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Chapter 7Resource Masters
Geometry
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3
ANSWERS FOR WORKBOOKS The answers for Chapter 7 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-860184-3 GeometryChapter 7 Resource Masters
1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03
© Glencoe/McGraw-Hill iii Glencoe Geometry
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix
Lesson 7-1Study Guide and Intervention . . . . . . . . 351–352Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 353Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 354Reading to Learn Mathematics . . . . . . . . . . 355Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 356
Lesson 7-2Study Guide and Intervention . . . . . . . . 357–358Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 359Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 360Reading to Learn Mathematics . . . . . . . . . . 361Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 362
Lesson 7-3Study Guide and Intervention . . . . . . . . 363–364Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 365Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 366Reading to Learn Mathematics . . . . . . . . . . 367Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 368
Lesson 7-4Study Guide and Intervention . . . . . . . . 369–370Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 371Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 372Reading to Learn Mathematics . . . . . . . . . . 373Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 374
Lesson 7-5Study Guide and Intervention . . . . . . . . 375–376Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 377Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 378Reading to Learn Mathematics . . . . . . . . . . 379Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 380
Lesson 7-6Study Guide and Intervention . . . . . . . . 381–382Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 383Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 384Reading to Learn Mathematics . . . . . . . . . . 385Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 386
Lesson 7-7Study Guide and Intervention . . . . . . . . 387–388Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 389Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 390Reading to Learn Mathematics . . . . . . . . . . 391Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 392
Chapter 7 AssessmentChapter 7 Test, Form 1 . . . . . . . . . . . . 393–394Chapter 7 Test, Form 2A . . . . . . . . . . . 395–396Chapter 7 Test, Form 2B . . . . . . . . . . . 397–398Chapter 7 Test, Form 2C . . . . . . . . . . . 399–400Chapter 7 Test, Form 2D . . . . . . . . . . . 401–402Chapter 7 Test, Form 3 . . . . . . . . . . . . 403–404Chapter 7 Open-Ended Assessment . . . . . . 405Chapter 7 Vocabulary Test/Review . . . . . . . 406Chapter 7 Quizzes 1 & 2 . . . . . . . . . . . . . . . 407Chapter 7 Quizzes 3 & 4 . . . . . . . . . . . . . . . 408Chapter 7 Mid-Chapter Test . . . . . . . . . . . . 409Chapter 7 Cumulative Review . . . . . . . . . . . 410Chapter 7 Standardized Test Practice . 411–412Unit 2 Test/Review (Ch. 4–7) . . . . . . . . 413–414First Semester Test (Ch. 1–7) . . . . . . . 415–416
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A34
© Glencoe/McGraw-Hill iv Glencoe Geometry
Teacher’s Guide to Using theChapter 7 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 7 Resource Masters includes the core materials neededfor Chapter 7. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theGeometry TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 7-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.
Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.
WHEN TO USE Give these pages tostudents before beginning Lesson 7-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.
Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
© Glencoe/McGraw-Hill v Glencoe Geometry
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
Assessment OptionsThe assessment masters in the Chapter 7Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 398–399. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
77
© Glencoe/McGraw-Hill vii Glencoe Geometry
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 7.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
ambiguous case
angle of depression
angle of elevation
cosine
geometric mean
Law of Cosines
Law of Sines
Pythagorean identity
puh·thag·uh·REE·ahn
(continued on the next page)
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
© Glencoe/McGraw-Hill viii Glencoe Geometry
Vocabulary Term Found on Page Definition/Description/Example
Pythagorean triple
reciprocal identity
ri·SIP·ruh·kuhl
sine
solve a triangle
tangent
trigonometric identity
trig·uh·nuh·MET·rik
trigonometric ratio
trigonometry
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
77
Learning to Read MathematicsProof Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
77
© Glencoe/McGraw-Hill ix Glencoe Geometry
Proo
f Bu
ilderThis is a list of key theorems and postulates you will learn in Chapter 7. As you
study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 7.1
Theorem 7.2
Theorem 7.3
Theorem 7.4Pythagorean Theorem
Theorem 7.5Converse of the Pythagorean Theorem
Theorem 7.6
Theorem 7.7
Study Guide and InterventionGeometric Mean
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
© Glencoe/McGraw-Hill 351 Glencoe Geometry
Less
on
7-1
Geometric Mean The geometric mean between two numbers is the square root oftheir product. For two positive numbers a and b, the geometric mean of a and b is the positive number x in the proportion !
ax! " !b
x!. Cross multiplying gives x2 " ab, so x " !ab".
