20, p 10
55
22
19. m 11, p 6, n 5, o 4 k , ᐉ 6 112
3 84
20. k 16, ᐉ 13, m 12, o 7 n , p 13
13
21. n 6, p 2.5, ᐉ 4, m 1.25 k 3, o 8
m
k
K
Answers
M
l
O
22. p 5, k 10.5, ᐉ 15, m 7.5 n 7, o 10
p
23. n 2.1, ᐉ 4.5, p 3.2, o 3.4 k 2.8, m 4.2
N
24. m 5, k 12.6, o 8.1, p 2.5 ᐉ 16.2, n 6.3
About the Exercises …
n
P
o
25. Determine whether the following statement is sometimes, always, or never true.
If the measures of the sides of a triangle are multiplied by 3, then the measures of the
angles of the enlarged triangle will have the same measures as the angles of the original
triangle. always
11. The angle measures are equal.
12. The angle measures are not equal.
13. The angle measures are not equal.
14. The angle measures are equal.
15. The angle measures are not equal.
16. The angle measures are equal.
26. PHOTOGRAPHY Refer to the diagram of a camera at the beginning of the
lesson. Suppose the image of a man who is 2 meters tall is 1.5 centimeters tall
on film. If the film is 3 centimeters from the lens of the camera, how far is the
man from the camera? 4 m
1
3
27. 3 in.
27. BRIDGES Truss bridges use triangles in their support beams. Mark plans to
make a model of a truss bridge in the scale 1 inch 12 feet. If the height of the
triangles on the actual bridge is 40 feet, what will the height be on the model?
28. BILLIARDS Lenno is playing billiards on
a table like the one shown at the right. He
wants to strike the cue ball at D, bank it
at C, and hit another ball at the mouth
of pocket A. Use similar triangles to find
where Lenno’s cue ball should strike
the rail. 24 in. from pocket B
A
84 in.
B
x
42 in.
D
10 in.
28 in.
C
E
F
Lesson 11-6 Similar Triangles 619
Lesson 11-6 Similar Triangles 619
NAME ______________________________________________ DATE
CRAFTS For Exercises 29 and 30, use the following information.
Melinda is working on a quilt pattern containing isosceles triangles whose sides
measure 2 inches, 2 inches, and 2.5 inches.
____________ PERIOD _____
Study
Guide
andIntervention
Intervention,
11-6
Study
Guide and
Similar Triangles RST is similar to XYZ. The angles of the
two triangles have equal measure. They are called corresponding
angles. The sides opposite the corresponding angles are called
corresponding sides.
Z
X
30
S
60
60
Y
30
R
T
E
If two triangles are similar, then the
measures of their corresponding sides
are proportional and the measures of
their corresponding angles are equal.
Similar Triangles
ABC DEF
B
Lesson 11-6
p. 673
and p. 674
Similar(shown)
Triangles
★ 29. She has several square pieces of material that measure 4 inches on each side.
From each square piece, how many triangles with the required dimensions
can she cut? 8
D
F
AB
BC
AC
DE
EF
DF
★ 30. She wants to enlarge the pattern to make similar triangles for the center of
A
C
Example 1
S
Determine whether the pair
of triangles is similar. Justify your answer.
F
90
X
G
45
R 75
50
45
T
the quilt. What is the largest similar triangle she can cut from the square
material? 4 by 4 by 5 in.
Example 2
Determine whether
the pair of triangles is similar.
Justify your answer.
H
I
45
45
E
89 Y
MIRRORS For Exercises 31 and 32,
use the diagram and the following
information.
Viho wanted to measure the height of
a nearby building. He placed a mirror
on the pavement at point P, 80 feet
from the base of the building. He then
backed away until he saw an image of
the top of the building in the mirror.
J
Z
The measure of G 180° (90° 45°) 45°.
The measure of I 180° (45° 45°) 90°.
Since corresponding angles have equal
measures, EFG HIJ.
Since corresponding angles do not have
the equal measures, the triangles are
not similar.
Exercises
Determine whether each pair of triangles is similar. Justify your answer.
1.
2.
120
3.
90
30
45
30
30
60
60
Yes; corresponding
angles have equal
measures.
No; corresponding
angles do not have
equal measures.
4.
Yes; corresponding
angles have equal
measures.
5.
40
6.
45
30
Yes; corresponding
angles have equal
measures.
★ 31. If Viho is 6 feet tall and he is standing 9 feet from the mirror, how tall is the
building? about 53 ft
120
110
30
20
80
45
30 115
55
No; corresponding
angles do not have
equal measures.
No; corresponding
angles do not have
equal measures.
NAME ______________________________________________ DATE
____________ PERIOD _____
Skills
Practice,
p. 675 and
11-6
Practice
(Average)
Practice,
p. 676 (shown)
Similar Triangles
Determine whether each pair of triangles is similar. Justify your answer.
1.
2.
U
D
P
R
Q
56
47
59
S
G
80
C
31
T
47
F
E H
Yes; Q T 90°; P 180° (90° 31°) 59° S;
U 180° (90° 59°) 31° R. Since the corresponding
angles have equal measures,
PQR STU.
No; C 180° (47° 80°) 53°.
Since FGH has a 56° angle, but
CDE does not, corresponding
angles do not all have equal
measures, and the triangles are not
similar.
For each set of measures given, find the measures
of the missing sides if ABC DEF.
E
f
3. c 4, d 12, e 16, f 8 a 6, b 8
D
B
d
c
F
e
A
a
b
C
4. e 20, a 24, b 30, c 15 d 16, f 10
★ 32. What assumptions did you make in solving the problem?
32. Viho’s eyes are
6 feet off the ground,
Viho and the building
CRITICAL THINKING For Exercises 33–35, use the following information.
each create right
The radius of one circle is twice the radius of another. 34–35. See margin.
angles with the
33.
Are the circles similar? Explain your reasoning.
ground, and the
two angles with
34. What is the ratio of their circumferences? Explain your reasoning.
the ground at P have
35. What is the ratio of their areas? Explain your reasoning.
equal measure.
33. Yes; all circles are
similar because they
36. WRITING IN MATH Answer the question that was posed at the beginning of
have the same shape.
the lesson. See margin.
How are similar triangles related to photography?
5. a 10, b 12, c 6, d 4 e 4.8, f 2.4
Include the following in your answer:
• an explanation of the effect of moving a camera with a zoom lens closer to the
object being photographed, and
• a description of what you could do to fit the entire image of a large object on
the picture.
8
3
6. a 4, d 6, e 4, f 3 c 2, b 15
2
7. b 15, d 16, e 20, f 10 a 12, c 32
3
44
3
8. a 16, b 22, c 12, f 8 d , e 5
2
33
14
11
2
35
6
9. a , b 3, f , e 7 c , d 10. c 4, d 6, e 5.625, f 12 a 2, b 1.875
11. SHADOWS Suppose you are standing near a building and you want to know its height.
The building casts a 66-foot shadow. You cast a 3-foot shadow. If you are 5 feet 6 inches
tall, how tall is the building? 121 ft
Standardized
Test Practice
12. MODELS Truss bridges use triangles in their support beams. Molly made a model of a
truss bridge in the scale of 1 inch 8 feet. If the height of the triangles on the model is
4.5 inches, what is the height of the triangles on the actual bridge? 36 ft
NAME ______________________________________________ DATE
For Exercises 37 and 38, use the figure at
the right.
37. Which statement is true? D
____________ PERIOD _____
ABC ADC
ABC ACD
ABC CAD
none of the above
A
Reading
11-6
Readingto
to Learn
Learn MathematicsELL
Mathematics,
p. 677
Similar Triangles
Pre-Activity
B
C
How are similar triangles related to photography?
Read the introduction to Lesson 11-6 at the top of page 616 in your textbook.
D
How would you describe the shapes and sizes of the figures in the diagram?
Sample answer: The figures have the same shape, but they do
not have the same size.
Reading the Lesson
1. In similar triangles, the angles of the two triangles can be matched so that
angles have equal
measures
triangle, then the two angles are not
similar
C
.
3. If two triangles have the same size and shape, then the measures of the corresponding
proportional
AB DC
AC BC
A
.
2. If the angles of one triangle do not have the same measures as the angles of a second
sides are
620
Chapter 11 Radical Expressions and Triangles
.
Determine whether each pair of triangles is similar. Explain how you know that
your answer is correct.
Not similar; the angles cannot be matched so that
corresponding angles have the same measures.
4.
58
30
NAME ______________________________________________ DATE
11-6
Enrichment
Enrichment,
32
____________ PERIOD _____
p. 678
60
Not similar; the sides cannot be matched to
have corresponding sides proportional.
5.
7.5
6
3
8
6.
Similar; the angles can be matched so that
corresponding angles have equal measures.
125
35
35
125
A Curious Construction
Many mathematicians have been interested in
ways to construct the number . Here is one such
geometric construction.
3.5
4
20
A
20
Helping You Remember
7. How can you use the idea that the corresponding sides of similar triangles are
proportional to help you remember how to find the unknown lengths of the sides of
similar triangles? Sample answer: Since the corresponding sides of similar
C
7
–
8
In the drawing, triangles ABC and ADE are right
triangles. The length of A
D
equals the length of AC
and F
B
is parallel to EG
.
G
gives a decimal approximation
The length of B
of the fractional part of to six decimal places.
.
Follow the steps to find the length of BG
1. Use the length of B
C
and the Pythagorean
Theorem to find the length of AC
.
AC triangles are proportional, use a proportion to find the lengths of the
unknown sides.
1 1.3287682
2
7 2
8
2. Find the length of A
D
.
AD
620 Chapter 11 Radical Expressions and Triangles
AC
1 3287682
F
E
1
–
2
A
1
B G
D
C
D
38. Which statement is always true? A
Complete each sentence.
corresponding
B
B
D
CB AD
AC AB
Maintain Your Skills
Mixed Review
Find the distance between each pair of points whose coordinates are given.
Express answers in simplest radical form and as decimal approximations
rounded to the nearest hundredth if necessary. (Lesson 11-5)
40. (6, 3), (12, 5) 10
39. (1, 8), (2, 4) 5
41. (4, 7), (3, 12)
26
5.1
42. 1, 56 , 6, 76 7
Determine whether the following side measures form right triangles. Justify
your answer. (Lesson 11-4)
43. 25, 60, 65 Yes; 252 602 652.
45. 49, 168, 175 Yes;
492
1682
1752.
44. 20, 25, 35 No; 202 252 352.
46. 7, 9, 12 No; 72 92 122.
Arrange the terms of each polynomial so that the powers of the variable are in
descending order. (Lesson 8-4) 48. 5x3 2x2 4x 7
47. 1 3x2 7x 3x2 7x 1
48. 7 4x 2x2 5x3
49. 6x 3 3x2 3x2 6x 3
50. abx2 bcx 34 x7
x7 abx2 bcx 34
Use elimination to solve each system of equations. (Lesson 7-3)
51. 2x y 4 (3, 2)
xy5
52. 3x 2y 13 (5, 1)
2x 5y 5
53. 0.6m 0.2n 0.9 (1.5, 0)
0.3m 0.45 0.1n
1
1
54. x y 8 (6, 12)
3
2
1
1
x y 0
2
4
55. AVIATION An airplane passing over Sacramento at an elevation of 37,000 feet
begins its descent to land at Reno, 140 miles away. If the elevation of Reno is
4500 feet, what should be the approximate slope of descent?
(Hint: 1 mi 5280 ft) (Lesson 5-1) 232 ft/mi or about 0.044
Getting Ready for
the Next Lesson
PREREQUISITE SKILL Evaluate if a 6, b 5, and c 1.5.
(To review evaluating expressions, see Lesson 1-2.)
a
56. 4
c
ac 9
59. b 5
or 1.8
5
b
57. or 0.83
a
6
10
b
60. or 1.1
ac
9
P ractice Quiz 2
2
ab
58. or 0.6
c
3
1
c
61. or 0.3
ac
3
Lessons 11-4 through 11-6
If c is the measure of the hypotenuse of a right triangle, find each missing measure.
If necessary, round to the nearest hundredth. (Lesson 11-4)
1. a 14, b 48, c ? 50
2. a 40, c 41, b ? 9
3. b 8, c 84
, a ? 25 4.47
4. a 5, b 8, c ?
4 Assess
Open-Ended Assessment
Modeling Construct a triangle
using unsharpened pencils, uncooked spaghetti or other items
that have a uniform length. Have
students construct a triangle
similar to yours and explain how
they would show that the two
triangles are similar.
Getting Ready for
Lesson 11-7
PREREQUISITE SKILL Students
will learn about trigonometric
ratios in Lesson 11-7. In order to
compute trigonometric ratios,
students must be able to divide
rational numbers. Use Exercises
56–61 to determine your students’
familiarity with dividing rational
numbers.
Assessment Options
Practice Quiz 2 The quiz
provides students with a brief
review of the concepts and skills
in Lessons 11-4 through 11-6.
Lesson numbers are given to the
right of exercises or instruction
lines so students can review
concepts not yet mastered.
Quiz (Lessons 11-5 and 11-6) is
available on p. 700 of the Chapter
11 Resource Masters.
3.61
13
Find the distance between each pair of points whose coordinates are given. (Lesson 11-5)
17.49
306
22 2.83
5. (6, 12), (3, 3)
6. (1, 3), (5, 11) 10
7. (2, 5), (4, 7)
8. (2, 9), (5, 4)
178 13.34
Find the measures of the missing sides if BCA EFD. (Lesson 11-6)
9. b 10, d 7, e 2, f 3 a 35, c 15
10. a 12, c 9, d 8, e 12 b 18, f 6
www.algebra1.com/self_check_quiz
Lesson 11-6 Similar Triangles 621
Answers
34. 2:1; Let the first circle have radius r and the larger have
radius 2r. The circumference of the first is 2r and the other
has circumference 2(2r ) 4r.
35. 4:1; The area of the first is r 2 and the area of the other is
(2r)2 4r 2.
36. The size of an object on the film of a camera can be related to
its actual size using similar triangles. Answers should include
the following.
• Moving the lens closer to the object (and farther from the
film) makes the object appear larger.
• Taking a picture of a building; you would need to be a great
distance away to fit the entire building in the picture.
Lesson 11-6 Similar Triangles 621
Algebra
Activity
A Preview of Lesson 11-7
A Preview of Lesson 11-7
Investigating Trigonometric Ratios
Getting Started
You can use paper triangles to investigate trigonometric ratios.
Objective Construct right
triangles with legs whose lengths
are equal to a specific ratio, in
order to investigate the relationship between the sides and the
angles as a preview of trigonometric ratios.
Collect the Data
Materials
ruler
grid paper
protractor
Teach
• Remind students that the
triangles they are drawing are
similar because the ratio of the
corresponding sides is the same
for each triangle.
• Point out to students that because they drew the triangles
by hand, the measures of the
angles may not be exact,
according to the protractor.
However, the measures should
be close to 35°, 55°, and 90°.
Use a ruler and grid paper to draw several right
triangles whose legs are in a 7:10 ratio. Include a
right triangle with legs 3.5 units and 5 units, a
right triangle with legs 7 units and 10 units,
another with legs 14 units and 20 units, and
several more right triangles similar to these three.
Label the vertices of each triangle as A, B, and C,
where C is at the right angle, B is opposite the
longest leg, and A is opposite the shortest leg.
Step 2
Copy the table below. Complete the first three columns by measuring the
hypotenuse (side AB) in each right triangle you created and recording its length.
Step 3
Calculate and record the ratios in the middle two columns. Round to the
nearest tenth, if necessary.
Step 4
Use a protractor to carefully measure angles A and B in each right triangle.
Record the angle measures in the table.
Side Lengths
side BC
side AC
side AB
Ratios
BC:AC
Angle Measures
BC:AB
angle A
angle B
angle C
0.6
0.6
0.6
0.6
0.6
0.6
35°
35°
35°
35°
35°
35°
55°
55°
55°
55°
55°
55°
90°
6.1
7:10
7
10
12.2
7:10
14
20
24.4
7:10
28
40
48.8
7:10
35
50
61
7:10
42
60
73.2
7:10
Sample answers are given in table.
3.5
5
90°
90°
90°
90°
90°
Analyze the Data
Assess
Ask students to conjecture why
knowing the relationship
between the ratios of the sides of
a triangle to each other and the
angles might be useful. Sample
answer: Knowing these relationships
might allow you to calculate a missing side length or angle measure.
Step 1
1. Examine the measures and ratios in the table. What do you notice? Write a
sentence or two to describe any patterns you see.
All ratios and angle measures are the same for any 7:10 right triangle.
Make a Conjecture
2. For any right triangle similar to the ones you have drawn here, what will be the
value of the ratio of the length of the shortest leg to the length of the longest leg? 7:10
3. If you draw a right triangle and calculate the ratio of the length of the shortest
leg to the length of the hypotenuse to be approximately 0.573, what will be the
measure of the larger acute angle in the right triangle? 55°
622 Investigating Slope-Intercept Form
622 Chapter 11 Radical Expressions and Triangles
Resource Manager
Teaching Algebra with
Manipulatives
Glencoe Mathematics Classroom
Manipulative Kit
• p. 1 (master for grid paper)
• p. 23 (master for protractors)
• p. 24 (master for rulers)
• p. 191 (student recording sheet)
• coordinate grid stamp
• protractors
• rulers
622 Chapter 11 Radical Expressions and Triangles
Lesson
Notes
Trigonometric Ratios
• Define the sine, cosine, and tangent ratios.
1 Focus
• Use trigonometric ratios to solve right triangles.
Vocabulary
•
•
•
•
•
•
•
trigonometric ratios
sine
cosine
tangent
solve a triangle
angle of elevation
angle of depression
are trigonometric ratios used in surveying?
5-Minute Check
Transparency 11-7 Use as
a quiz or review of Lesson 11-6.
Surveyors use triangle ratios called
trigonometric ratios to determine
distances that cannot be measured
directly.
Mathematical Background notes
are available for this lesson on
p. 584D.
• In 1852, British surveyors
measured the altitude of the peak
of Mt. Everest at 29,002 feet using
these trigonometric ratios.
• In 1954, the official height
became 29,028 feet, which was
also calculated using surveying
techniques.
• On November 11, 1999, a team using advanced technology and the Global
Positioning System (GPS) satellite measured the mountain at 29,035 feet.
TRIGONOMETRIC RATIOS Trigonometry is an area of mathematics that
involves angles and triangles. If enough information is known about a right
triangle, certain ratios can be used to find the measures of the remaining parts
of the triangle. Trigonometric ratios are ratios of the measures of two sides of a
right triangle. Three common trigonometric ratios are called sine, cosine,
and tangent.
Trigonometric Ratios
• Words
measure of leg opposite A
measure of hypotenuse
sine of A measure of leg adjacent to A
cosine of A Study Tip
measure of hypotenuse
Reading Math
Notice that sine, cosine,
and tangent are
abbreviated sin, cos,
and tan respectively.