Find the geometric mean between each pair of numbers.
a. 12 and 3Let x represent the geometric mean.
!1x2! " !3
x! Definition of geometric mean
x2 " 36 Cross multiply.
x " !36" or 6 Take the square root of each side.
b. 8 and 4Let x represent the geometric mean.
!8x! " !4
x!
x2 " 32x " !32"
# 5.7
ExercisesExercises
Find the geometric mean between each pair of numbers.
1. 4 and 4 2. 4 and 6
3. 6 and 9 4. !12! and 2
5. 2!3" and 3!3" 6. 4 and 25
7. !3" and !6" 8. 10 and 100
9. !12! and !
14! 10. and
11. 4 and 16 12. 3 and 24
The geometric mean and one extreme are given. Find the other extreme.
13. !24" is the geometric mean between a and b. Find b if a " 2.
14. !12" is the geometric mean between a and b. Find b if a " 3.
Determine whether each statement is always, sometimes, or never true.
15. The geometric mean of two positive numbers is greater than the average of the twonumbers.
16. If the geometric mean of two positive numbers is less than 1, then both of the numbersare less than 1.
3!2"!5
2!2"!5
ExampleExample
© Glencoe/McGraw-Hill 352 Glencoe Geometry
Altitude of a Triangle In the diagram, !ABC $ !ADB $ !BDC.An altitude to the hypotenuse of a right triangle forms two right triangles. The two triangles are similar and each is similar to the original triangle. CD
B
A
Study Guide and Intervention (continued)
Geometric Mean
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
Use right !ABC with B!D! ⊥ A!C!. Describe two geometricmeans.
a. !ADB " !BDC so !BAD
D! " !BC
DD!.
In !ABC, the altitude is the geometricmean between the two segments of thehypotenuse.
b. !ABC " !ADB and !ABC " !BDC,
so !AACB! " !A
ADB! and !B
ACC! " !D
BCC!.
In !ABC, each leg is the geometricmean between the hypotenuse and thesegment of the hypotenuse adjacent tothat leg.
Find x, y, and z.
!PP
QR! " !
PP
QS!
!21
55! " !
1x5! PR " 25, PQ " 15, PS " x
25x " 225 Cross multiply.x " 9 Divide each side by 25.
Theny " PR # SP
" 25 # 9" 16
!QPR
R! " !QR
RS!
!2z5! " !y
z! PR " 25, QR " z, RS " y
!2z5! " !1
z6! y " 16
z2 " 400 Cross multiply.z " 20 Take the square root of each side.
zy
x
15
R
Q P
S25
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find x, y, and z to the nearest tenth.
1. 2. 3.
4. 5. 6.x zy
62
x
z y
2
2xy
1
!%3
!"12
zx y
81
z
xy 5
2
x
1 3
Skills PracticeGeometric Mean
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
© Glencoe/McGraw-Hill 353 Glencoe Geometry
Less
on
7-1
Find the geometric mean between each pair of numbers. State exact answers andanswers to the nearest tenth.
1. 2 and 8 2. 9 and 36 3. 4 and 7
4. 5 and 10 5. 2!2" and 5!2" 6. 3!5" and 5!5"
Find the measure of each altitude. State exact answers and answers to the nearesttenth.
7. 8.
9. 10.
Find x and y.
11. 12.
13. 14.
2
5y
x15
4
y
x
10
4
yx
3 9
yx
R T
S
U4.5 8G
E H
F
2
9
L
M
N
P 2
12
C
D
B
A 2
7
© Glencoe/McGraw-Hill 354 Glencoe Geometry
Find the geometric mean between each pair of numbers to the nearest tenth.
1. 8 and 12 2. 3!7" and 6!7" 3. !45! and 2
Find the measure of each altitude. State exact answers and answers to the nearesttenth.
4. 5.
Find x, y, and z.
6. 7.
8. 9.