• Symbols
measure of leg opposite A
tangent of A measure of leg adjacent to A
BC
sin A AB
AC
cos A AB
BC
tan A AC
B
• Model
leg
opposite
hypotenuse
A
A
leg adjacent
A
are trigonometric ratios
used in surveying?
Ask students:
• In the previous lesson, you
calculated the height of the
Washington Monument using
ratios. What is required in
order to make this calculation?
Two similar triangles formed by the
shadow of the monument and the
shadow of a person standing next
to it. You know the length of both
shadows and the height of the person. You can use ratios to calculate the height of the monument.
• Based on the Algebra Activity
on the previous page, how do
you suppose a surveyor could
calculate the height of a
mountain? Sample answer: A
surveyor could use the angle between him or herself and the top of
the mountain, and the distance to
the mountain to construct a similar
right triangle. Then based on the
side measures of that similar
triangle, the height of the mountain
could be calculated using ratios.
C
Lesson 11-7 Trigonometric Ratios
623
Resource Manager
Workbook and Reproducible Masters
Chapter 11 Resource Masters
• Study Guide and Intervention, pp. 679–680
• Skills Practice, p. 681
• Practice, p. 682
• Reading to Learn Mathematics, p. 683
• Enrichment, p. 684
• Assessment, p. 700
Parent and Student Study Guide
Workbook, p. 89
Teaching Algebra With Manipulatives
Masters, pp. 23, 24, 192
Transparencies
5-Minute Check Transparency 11-7
Answer Key Transparencies
Technology
Interactive Chalkboard
Lesson x-x Lesson Title 623
Example 1 Sine, Cosine, and Tangent
2 Teach
Find the sine, cosine, and tangent of each acute angle
of RST. Round to the nearest ten thousandth.
TRIGONOMETRIC RATIOS
The trigonometric ratios are
a concept that
students will
use in other
classes such as Geometry,
Algebra II, and Trigonometry.
Any time students are introduced to such important concepts, have them record the
concepts in their Study Notebooks for future reference.
Additionally, the act of recording the concepts will help the
students remember them.
18
17
Write each ratio and substitute the measures. Use a
calculator to find each value.
S 35 T
New
In-Class Example
R
opposite leg
hypotenuse
adjacent leg
hypotenuse
opposite leg
adjacent leg
cos R sin R 35
or 0.3287
18
opposite leg
hypotenuse
tan R adjacent leg
hypotenuse
17
opposite leg
adjacent leg
cos T sin T 35
or 0.3480
17
18
or 0.9444
tan T 35
or 0.3287
17
18
or 0.9444
17
or 2.8735
35
18
You can use a calculator to find the values of trigonometric functions or to find
the measure of an angle. On a graphing calculator, press the trigometric function
key, and then enter the value. On a nongraphing scientific calculator, enter the value,
and then press the function key. In either case, be sure your calculator is in degree
mode. Consider cos 50°.
Graphing Calculator
COS 50 ENTER .6427876097
Power
Point®
Nongraphing Scientific Calculator
KEYSTROKES: 50 COS
.642787609
KEYSTROKES:
Teaching Tip
While it is true
that a fraction with radicals in the
denominator is not in simplest
form, it is not necessary to
rationalize the denominator if
students are going to use a calculator to simplify the expression.
Suggest that students use a cal10
3
Example 2 Find the Sine of an Angle
Find sin 35° to the nearest ten thousandth.
KEYSTROKES:
SIN 35 ENTER
.5735764364
Rounded to the nearest ten thousandth, sin 35° 0.5736.
30
3
culator to find and .
Example 3 Find the Measure of an Angle
Both produce the same result,
and rationalization was not
necessary.
Find the measure of J to the nearest degree.
Since the lengths of the opposite and adjacent sides
are known, use the tangent ratio.
1 Find the sine, cosine, and
opposite leg
adjacent leg
6
9
tangent of each acute angle of
DEF. Round to the nearest
ten-thousandth.
9
KL 6 and JK 9
L
Now use the TAN on a calculator to find the
6
measure of the angle whose tangent ratio is .
–1
9
19
KEYSTROKES: 2nd
[TAN1] 6 ⫼ 9 ENTER
33.69006753
To the nearest degree, the measure of J is 34°.
E
F
83
sin D 0.7293; cos D 0.6842;
tan D 1.0659; sin F 0.6842;
cos F 0.7293; tan F 0.9382
624
Chapter 11 Radical Expressions and Triangles
Differentiated Instruction
Auditory/Musical Divide the class into small groups. Challenge each
group to devise a rap verse, limerick, poem, or a song that they can use
to remember the trigonometric ratios.
624
Chapter 11 Radical Expressions and Triangles
K
6
tan J Definition of tangent
D
13
J
SOLVE TRIANGLES You can find the missing measures of a right triangle if
you know the measure of two sides of a triangle or the measure of one side and
one acute angle. Finding all of the measures of the sides and the angles in a right
triangle is called solving the triangle .
Example 4 Solve a Triangle
Find all of the missing measures in ABC.
You need to find the measures of B, A
C
, and B
C
.
A
Step 1 Find the measure of B. The sum of the
measures of the angles in a triangle is 180.
38˚
y in.
180° 90° 38° 52°
Study Tip
Verifying Right
Triangles
You can use the
Pythagorean Theorem to
verify that the sides are
sides of a right triangle.
C
x in.
B
If students have
trouble understanding the concept of the inverse of a trigono6
9
B
C
is about 7.4 inches long.
Step 3 Find the value of y, which is the measure of the side adjacent to A.
Use the cosine ratio.
9.5 y
Teaching Tip
metric function, write 4J on
Evaluate sin 38°.
Multiply by 12.
y
12
y
0.7880 12
To make sure that
their calculators are in degree
mode, tell students to press
MODE and then look at the
third row to see whether Radian
or Degree is highlighted. If Radian
is highlighted, students should
use the cursor to select Degree,
then press ENTER .
2 Find cos 65° to the nearest
sin 38° Definition of sine
cos 38° Teaching Tip
ten thousandth. 0.4226
Step 2 Find the value of x, which is the measure of
the side opposite A. Use the sine ratio.
7.4 x
Power
Point®
12 in.
The measure of B is 52°.
x
12
x
0.6157 12
In-Class Examples
Definition of cosine
the chalkboard. Ask students to
explain how you would solve
this problem. They will say to
divide both sides by 4. Point out
that dividing by 4 is the inverse
operation of multiplying J by 4.
Therefore, to solve for J in the
6
9
Evaluate cos 38°.
equation tan J , you perform
Multiply by 12.
the inverse of the tangent
operation on both sides of the
equation.
A
C
is about 9.5 inches long.
So, the missing measures are 52°, 7.4 in., and 9.5 in.
3 Find the measure of B to
Trigonometric ratios are often used to find distances or lengths that cannot be
measured directly. In these situations, you will sometimes use an angle of elevation
or an angle of depression. An angle of elevation is formed by a horizontal line of
sight and a line of sight above it. An angle of depression is formed by a horizontal
line of sight and a line of sight below it.
the nearest degree.
12
A
B
20
C
To the nearest degree, the
measure of B is 53°.
line of sight
angle of
depression
angle of
elevation
SOLVE TRIANGLES
In-Class Example
www.algebra1.com/extra_examples
Power
Point®
4 Find all of the missing
Lesson 11-7 Trigonometric Ratios 625
measures in DEF.
D
15 cm
x cm
E
62
y cm
F
The missing measures are
mF 28°, x 8 cm, and
y 17 cm.
Lesson 11-7 Trigonometric Ratios 625
Power
Point®
Make a Hypsometer
Teaching Tip
Problems
involving angles of depression
are solved in the same manner
as problems with angles of
elevation. It is just a different
orientation of the angle.
110
170
120
130 140 150 160
• Tie one end of a piece of string to the middle of
a straw. Tie the other end of string to a paper clip.
• Tape a protractor to the side of the straw. Make
Straw
sure that the string hangs freely to create a
vertical or plumb line.
• Find an object outside that is too tall to measure
90
directly, such as a basketball hoop, a flagpole, or
the school building.
String
• Look through the straw to the top of the object
you are measuring. Find the angle measure where
Paper Clip
the string and protractor intersect. Determine the
angle of elevation by subtracting this
measurement from 90°.
• Measure the distance from your eye level to the ground and from your foot
to the base of the object you are measuring.
10
5 INDIRECT MEASUREMENT
In the diagram, Barone is
flying his model airplane
400 feet above him. An angle
of depression is formed by a
horizontal line of sight and a
line of sight below it. Find
the angles of depression at
points A and B.
400 ft
10
0
In-Class Example
20
30
40 50 60 70
80
Analyze 1. See students’ work.
1. Make a sketch of your measurements. Use the equation
130 ft
B
A
height of object x
tan (angle of elevation) , where x represents distance
distance of object
from the ground to your eye level, to find the height of the object.
400 ft
The angle of depression at point
A is 45° and the angle of
depression at point B is 37°.
2. The angle measured
by the hypsometer
is not the angle of
elevation. It is the
other acute angle
formed in the triangle.
So, to find the measure
of the angle of
elevation, subtract
the reading on the
hypsometer from
90 since the sum of
the two acute angles
in a right triangle is 90.
2. Why do you have to subtract the angle measurement on the hypsometer
from 90° to find the angle of elevation?
3. Compare your answer with someone who measured the same object. Did
your heights agree? Why or why not? See students’ work.
Example 5 Angle of Elevation
INDIRECT MEASUREMENT At point A, Umeko measured the angle of elevation
to point P to be 27 degrees. At another point B, which was 600 meters closer to
the cliff, Umeko measured the angle of elevation to point P to be 31.5 degrees.
Determine the height of the cliff.
Explore
Draw a diagram to model the
situation. Two right triangles,
BPC and APC, are formed.
You know the angle of elevation
for each triangle. To determine
the height of the cliff, find the
length of P
C
, which is shared
by both triangles.
y
27˚ B
A
31.5˚
600 m
xm
Let y represent the distance from the top of the cliff P to its base C.
Let x represent BC in the first triangle and let x 600 represent AC.
Solve
Write two equations involving the tangent ratio.
y
x
and
x tan 31.5° y
y
600 x
tan 27° (600 x)tan 27° y
Chapter 11 Radical Expressions and Triangles
Algebra Activity
Materials string, drinking straw, paper clip, protractor, tape, meter sticks or
tape measure
Remind students that the angle of elevation is being measured from their eye
level, not the ground. They must account for this distance when calculating
the height of the object that they are measuring with their hypsometer.
626
Chapter 11 Radical Expressions and Triangles
C
Plan
tan 31.5° 626
P
Since both expressions are equal to y, use substitution to solve for x.
x tan 31.5° (600 x) tan 27°
Substitute.
3 Practice/Apply
x tan 31.5° 600 tan 27° x tan 27° Distributive Property
x tan 31.5° x tan 27° 600 tan 27°
Subtract.
x(tan 31.5° tan 27°) 600 tan 27°
Isolate x.
600 tan 27°
x tan 31.5° tan 27°
Divide.
x 2960 feet
Use a calculator.
Use this value for x and the equation x tan 31.5° y to solve for y.
x tan 31.5° y Original equation
2960 tan 31.5° y
Replace x with 2960.
1814 y Use a calculator.
The height of the cliff is about 1814 feet.
Examine Examine the solution by finding the angles of elevation.
y
x
1814
tan B 2960
y
600 x
1814
tan A 600 2960
tan A tan B B 31.5°
A 27°
Study Notebook
Have students—
• complete the definitions/examples
for the remaining terms on their
Vocabulary Builder worksheets for
Chapter 11.
• include examples of how to derive
the trigonometric ratios from a
right triangle, and how to use the
ratios to solve triangles.
• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
The solution checks.
Answers
Concept Check
2. See margin.
1. Explain how to determine which trigonometric ratio to use when solving for an
unknown measure of a right triangle. See margin.
2. OPEN ENDED Draw a right triangle and label the measure of the hypotenuse
and the measure of one acute angle. Then solve for the remaining measures.
3. Compare the measure of the angle of elevation and the measure of the angle of
depression for two objects. What is the relationship between their measures?
They are equal.
Guided Practice
GUIDED PRACTICE KEY
Exercises
Examples
4, 5
6–8
9–14
15–17
18
1
2
3
4
5
For each triangle, find sin Y, cos Y, and tan Y to the nearest ten thousandth.
4. Y
27
X
sin Y 0.8,
cos Y 0.6,
tan Y 1.3333
36
45
5.
W
10
26
sin Y 0.3846,
cos Y 0.9231,
tan Y 0.4167
1. If you know the measure of the
hypotenuse, use sine or cosine,
depending on whether you know
the measure of the adjacent side
or the opposite side. If you know
the measures of the two sides,
use tangent.
2. Sample answer:
A 180° (90° 50°) or 40°
AC
10
AC 7.66
X
24
Y
BC 6.43
A
Z
10
Use a calculator to find the value of each trigonometric ratio to the nearest
ten thousandth.
6. sin 60° 0.8660
BC
10
sin 50° cos 50° 7. cos 75° 0.2588
8. tan 10° 0.1763
C
50˚
B
Use a calculator to find the measure of each angle to the nearest degree.
9. sin W 0.9848 80°
10. cos X 0.6157 52°
11. tan C 0.3249 18°
Lesson 11-7 Trigonometric Ratios 627
Lesson 11-7 Trigonometric Ratios 627
For each triangle, find the measure of the indicated angle to the nearest degree.
About the Exercises …
12.
Organization by Objective
• Trigonometric Ratios: 19–51
• Solve Triangles: 52–65
19. sin R 0.6, cos R 0.8,
tan R 0.75
20. sin R 0.3243, cos R 0.9459, tan R 0.3429
21. sin R 0.7241, cos R 0.6897, tan R 1.05
22. sin R 0.4706, cos R 0.8824, tan R 0.5333
23. sin R 0.5369, cos R 0.8437, tan R 0.6364
24. sin R 0.8182, cos R 0.5750, tan R 1.4230
9.7
7
?
6
17°
22°
9.3
Solve each right triangle. State the side lengths to the nearest tenth and the angle
measures to the nearest degree.
15. A
16. C
17.
B
18 m
42 in.
30˚
C
A
B
A 60°, AC 21 in.,
BC 36.4 in.
Application
B
B 35°,
BC 5.7 in.,
AB 7.0 in.
35˚
Assignment Guide
Answers
14.
?
?
Odd/Even Assignments
Exercises 19–60 are structured
so that students practice the
same concepts whether they
are assigned odd or even
problems.
Basic: 19–49 odd, 53–59 odd,
61, 62, 66–79
Average: 19–59 odd, 61–64,
66–79
Advanced: 20–60 even, 63–79
15
13.
33°
13
A 55°, AC 12.6 m,
A
AB 22 m
18. DRIVING The percent grade of a road is the
ratio of how much the road rises or falls in a
given horizontal distance. If a road has a vertical
rise of 40 feet for every 1000 feet horizontal
distance, calculate the percent grade of the road
and the angle of elevation the road makes with
the horizontal. 4% grade, 2.3°
55˚
C
4 in.
Trucks
check brakes
6% grade
★ indicates increased difficulty
Practice and Apply
Homework Help
For
Exercises
See
Examples
19–24
25–33
34–51
52–60
61–65
1
2
3
4
5
19–24. See margin.
For each triangle, find sin R, cos R, and tan R to the nearest ten thousandth.
19.
20. R
T
10
F
35
8
S
Extra Practice
See page 846.
29
20
G
R
R
12
37
6
21.
M
22. R
30
34
P
23.
24.
K
N
21
R
410
16
7
170
O
22
T
18
H
11
R
U
Use a calculator to find the value of each trigonometric ratio to the nearest
ten thousandth.
25. sin 30° 0.5
26. sin 80° 0.9848
27. cos 45° 0.7071
28. cos 48° 0.6691
29. tan 32° 0.6249
30. tan 15° 0.2679
31. tan 67° 2.3559
32. sin 53° 0.7986
33. cos 12° 0.9781
Use a calculator to find the measure of each angle to the nearest degree.
628
628
Chapter 11 Radical Expressions and Triangles
34. cos V 0.5000 60°
35. cos Q 0.7658 40°
36. sin K 0.9781 78°
37. sin A 0.8827 62°
38. tan S 1.2401 51°
39. tan H 0.6473 33°
40. sin V 0.3832 23°
41. cos M 0.9793 12°
42. tan L 3.6541 75°
Chapter 11 Radical Expressions and Triangles
For each triangle, find the measure of the indicated angle to the nearest degree.
44.
45.
14
2
16
10
For each acute
angle of a right triangle, certain ratios of
side lengths are useful. These ratios are
called trigonometric ratios. Three common
ratios are the sine, cosine, and tangent,
as defined at the right.
16
51°
47.
21
Trigonometric Ratios
?
39°
46.
p. 679
(shown)
Trigonometric
Ratios and p. 680
10
?
8°
?
57°
219
?
sin R 0.436
20
18
20
cos R 0.9
219
49.
15
★ 50.
9
45
16
?
tan R 0.484
24
30
?
56°
1.
45˚
C
8 ft
A
B 69°,
AB 13.9 cm,
BC 5 cm
20 in.
B
13 cm
2.
A
13
5
A
9 cm
C
A: 0.4455, 0.8911, 0.5000
B: 0.8911, 0.4455, 2.0000
A
56.
A: 0.9231, 0.3846, 2.4000
B: 0.3846, 0.9231, 0.4167
Use a calculator to find the value of each trigonometric ratio to the nearest ten
thousandth.
4. cos 47° 0.6820
5. tan 48° 1.1106
6. sin A 0.7547 49°
7. tan C 2.3456 67°
8. cos B 0.6947 46°
9. sin A 0.6589 41°
10. tan C 1.9832 63°
11. cos B 0.0136 89°
NAME ______________________________________________ DATE
p. 681 and
Practice,
p. Ratios
682 (shown)
Trigonometric
2. P
Y
3.
25
24
C
In emergency situations,
modern submarines are
built to allow for a rapid
surfacing, a technique
called an emergency main
ballast blow.
58. A 53°,
B 37°,
AB 10 ft
B 50°,
AC 12.3 ft,
BC 10.3 ft
B
A
A
C
B
38˚
C
Source: www.howstuffworks.com
T
sin T 0.8944,
cos T 0.4472,
tan T 2
Use a calculator to find the value of each trigonometric ratio to the nearest ten
thousandth.
24 in.
B 52°,
AC 30.7 in.,
AB 39 in.
4. sin 85° 0.9962
5. cos 5° 0.9962
6. tan 32.5° 0.6371
Use a calculator to find the measure of each angle to the nearest degree.
7. sin D 0.5000 30°
8. cos Q 0.1123 84°
9. tan B 4.7465 78°
For each triangle, find the measure of the indicated angle to the nearest degree.
B
10.
55°
13
11.
72° 12.
?
42
A 30°,
B 60°,
AB 5.2 cm
60. A
6 cm
C
12 ft
A 23°,
B 67°,
AB 13 ft
41°
12
9
B
B
5
8
T
sin T 0.96,
cos T 0.28,
tan T 3.4286
?