10. CIVIL ENGINEERING An airport, a factory, and a shopping center are at the vertices of aright triangle formed by three highways. The airport and factory are 6.0 miles apart. Theirdistances from the shopping center are 3.6 miles and 4.8 miles, respectively. A service roadwill be constructed from the shopping center to the highway that connects the airport andfactory. What is the shortest possible length for the service road? Round to the nearesthundredth.
x y
10z
20x
y
2
3
z
zx y
625
23
z
xy
8
17
6
KL
J M
125
U
T A V
Practice Geometric Mean
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
Reading to Learn MathematicsGeometric Mean
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
© Glencoe/McGraw-Hill 355 Glencoe Geometry
Less
on
7-1
Pre-Activity How can the geometric mean be used to view paintings?
Read the introduction to Lesson 7-1 at the top of page 342 in your textbook.
• What is a disadvantage of standing too close to a painting?
• What is a disadvantage of standing too far from a painting?
Reading the Lesson1. In the past, when you have seen the word mean in mathematics, it referred to the
average or arithmetic mean of the two numbers.
a. Complete the following by writing an algebraic expression in each blank.
If a and b are two positive numbers, then the geometric mean between a and b is
and their arithmetic mean is .
b. Explain in words, without using any mathematical symbols, the difference betweenthe geometric mean and the algebraic mean.
2. Let r and s be two positive numbers. In which of the following equations is z equal to thegeometric mean between r and s?
A. !zs
! " !zr! B. !z
r! " !z
s! C. s : z " z: r D. !z
r! " !
zs! E. !
zr! " !
zs! F. !
zs! " !z
r!
3. Supply the missing words or phrases to complete the statement of each theorem.
a. The measure of the altitude drawn from the vertex of the right angle of a right triangle
to its hypotenuse is the between the measures of the two
segments of the .
b. If the altitude is drawn from the vertex of the angle of a right
triangle to its hypotenuse, then the measure of a of the triangle
is the between the measure of the hypotenuse and the segment
of the adjacent to that leg.
c. If the altitude is drawn from the of the right angle of a right
triangle to its , then the two triangles formed are
to the given triangle and to each other.
Helping You Remember4. A good way to remember a new mathematical concept is to relate it to something you
already know. How can the meaning of mean in a proportion help you to remember howto find the geometric mean between two numbers?
© Glencoe/McGraw-Hill 356 Glencoe Geometry
Mathematics and MusicPythagoras, a Greek philosopher who lived during the sixth century B.C.,believed that all nature, beauty, and harmony could be expressed by whole-number relationships. Most people remember Pythagoras for his teachingsabout right triangles. (The sum of the squares of the legs equals the square ofthe hypotenuse.) But Pythagoras also discovered relationships between themusical notes of a scale. These relationships can be expressed as ratios.
C D E F G A B C$
!11! !
89! !
45! !
34! !
23! !
35! !1
85! !
12!
When you play a stringed instrument, The C string can be usedyou produce different notes by placing to produce F by placingyour finger on different places on a string. a finger !
34! of the way
This is the result of changing the length along the string.of the vibrating part of the string.
Suppose a C string has a length of 16 inches. Write and solve proportions to determine what length of string would have to vibrate to produce the remaining notes of the scale.
1. D 2. E 3. F
4. G 5. A 6. B
7. C$
8. Complete to show the distance between finger positions on the 16-inch
C string for each note. For example, C(16) # D&14!29!' " 1!
79!.
C D E F G A B C$
9. Between two consecutive musical notes, there is either a whole step or a half step. Using the distances you found in Exercise 8, determine what two pairs of notes have a half step between them.
1!79! in.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
34 of C string
Study Guide and InterventionThe Pythagorean Theorem and Its Converse
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
© Glencoe/McGraw-Hill 357 Glencoe Geometry
Less
on
7-2
The Pythagorean Theorem In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse.
!ABC is a right triangle, so a2 % b2 " c2.
Prove the Pythagorean Theorem.With altitude C"D", each leg a and b is a geometric mean between hypotenuse c and the segment of the hypotenuse adjacent to that leg.
!ac
! " !ay! and !b
c! " !
bx!, so a2 " cy and b2 " cx.