59.
A
8
7
R
sin T 0.3243,
cos T 0.9459,
tan T 0.3429
16
C
T
24
74
C
5 ft
8 ft
6 ft
9m
X
40
16
?
Solve each right triangle. State the side lengths to the nearest tenth and the angle
measures to the nearest degree.
13. C
14. A
B
51
15. B
64
C
13 cm
15 f
26 yd
30 ft
C
A
A 39°,
AC 20.2 yd,
BC 16.4 yd
A
B
B 26°,
AC 5.7 cm,
BC 11.7 cm
A 60°,
B 30°,
BC 26.0 ft
HIKING For Exercises 16 and 17, use the following information.
The 10-mile Lower West Rim trail in Mt. Zion National Park ascends 2640 feet.
C
3 cm
16. What is the average angle of elevation of the hike from the canyon floor to the top of the
canyon? (Hint: Draw a diagram in which the length of the trail forms the hypotenuse of
a right triangle. Convert feet to miles.) about 2.9°
B
17. What is the horizontal distance covered on the hike? about 10 mi
SUBMARINES For Exercises 61 and 62, use the following information.
A submarine is traveling parallel to the
surface of the water 626 meters below the
surface. The sub begins a constant ascent
to the surface so that it will emerge on the
?
surface after traveling 4420 meters from
the point of its initial ascent.
18. RAMP DESIGN An engineer designed the entrance to a museum to include a wheelchair
ramp that is 15 feet long and forms a 6° angle with a sidewalk in front of the museum.
How high does the ramp rise? about 1.6 ft
NAME ______________________________________________ DATE
Mathematics,
p. 683
Trigonometric Ratios
626 m
Pre-Activity
ELL
How are trigonometric ratios used in surveying?
Read the introduction to Lesson 11-7 at the top of page 623 in your textbook.
In which years were the measurements of the height of Mt. Everest in
closest agreement?
4420 m
1954 and 1999
61. What angle of ascent did the submarine
make? 8.1°
Reading the Lesson
62. What horizontal distance did the
submarine travel during its ascent? 4375 m
1. Complete each sentence.
N
MR
a. The legs of triangle MNR are segments
b. The hypotenuse of triangle MNR is
www.algebra1.com/self_check_quiz
____________ PERIOD _____
Reading
11-7
Readingto
to Learn
Learn Mathematics
Lesson 11-7 Trigonometric Ratios 629
MN
and
NR
M
.
c. The leg opposite N is
MR
, and the leg adjacent to N is
NR
d. The leg opposite M is
NR
, and the leg adjacent to M is
MR
2. Write K or S to complete each equation.
NAME ______________________________________________ DATE
11-7
Enrichment
Enrichment,
____________ PERIOD _____
p. 684
Modern Art
The painting below, aptly titled, “Right Triangles,” was painted by
that well-known artist Two-loose La-Rectangle. Using the information
below, find the dimensions of this masterpiece. (Hint: The triangle
that includes F is isosceles.)
R
.
.
S
a. tan
S
KT
TS
b. sin
S
c. sin
K
ST
KS
d. cos
S
e. cos
K
KT
KS
f. tan
K
KT
KS
ST
KS
T
ST
KT
3. If you look straight in front of you and then up, the two lines of sight form an angle of
elevation
(depression/elevation). If you look straight in front of you and then
down, the two lines of sight form an angle of
6 ft
.
depression
(depression/elevation).
E
B
Helping You Remember
4. What is one way you can remember the trigonometric ratios sine, cosine, and tangent?
Sample answer: A well-known mnemonic device is SOH-CAH-TOA
(sin opposite/hypotenuse, cos adjacent/hypotenuse,
tan opposite/adjacent).
A
C
Lesson 11-7 Trigonometric Ratios 629
K
Lesson 11-7
Submarines
AC 3.3 m,
AB 9.6 m
20˚
40˚
____________ PERIOD _____
Skills
Practice,
11-7
Practice
(Average)
1.
A 70°, 57. A
B
B
12
For each triangle, find sin T, cos T, and tan T to the nearest ten thousandth.
C
16 ft
T
From part a above
70
55. B
2
19
Use a calculator to find the measure of each angle to the nearest degree.
21˚
A 63°,
AC 9.1 in.,
BC 17.8 in.
S
r 219
, t 18
3. sin 45° 0.7071
54.
27˚
A
20
18
10.1 cm
C
Solve each right triangle. State the side lengths to the nearest tenth and the angle
measures to the nearest degree.
B
R
r 219
, s 20
B
4.5 cm
23
53. C
A 45°,
AB 11.3 ft,
AC 8 ft
a
90 C
b
Find the sine, cosine, and tangent of each acute angle. Round your answers to the
nearest ten thousandth.
7°
52. A
A
b
Exercises
25
37°
tan A a
c
c
Use TAN –1 on a calculator to find the measure of the angle whose tangent ratio is 0.484.
KEYSTROKES: 2nd [TAN –1] .484 ENTER 25.82698212 or about 26°
★ 51.
?
tangent of A
B
c
b. Find the measure of R to the nearest degree.
32°
17
cos A b
t 18, s 20
tan R 0.484
18
?
36°
sin A a
cosine of A
a. Find the sine, cosine, and tangent of R of RST.
Round to the nearest thousandth.
8
5
?
sine of A
Example
48.
21
25
____________ PERIOD _____
Lesson 11-7
43.
NAME ______________________________________________ DATE
Study
Guide
andIntervention
Intervention,
11-7
Study
Guide and
AVIATION For Exercises 63 and 64, use the following information.
Germaine pilots a small plane on weekends. During a recent flight, he determined
that he was flying 3000 feet parallel to the ground and that the ground distance to
the start of the landing strip was 8000 feet.
4 Assess
Open-Ended Assessment
63. What is Germaine’s angle of depression to the start of the landing strip? 20.6°
Writing Ask students to sketch a
right triangle, label the sides and
angles with their measures, and
find the trigonometric ratios for
the two acute angles.
64. What is the distance between the plane in the air and the landing strip on
the ground? 8544 ft
Assessment Options
Quiz (Lesson 11-7) is available
on p. 700 of the Chapter 11
Resource Masters.
★ 65. FARMING Leonard and Alecia are building a new feed storage system on their
farm. The feed conveyor must be able to reach a range of heights. It has a length
of 8 meters, and its angle of elevation can be adjusted from 20° to 5°. Under
these conditions, what range of heights is possible for an opening in the
building through which feed can pass? 2.74 m to 0.7 m
66. See margin.
66. CRITICAL THINKING An important trigonometric identity is sin2 A cos2 A 1.
Use the sine and cosine ratios and the Pythagorean Theorem to prove this identity.
67. WRITING IN MATH
Answer the question that was posed at the beginning of
the lesson. See margin.
How are trigonometric ratios used in surveying?
Include the following in your answer:
• an explanation of how trigonometric ratios are used to measure the height
of a mountain, and
• any additional information you need to know about the point from which
you are measuring in order to find the altitude of a mountain.
Answers
a
c
b
c
66. Let sin A and let cos A ,
where a and b are legs of a right
triangle and c is the hypotenuse.
Then sin2 A cos2 A Standardized
Test Practice
a2
b2
a2 b2
2 . Since the
2
c
c
c2
26
B
23
C
43
D
22
45°
D
60°
4
69. What is the measure of Q? D
c2
c
becomes 2 or 1. Thus
630
R
68. RT is equal to TS. What is RS? A
A
Pythagorean Theorem states that
a 2 b 2 c2, the expression
sin2 A cos2 A 1.
67. If you know the distance between
two points and the angles from
these two points to a third point,
you can determine the distance to
the third point by forming a triangle
and using trigonometric ratios.
Answers should include the
following.
• If you measure your distance
from the mountain and the angle
of elevation to the peak of the
mountain from two different
points, you can write an equation using trigonometric ratios
to determine its height, similar
to Example 5.
• You need to know the altitude of
the two points you are
measuring.
For Exercises 68 and 69, use the figure at the right.
A
25°
B
30°
C
2
Q
T
S
Maintain Your Skills
Mixed Review
For each set of measures given, find
the measures of the missing sides if
KLM ∼ NOP. (Lesson 11-6)
N
70. k 5, ᐉ 3, m 6, n 10 o 6, p 12
71. ᐉ 9, m 3, n 12, p 4.5
k 8, o 13.5
p
K
o
m
ᐉ
M
k
L
P
n
O
Find the possible values of a if the points with the given coordinates are the
indicated distance apart. (Lesson 11-5)
72. (9, 28), (a, 8); d 39 6 or 24
73. (3, a), (10, 1); d 65
5 or 3
Find each product. (Lesson 8-6)
74. c2(c2 3c) c4 3c3
75. s(4s2 9s 12)
4s3 9s2 12s
76. xy2(2x2 5xy 7y2)
2x3y2 5x2y3 7xy4
Use substitution to solve each system of equations. (Lesson 7-2)
77. a 3b 2 (11, 3)
4a 7b 23
630
Chapter 11 Radical Expressions and Triangles
Chapter 11 Radical Expressions and Triangles
78. p q 10 (3, 7)
3p 2q 5
79. 3r 6s 0 (2, 1)
4r 10s 2
Reading
Mathematics
The Language of Mathematics
Getting Started
The language of mathematics is a specific one, but it borrows from everyday
language, scientific language, and world languages. To find a word’s correct
meaning, you will need to be aware of some confusing aspects of language.
Confusing Aspect
Have students think of words
that have more than one meaning.
As students respond, make a list
of the words on the chalkboard
along with their multiple
definitions.