Add the two equations and substitute c " y % x to geta2 % b2 " cy % cx " c( y % x) " c2.
c yx a
b
hA C
BD
c a
bA C
B
Example 1Example 1
Example 2Example 2
a. Find a.
a2 % b2 " c2 Pythagorean Theorem
a2 % 122 " 132 b " 12, c " 13
a2 % 144 " 169 Simplify.a2 " 25 Subtract.
a " 5 Take the square root of each side.
a
12
13
AC
B
b. Find c.
a2 % b2 " c2 Pythagorean Theorem
202 % 302 " c2 a " 20, b " 30
400 % 900 " c2 Simplify.
1300 " c2 Add.
!1300" " c Take the square root of each side.
36.1 # c Use a calculator.
c
30
20
AC
B
ExercisesExercises
Find x.
1. 2. 3.
4. 5. 6. x
1128
x
33
16x
59
49
x
6525
x
159
x
3 3
© Glencoe/McGraw-Hill 358 Glencoe Geometry
Converse of the Pythagorean Theorem If the sum of the squares of the measures of the two shorter sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle.
If the three whole numbers a, b, and c satisfy the equation a2 % b2 " c2, then the numbers a, b, and c form a If a2 % b2 " c2, then Pythagorean triple. !ABC is a right triangle.
Determine whether !PQR is a right triangle.a2 % b2 " c2 Pythagorean Theorem
102 % (10!3")2 " 202 a " 10, b " 10!3", c " 20
100 % 300 " 400 Simplify.
400 " 400✓ Add.
The sum of the squares of the two shorter sides equals the square of the longest side, so thetriangle is a right triangle.
Determine whether each set of measures can be the measures of the sides of aright triangle. Then state whether they form a Pythagorean triple.
1. 30, 40, 50 2. 20, 30, 40 3. 18, 24, 30
4. 6, 8, 9 5. !37!, !
47!, !
57! 6. 10, 15, 20
7. !5", !12", !13" 8. 2, !8", !12" 9. 9, 40, 41
A family of Pythagorean triples consists of multiples of known triples. For eachPythagorean triple, find two triples in the same family.
10. 3, 4, 5 11. 5, 12, 13 12. 7, 24, 25
10!%3
20 10
QR
P
c
ab
A
C
B
Study Guide and Intervention (continued)
The Pythagorean Theorem and Its Converse
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
ExampleExample
ExercisesExercises
Skills PracticeThe Pythagorean Theorem and Its Converse
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
© Glencoe/McGraw-Hill 359 Glencoe Geometry
Less
on
7-2
Find x.
1. 2. 3.
4. 5. 6.
Determine whether !STU is a right triangle for the given vertices. Explain.
7. S(5, 5), T(7, 3), U(3, 2) 8. S(3, 3), T(5, 5), U(6, 0)
9. S(4, 6), T(9, 1), U(1, 3) 10. S(0, 3), T(#2, 5), U(4, 7)
11. S(#3, 2), T(2, 7), U(#1, 1) 12. S(2, #1), T(5, 4), U(6, #3)
Determine whether each set of measures can be the measures of the sides of aright triangle. Then state whether they form a Pythagorean triple.
13. 12, 16, 20 14. 16, 30, 32 15. 14, 48, 50
16. !25!, !
45!, !
65! 17. 2!6", 5, 7 18. 2!2", 2!7", 6
x
31
14x9 9
8
x12.5
25
x
1232
x
12
13x
12
9
© Glencoe/McGraw-Hill 360 Glencoe Geometry
Find x.
1. 2. 3.
4. 5. 6.
Determine whether !GHI is a right triangle for the given vertices. Explain.
7. G(2, 7), H(3, 6), I(#4, #1) 8. G(#6, 2), H(1, 12), I(#2, 1)
9. G(#2, 1), H(3, #1), I(#4, #4) 10. G(#2, 4), H(4, 1), I(#1, #9)
Determine whether each set of measures can be the measures of the sides of aright triangle. Then state whether they form a Pythagorean triple.
11. 9, 40, 41 12. 7, 28, 29 13. 24, 32, 40
14. !95!, !
152!, 3 15. !
13!, , 1 16. , , !
47!
17. CONSTRUCTION The bottom end of a ramp at a warehouse is 10 feet from the base of the main dock and is 11 feet long. How high is the dock?