Words
Some words are used in English and in
mathematics, but have distinct meanings.
factor, leg, prime, power,
rationalize
Some words are used in English and in
mathematics, but the mathematical meaning
is more precise.
difference, even, similar,
slope
Some words are used in science and in
mathematics, but the meanings are different.
divide, radical, solution,
variable
Some words are only used in mathematics.
decimal, hypotenuse,
integer, quotient
Some words have more than one mathematical
meaning.
base, degree, range, round,
square
Sometimes several words come from the same
root word.
polygon and polynomial,
radical and radicand
Some mathematical words sound like English
words.
cosine and cosign, sine
and sign, sum and some
Some words are often abbreviated, but you
must use the whole word when you read
them.
cos for cosine, sin for sine,
tan for tangent
Teach
• Place students in small groups.
Assign each group several of
the words listed in this activity
to define. Have students use
their textbooks, dictionaries, or
the Internet to find the definitions. Then, as you discuss
each group of words, ask the
groups to give the definitions.
Assess
Words in boldface are in this chapter.
Study Notebook
Reading to Learn 1–3. See margin.
Ask students to summarize what
they have learned about the
language of mathematics.
1. How do the mathematical meanings of the following words compare to the
everyday meanings?
a. factor
b. leg
c. rationalize
2. State two mathematical definitions for each word. Give an example for each
definition.
a. degree
b. range
c. round
3. Each word below is shown with its root word and the root word’s meaning. Find
ELL English Language
Learners may benefit from
writing key concepts from this
activity in their Study Notebooks
in their native language and then
in English.
three additional words that come from the same root.
a. domain, from the root word domus, which means house
b. radical, from the root word radix, which means root
c. similar, from the root word similis, which means like
Reading Mathematics
The Language of Mathematics
631
Reading Mathematics The Language of Mathematics 631
Change art for numerals as necessary
should come in at x=p6, y=1p6
reduce to 65%
Study Guide
and Review
Vocabulary and Concept Check
Vocabulary and
Concept Check
angle of depression (p. 625)
angle of elevation (p. 625)
conjugate (p. 589)
corollary (p. 607)
cosine (p. 623)
Distance Formula (p. 611)
extraneous solution (p. 599)
• This alphabetical list of
vocabulary terms in Chapter 11
includes a page reference
where each term was
introduced.
• Assessment A vocabulary
test/review for Chapter 11 is
available on p. 698 of the
Chapter 11 Resource Masters.
hypotenuse (p. 605)
leg (p. 605)
Pythagorean triple (p. 606)
radical equation (p. 598)
radical expression (p. 586)
radicand (p. 586)
State whether each sentence is true or false. If false, replace the underlined
word, number, expression, or equation to make a true sentence.
1. The binomials 3 7 and 3 7 are conjugates. false, 3 7
2. In the expression 45, the radicand is 5. true
3. The sine of an angle is the measure of the opposite leg divided by the measure of
the hypotenuse. true
4. The longest side of a right triangle is the hypotenuse. true
Lesson-by-Lesson
Review
5. After the first step in solving 3x 19 x 3, you would have
3x 19 x2 9. false, 3x 19 x 2 6x 9
6. The two sides that form the right angle in a right triangle are called the legs of
the triangle. true
2x3x
x
2xy
7. The expression is in simplest radical form. false, y
6y
For each lesson,
• the main ideas are
summarized,
• additional examples review
concepts, and
• practice exercises are provided.
8. A triangle with sides having measures of 25, 20, and 15 is a right triangle. true
Vocabulary
PuzzleMaker
11-1 Simplifying Radical Expressions
See pages
586–592.
The Vocabulary PuzzleMaker
software improves students’ mathematics
vocabulary using four puzzle formats—
crossword, scramble, word search using a
word list, and word search using clues.
Students can work on a computer screen
or from a printed handout.
ELL
Example
Concept Summary
• A radical expression is in simplest form when no radicands have perfect
square factors other than 1, no radicands contain fractions, and no radicals
appear in the denominator of a fraction.
3
Simplify .
5 2
5 2
to rationalize the denominator.
3
5 2 Multiply by 3
5 2
5 2
5 2
5 2
3(5) 32
2
2
(a b)(a b) a2 b2
15 32
2
2 2
15 32
Simplify.
5 2 MindJogger
Videoquizzes
25 2
ELL MindJogger Videoquizzes
provide an alternative review of concepts
presented in this chapter. Students work
in teams in a game show format to gain
points for correct answers. The questions
are presented in three rounds.
23
632
Chapter 11 Radical Expressions and Triangles
Round 1 Concepts (5 questions)
Round 2 Skills (4 questions)
Round 3 Problem Solving (4 questions)
www.algebra1.com/vocabulary_review
TM
Have students flip back through their Foldables to make sure they
have included information for every lesson.
For more information
about Foldables, see
Teaching Mathematics
with Foldables.
632
rationalizing the denominator (p. 588)
similar triangles (p. 616)
sine (p. 623)
solve a triangle (p. 625)
tangent (p. 623)
trigonometric ratios (p. 623)
Chapter 11 Radical Expressions and Triangles
Encourage students to refer to their Foldables while completing
the Study Guide and Review and use them in preparing for the
Chapter Test.
Study Guide and Review
Chapter 11 Study Guide and Review
Exercises Simplify. See Examples 1–5 on pages 586–589.
60
y
215
y
9
27 92
12. 3 2
7
9.
2
10. 44a2b5 2ab2
11b 11. 3 212
2 57 243
27
13. 3a3b4 a
6
14. 8ab10 4b3
35
53
521
35
3
15
11-2 Operations with Radical Expressions
See pages
593–597.
Concept Summary
• Radical expressions with like radicands can be added or subtracted.
• Use the FOIL Method to multiply radical expressions.
Examples 1
Simplify 6 54
312
53.
312
53
6 54
6 32 6 3
22 3 53
Simplify radicands.
6 32 6 3
22 3 53
Product Property of Square Roots
6 36 323 53
Evaluate square roots.
6 36 63 53
Simplify.
26 113
Add like radicands.
2 Find 23 5 10
46 .
23 5
10 46
F irst terms
Outer terms
I nner terms
Last terms
23
10 2346 5
10 546
230
818
50
430
Multiply.
230
32 2 52 2 430
8
Prime factorization
230
242 52 430
Simplify.
230
192
Combine like terms.
Exercises Simplify each expression. See Examples 1 and 2 on pages 593 and 594.
15. 23
85 35 33
16. 26 48
26 43
17. 427
648
363
18. 47k
77k
27k
7k
19. 518
3112
398
20. 8 15. 53 55
19. 62 127
18
9
2
4
Find each product. See Example 3 on page 594. 24. 1810
30 62 25
21. 2
3 33 32 36
23. 3
2 22 3 6 1
22. 525 7 10 35
24. 65 232 5 Chapter 11
Study Guide and Review
633
Chapter 11 Study Guide and Review 633
Study Guide and Review
Chapter 11 Study Guide and Review
11-3 Radical Expressions
Answers
See pages
598–603.
31. 34
32. 234
11.66
33. 55 11.18
34. 4
195 55.86
35. 24
36. 2
Example
Concept Summary
• Solve radical equations by isolating the radical on one side of the equation.
Square each side of the equation to eliminate the radical.
4x 6 7.
Solve 5
5 4x 6 7
5 4x 13
Original equation
Add 6 to each side.
5 4x 169
Square each side.
4x 164
Subtract 5 from each side.
x 41
Divide each side by 4.
Exercises Solve each equation. Check your solution. See Examples 2 and 3 on page 599.
no
26
25. 10 2b 0 solution 26. a 4 6 32
27. 7x 1 5 7
28.
4a
20 3
3
29. x 4 x 8 12
30. 3x 14
x6 5
11-4 The Pythagorean Theorem
See pages
605–610.
Concept Summary
• If a and b are the measures of the legs of a right triangle
and c is the measure of the hypotenuse, then c2 a2 b2.
• If a and b are measures of the shorter sides of a triangle,
c is the measure of the longest side, and c2 a2 b2,
then the triangle is a right triangle.
c
a
b
Example
15
Find the length of the missing side.
c2 a2 b2
Pythagorean Theorem
252 152 b2
c 25 and a 15
625 225 Evaluate squares.
b2
400 b2
20 b
25
b
Subtract 225 from each side.
Take the square root of each side.
Exercises If c is the measure of the hypotenuse of a right triangle, find each
missing measure. If necessary, round answers to the nearest hundredth.
See Example 2 on page 606.
31. a 30, b 16, c ?
34. b 4, c 56, a ?
31–36. See margin.
32. a 6, b 10, c ?
35. a 18, c 30, b ?
33. a 10, c 15, b ?
36. a 1.2, b 1.6, c ?
Determine whether the following side measures form right triangles.
See Example 4 on page 608.
37. 9, 16, 20 no
634
634
Chapter 11 Radical Expressions and Triangles
Chapter 11 Radical Expressions and Triangles
38. 20, 21, 29 yes
39. 9, 40, 41 yes
40. 18, 24
, 30 no
Study Guide and Review
Chapter 11 Study Guide and Review
11-5 The Distance Formula
See pages
611–615.
Concept Summary
• The distance d between any two points
with coordinates (x1, y1) and (x2, y2)
B (x2, y2)
(x2 x1)2 (y2 y1)2.
is given by d Example
y
A(x1, y1)
O
x
Find the distance between the points with coordinates (5, 1) and (1, 5).
x1)2 (y2 y1)2
d (x
2
(1 (5))2
(5 1)2
Distance Formula
(x1, y1) (5, 1) and (x2, y2) (1, 5)
Simplify.
36 16
Evaluate squares.
52
or about 7.21 units
Simplify.
62
42
42. 58 7.62 43. 205
14.32
Exercises Find the distance between each pair of points whose coordinates are
given. Express in simplest radical form and as decimal approximations rounded
to the nearest hundredth if necessary. See Example 1 on page 611.
41. (9, 2), (1, 13) 17
42. (4, 2), (7, 9)
43. (4, 6), (2, 7)
44. 25
, 9, 45, 3
45. (4, 8), (7, 12)
46. (2, 6), (5, 11)
214
7.48
137
11.70
74
8.60
Find the value of a if the points with the given coordinates are the indicated
distance apart. See Example 3 on page 612.
47. (3, 2), (1, a); d 5 5 or 1
48. (1, 1), (4, a); d 5 5 or 3
49. (6, 2), (5, a); d 145
10 or 14 50. (5, 2), (a, 3); d 170
18 or 8
11-6 Similar Triangles
See pages
616–621.
Concept Summary
• Similar triangles have congruent corresponding angles and proportional
corresponding sides.
B
E
• If ABC DEF, then
AB
BC
AC
.
DE
EF
DF
A
C
D
Example
Find the measure of side a if the
two triangles are similar.
10
6
5
a
Corresponding sides of similar
triangles are proportional.
10a 30 Find the cross products.
a 3 Divide each side by 10.
F
5 cm
a cm
10 cm
6 cm
Chapter 11
Study Guide and Review
635
Chapter 11 Study Guide and Review 635
• Extra Practice, see pages 844–846.
• Mixed Problem Solving, see page 863.
Study Guide and Review
Exercises For each set of measures given, find the measures of the remaining
sides if ABC DEF. See Example 2 on page 617. 51–54. See margin.
Answers
45
8
51.
52.
53.
54.
27
4
51. d , e 52. d 9.6, e 7.2
44
53. b , d 6
3
c 16, b 12, a 10, f 9
a 8, c 10, b 6, f 12
c 12, f 9, a 8, e 11
b 20, d 7, f 6, c 15
E
d
f
B
D
a
c
A
F
C
b
54. a 17.5, e 8
e
11-7 Trigonometric Ratios
See pages
623–630.
Concept Summary
Three common trigonometric ratios are sine, cosine, and tangent.
BC
• sin A AB
B
hypotenuse
AC
• cos A AB
BC
• tan A AC
Example
leg
opposite
A
A
A
leg adjacent to
C
Find the sine, cosine, and tangent of A. Round to the nearest ten thousandth.
opposite leg
hypotenuse
20
or 0.8000
25
sin A adjacent leg
hypotenuse
15
or 0.6000
25
A
25
15
cos A C
B
20
opposite leg
adjacent leg
20
or 1.3333
15
tan A Exercises For ABC, find each value of each trigonometric ratio to the nearest
ten thousandth. See Example 1 on page 624.
55.
56.
57.
58.
59.
60.
cos B 0.5283
tan A 0.6222
sin B 0.8491
cos A 0.8491
tan B 1.6071
sin A 0.5283
B
28 ft
C
53 ft
45 ft
A
Use a calculator to find the measure of each angle to the nearest degree.
See Example 3 on page 624.
61. tan M 0.8043 39°
64. cos F 0.7443 42°
636
636
Chapter 11 Radical Expressions and Triangles
Chapter 11 Radical Expressions and Triangles
62. sin T 0.1212 7°
65. sin A 0.4540 27°
63. cos B 0.9781 12°
66. tan Q 5.9080 80°
Practice Test
Vocabulary and Concepts
Assessment Options
Match each term and its definition.
1. measure of the opposite side divided by the measure of the hypotenuse b
2. measure of the adjacent side divided by the measure of the hypotenuse a
3. measure of the opposite side divided by the measure of the adjacent side c
Vocabulary Test A vocabulary
test/review for Chapter 11 can be
found on p. 698 of the Chapter 11
Resource Masters.
a. cosine
b. sine
c. tangent
Skills and Applications
37
Simplify. 4. 23
9. 3 2 33 6
4. 227
63
43
7.
4 2
10
30 3
3
23
46
3
8. 64 12 46
62
5. 6 6. 112x4y6 4x2y37
9. 1 3 3 2 Solve each equation. Check your solution.
10. 10x
20 40
13. x 6x 8 no solution
11. 4s
1 11 25
14. x 5x 14 7
Form
12. 4x 1 5 6
15. 4x
36x 3
If c is the measure of the hypotenuse of a right triangle, find each missing measure. 16.
If necessary, round to the nearest hundredth. 17. 72
8.49 18. 120
10.95
16. a 8, b 10, c ?
17. a 62, c 12, b ?
164 12.81
18. b 13, c 17, a ?
Find the distance between each pair of points whose coordinates are given.
Express in simplest radical form and as decimal approximations rounded to the
nearest hundredth if necessary.
20. (1, 1), (1, 5) 210
6.32 21. (9, 2), (21, 7)
19. (4, 7), (4, 2) 9
For each set of measures given, find the measures of
the missing sides if ABC JKH.
A
64
22. c 20, h 15, k 16, j 12 a 16, b c
65 3
b
23. c 12, b 13, a 6, h 10 j 5, k K
6
24. k 5, c 6.5, b 7.5, a 4.5 h 4.3
, j 3
C
B
a
3
1
1
1
25. h 1, c 4, k 2, a 3 b 6, j 1
2
2
4
4
Solve each right triangle. State the side lengths to the nearest tenth and the angle
measures to the nearest degree.
26.
B
29
A
a
21
mA 44°
mB 46°
a 20
27.
C
21
A
15
c
B
mA 36° 28. B
mB 54°
c 25.8
a
42˚
10
C
925
30.41
k
j
C
b
H
mA 48°
a 7.4
b 6.7
29. SPORTS A hiker leaves her camp in the morning. How far is she from camp
after walking 9 miles due west and then 12 miles due north? 15 mi
C
323
18 units2
D
232
18 units2
232 36
6
www.algebra1.com/chapter_test
1
2A
2B
2C
2D
3
Chapter 11 Tests
Type
Level
Pages
MC
MC
MC
FR
FR
FR
basic
average
average
average
average
advanced
685–686
687–688
689–690
691–692
693–694
695–696
MC = multiple-choice questions
FR = free-response questions
J
h
A
30. STANDARDIZED TEST PRACTICE Find the area of the rectangle. B
A 16
B 16
2 46
3 18 units2
units2
Chapter Tests There are six
Chapter 11 Tests and an OpenEnded Assessment task available
in the Chapter 11 Resource Masters.
Chapter 11
Open-Ended Assessment
Performance tasks for Chapter 11
can be found on p. 697 of the
Chapter 11 Resource Masters. A
sample scoring rubric for these
tasks appears on p. A27.
TestCheck and
Worksheet Builder
This networkable software has
three modules for assessment.
• Worksheet Builder to make
worksheets and tests.
• Student Module to take tests
on-screen.
• Management System to keep
student records.
Practice Test 637
Portfolio Suggestion
Introduction In this chapter, students are introduced to ways in which
professionals such as architects and surveyors can use math to solve problems.
Ask Students Pick a profession such as architecture, surveying, or engineering
and research to find out how triangles, ratios, or trigonometric ratios are used in
the field. Write a short report in which you explain how these concepts are
used, along with examples.
Chapter 11 Practice Test 637
Standardized
Test Practice
These two pages contain practice
questions in the various formats
that can be found on the most
frequently given standardized
tests.
Part 1 Multiple Choice
Record your answers on the answer sheet
provided by your teacher or on a sheet of
paper.
1. Which equation describes the data in the
table? (Lesson 4-8) D
A practice answer sheet for these
two pages can be found on p. A1
of the Chapter 11 Resource Masters.
NAME
DATE
x
y
PERIOD
Standardized
Practice
11 Standardized Test
Test Practice
Student
Recording
Sheet,
A1Edition.)
Student Record
Sheet (Use with pages
638–639 of p.
the Student
5
2
11
5
1
A
11
B
12
C
22
D
33
4
1 7
A
yx6
B
y 2x 1
C
y 2x 1
D
y 2x 1
Part 1 Multiple Choice
6. The function g t2 t represents the total
number of games played by t teams in a sports
league in which each team plays each of the
other teams twice. The Metro League plays a
total of 132 games. How many teams are in the
league? (Lesson 9-4) B
7. One leg of a right triangle is 4 inches longer
than the other leg. The hypotenuse is 20 inches
long. What is the length of the shorter leg?
(Lesson 11-4)
B
Select the best answer from the choices given and fill in the corresponding oval.
1
A
B
C
D
4
A
B
C
D
7
A
B
C
D
2
A
B
C
D
5
A
B
C
D
8
A
B
C
D
3
A
B
C
D
6
A
B
C
D
9
A
B
C
D
2. The length of a rectangle is 6 feet more than
the width. The perimeter is 92 feet. Which
system of equations will determine the length
in feet ᐉ and the width in feet w of the
rectangle? (Lesson 7-2) C
Part 2 Short Response/Grid In
Solve the problem and write your answer in the blank.
For Questions 13 and 15, also enter your answer by writing each number or
symbol in a box. Then fill in the corresponding oval for that number or symbol.
13
11
12
13
(grid in)
14
15
15
.
/
.
/
.
.
.
/
.
/
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
(grid in)
16
Part 3 Quantitative Comparison
Select the best answer from the choices given and fill in the corresponding oval.
A
B
C
D
19
A
B
C
D
20
A
B
C
wᐉ6
2ᐉ 2w 92
B
ᐉw6
ᐉw 92
C
ᐉw6
2ᐉ 2w 92
D
ᐉw6
ᐉ w 92
D
Part 4 Open-Ended
Record your answers for Question 21 on the back of this paper.
A
$450,000
B
$734,000
C
$900,000
D
$1,600,000
B
12 in.
16 in.
D
18 in.
8. What is the distance from
one corner of the garden
to the opposite corner?
5 yd
A
A
13 yards
B
14 yards
C
15 yards
D
17 yards
12 yd
9. How many points in the coordinate plane are
equidistant from both the x- and y-axes and
are 5 units from the origin? (Lesson 11-5) D
A
0
B
1
C
2
D
4
4. If 32,800,000 is expressed in the form 3.28 10n,
what is the value of n? (Lesson 8-3) C
Additional Practice
See pp. 703–704 in the Chapter 11
Resource Masters for additional
standardized test practice.
A
5
B
6
C
7
D
8
5. What are the solutions of the equation
x2 7x 18 0? (Lesson 9-4) A
638
A
2 or 9
B
2 or 9
C
2 or 9
D
2 or 9
The Princeton Review offers
additional test-taking tips and
practice problems at their web site. Visit
www.princetonreview.com or
www.review.com
Chapter 11 Radical Expressions and Triangles
Test-Taking Tip
Questions 7, 21, and 22 Be sure that you
know and understand the Pythagorean Theorem.
References to right angles, the diagonal of a
rectangle, or the hypotenuse of a triangle indicate
that you may need to use the Pythagorean Theorem
to find the answer to an item.
Chapter 11 Radical Expressions and Triangles
Log On for Test Practice
638
10 in.
(Lesson 11-4)
3. A highway resurfacing project and a bridge
repair project will cost $2,500,000 altogether.
The bridge repair project will cost $200,000 less
than twice the cost of the highway resurfacing.
How much will the highway resurfacing
project cost? (Lesson 7-2) C
17
18
A
Answers
10
A
C
TestCheck and
Worksheet Builder
Special banks of standardized test
questions similar to those on the SAT,
ACT, TIMSS 8, NAEP 8, and Algebra 1
End-of-Course tests can be found on
this CD-ROM.
Aligned and
verified by
Part 2 Short Response/Grid In
Record your answers on the answer sheet
provided by your teacher or on a sheet of
paper.
10. A line is parallel to the line represented by
1
2
3
2
the equation y x 4 0. What is the
slope of the parallel line?
(Lesson 5-6)
3
Compare the quantity in Column A and the
quantity in Column B. Then determine
whether:
A
the quantity in Column A is greater,
B
the quantity in Column B is greater,
C
the two quantities are equal, or
D
the relationship cannot be determined
from the information given.
11. Graph the solution of the system of linear
inequalities 2x y 2 and 3x 2y 4.
(Lesson 6-6)
See margin.
18.
12. The sum of two integers is 66. The second
integer is 18 more than half of the first. What
are the integers? (Lesson 7-2) 32 and 34
13. The function h(t) 16t2 v0t h0
describes the height in feet above the
ground h(t) of an object thrown vertically
from a height of h0 feet, with an initial
velocity of v0 feet per second, if there is no
air friction and t is the time in seconds that
it takes the ball to reach the ground. A ball
is thrown upward from a 100-foot tower at
a velocity of 60 feet per second. How many
seconds will it take for the ball to reach the
ground? (Lesson 9-5) 5
Column A
Column B
the value of x in
13x 12 10x 3
the value of y in
12y 16 8y
B
19.
the slope of
2x 3y 10
the measure of the
hypotenuse of a
right triangle if the
measures of the
legs are 10 and 11
the measure of the leg
of a right triangle if the
measure of the other
leg is 13 and the
hypotenuse is 390
C
x .
16. Simplify the expression x 2 3 x
(Lesson 11-1)
3
x 2 or x x
17. The area of a rectangle is 64. The length is
x3
x1
, and the width is . What is x?
x1
x
(Lesson 11-3)
8 or 8
A correct solution that is supported
by well-developed, accurate
explanations
A generally correct solution, but
may contain minor flaws in
reasoning or computation
A partially correct interpretation
and/or solution to the problem
A correct solution with no
supporting evidence or explanation
An incorrect solution indicating no
mathematical understanding of
the concept or task, or no solution
is given
3
Part 4 Open Ended
1
0
b. To the nearest tenth of a mile, how far (in
a straight line) is Haley from her starting
point? 11.4 mi
c. How did your diagram help you to find
Haley’s distance from her starting point?
See margin.
www.algebra1.com/standardized_test
Answers
4
a. Draw a diagram showing the direction
and distance of each segment of Haley’s
hike. Label Haley’s starting point, her
ending point, and the distance, in miles,
of each segment of her hike. See margin.
3
3 4
Criteria
2
21. Haley hikes 3 miles north, 7 miles east, and
then 6 miles north again. (Lesson 11-4)
3
Score
(Lesson 11-4)
7.16 and 0.84
(Lesson 11-1)
Sample Scoring Rubric: The following rubric is a sample scoring
device. You may wish to add
more detail to this sample to meet
your individual scoring needs.
(Lesson 5-3)
Record your answers on a sheet of paper.
Show your work.
15. Simplify the expression 381
.
Goal: Students use right
triangles to find a distance.
the y-intercept of
7x 4y 4
B
20.
Open-Ended Assessment
questions are graded by using a
multilevel rubric that guides you
in assessing a student’s knowledge of a particular concept.
(Lesson 3-5)
14. Find all values of x that satisfy the equation
x2 8x 6 0. Approximate irrational
numbers to the nearest hundredth.
(Lesson 10-4)
Evaluating Open-Ended
Assessment Questions
Part 3 Quantative Comparison
11.
Chapter 11 Standardized Test Practice 639
21a.
y
finish
6
O
x
21c. The sketch shows that there are
two right triangles. The
distance is equal to the sum of
the measures of the hypotenuse
of each triangle.
7
3
start
7
Chapter 11 Standardized Test Practice 639
Page 604, Follow-Up to Lesson 11-3
Graphing Calculator Investigation
1. {x | x 0}; shifted up 1
6. {x | x 0}; expanded
[10, 10] scl: 1 by [10, 10] scl: 1
[10, 10] scl: 1 by [10, 10] scl: 1
7. {x | x 0}; reflected across x-axis
Additional Answers for Chapter 11
2. {x | x 0}; shifted down 3
[10, 10] scl: 1 by [10, 10] scl: 1
[10, 10] scl: 1 by [10, 10] scl: 1
8. {x | x 1}; reflected across y-axis, shifted right 1, up 6
3. {x | x 2}; shifted left 2
[10, 10] scl: 1 by [5, 15] scl: 1
[10, 10] scl: 1 by [10, 10] scl: 1
9. {x | x 2.5}; shifted left 2.5, down 4
4. {x | x 5}; shifted right 5
[10, 10] scl: 1 by [10, 10] scl: 1
[5, 15] scl: 1 by [10, 10] scl: 1
5. {x | x 0}; reflected across y-axis
[10, 10] scl: 1 by [10, 10] scl: 1
639A
Chapter 11 Additional Answers
Pages 612–615, Lesson 11-5
44. You can determine the distance between two points by
forming a right triangle. Drawing a line through each
point parallel to the axes forms the legs of the triangle.
The hypotenuse of this triangle is the distance between
the two points. You can find the lengths of each leg by
subtracting the corresponding x- and y-coordinates,
then use the Pythagorean Theorem. Answers should
include the following.
• You can draw lines parallel to the axes through the
two points that will intersect at another point forming
a right triangle. The length of a leg of a triangle is the
difference in the x- or y-coordinates. The length of
the hypotenuse is the distance between the points.
Using the Pythagorean Theorem to solve for the
hypotenuse, you have the Distance Formula.
• The points are on a vertical line so you can calculate
distance by determining the absolute difference
between the y-coordinates.
57.
3
4
5
6
7
8
9
10
58.
8 7 6 5 4 3 2 1 0
59.
8 7 6 5 4 3 2 1 0
60.
4 3 2 1 0
1
2
3
4
7
8
9
10
61.
2
3
4
5
6
62.
4 3 2 1 0
1
2
3
4
B
C
A
E
F
D
corresponding angles
A and D
corresponding sides
AB
and DE
B and E
C and F
BC
and E
F
AC
and D
F
Page 631, Reading Mathematics
1a. Sample answer: A circumstance that brings about a
result; any of two or more quantities that form a
product when multiplied together; the quantities bring
about a result, a product.
1b. Sample answer: The legs of an animal support the
animal; one of the two shorter sides of a right triangle;
the legs of a triangle support the hypotenuse.
1c. Sample answer: To devise rational explanations for
one’s acts without being aware that these are not the
real motives; to remove the radical signs from an
expression without changing its value; to justify an
action without changing its intent.
2a. Sample answer: Rank as determined by the sum of a
term’s exponents; the degree of x 2y 2 is 4.
1
of a circle; a semicircle measures 180°.
360
2b. Sample answer: The difference between the greatest
and least values in a set of data; the range of 2, 3, 6 is
6 2 or 4. The set of all y-values in a function; the
range of {(2, 6), (1, 3)} is {6, 3}.
2c. Sample answer: Circular or spherical; circles are
round. To abbreviate a number by replacing its ending
digits with zeros; 235,611 rounded to the nearest
hundred is 235,600.
3a. Sample answer: dome, domestic, domicile
3b. Sample answer: eradicate, radicand, radius
3c. Sample answer: simile, similarity, similitude
Chapter 11 Additional Answers 639B
Additional Answers for Chapter 11
2
Pages 618–621, Lesson 11-6
2. Sample answer: ABC DEF