11 ft?
dock
ramp
10 ft
2!3"!7
!4"!7
2!2"!3
x 2424
42
x16
14
x
34
22
x26
2618
x
34 21x
13
23
Practice The Pythagorean Theorem and Its Converse
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
Reading to Learn MathematicsThe Pythagorean Theorem and Its Converse
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
© Glencoe/McGraw-Hill 361 Glencoe Geometry
Less
on
7-2
Pre-Activity How are right triangles used to build suspension bridges?
Read the introduction to Lesson 7-2 at the top of page 350 in your textbook.
Do the two right triangles shown in the drawing appear to be similar?Explain your reasoning.
Reading the Lesson
1. Explain in your own words the difference between how the Pythagorean Theorem is usedand how the Converse of the Pythagorean Theorem is used.
2. Refer to the figure. For this figure, which statements are true?
A. m2 % n2 " p2 B. n2 " m2 % p2
C. m2 " n2 % p2 D. m2 " p2 # n2
E. p2 " n2 # m2 F. n2 # p2 " m2
G. n " !m2 %"p2" H. p " !m2 #"n2"
3. Is the following statement true or false?A Pythagorean triple is any group of three numbers for which the sum of the squares of thesmaller two numbers is equal to the square of the largest number. Explain your reasoning.
4. If x, y, and z form a Pythagorean triple and k is a positive integer, which of the followinggroups of numbers are also Pythagorean triples?
A. 3x, 4y, 5z B. 3x, 3y, 3z C. #3x, #3y, #3z D. kx, ky, kz
Helping You Remember
5. Many students who studied geometry long ago remember the Pythagorean Theorem as theequation a2 % b2 " c2, but cannot tell you what this equation means. A formula is uselessif you don’t know what it means and how to use it. How could you help someone who hasforgotten the Pythagorean Theorem remember the meaning of the equation a2 % b2 " c2?
pm
n
© Glencoe/McGraw-Hill 362 Glencoe Geometry
Converse of a Right Triangle TheoremYou have learned that the measure of the altitude from the vertex ofthe right angle of a right triangle to its hypotenuse is the geometricmean between the measures of the two segments of the hypotenuse.Is the converse of this theorem true? In order to find out, it will helpto rewrite the original theorem in if-then form as follows.
If !ABQ is a right triangle with right angle at Q, then QP is the geometric mean between AP and PB, where Pis between A and B and Q"P" is perpendicular to A"B".
1. Write the converse of the if-then form of the theorem.
2. Is the converse of the original theorem true? Refer to the figure at the right to explain your answer.
You may find it interesting to examine the other theorems inChapter 7 to see whether their converses are true or false. You willneed to restate the theorems carefully in order to write theirconverses.
Q
BPA
Q
BPA
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
Study Guide and InterventionSpecial Right Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
© Glencoe/McGraw-Hill 363 Glencoe Geometry
Less
on
7-3
Properties of 45°-45°-90° Triangles The sides of a 45°-45°-90° right triangle have aspecial relationship.
If the leg of a 45°-45°-90°right triangle is x units, show that the hypotenuse is x#2! units.
Using the Pythagorean Theorem with a " b " x, then
c2 " a2 % b2
" x2 % x2
" 2x2
c " !2x2"" x!2"
x!%
x
x 245#
45#
In a 45°-45°-90° right triangle the hypotenuse is #2! times the leg. If the hypotenuse is 6 units,find the length of each leg.The hypotenuse is !2" times the leg, sodivide the length of the hypotenuse by !2".
a "
"
"
" 3!2" units
6!2"!2
6!2"!!2"!2"
6!!2"
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find x.
1. 2. 3.
4. 5. 6.
7. Find the perimeter of a square with diagonal 12 centimeters.
8. Find the diagonal of a square with perimeter 20 inches.
9. Find the diagonal of a square with perimeter 28 meters.
x 3!%2x 18x x
18
x10x
45#3!%2
x
8
45#
45#
© Glencoe/McGraw-Hill 364 Glencoe Geometry
Properties of 30°-60°-90° Triangles The sides of a 30°-60°-90° right triangle alsohave a special relationship.
In a 30°-60°-90° right triangle, show that the hypotenuse is twice the shorter leg and the longer leg is #3! times the shorter leg.
!MNQ is a 30°-60°-90° right triangle, and the length of the hypotenuse M"N" is two times the length of the shorter side N"Q".Using the Pythagorean Theorem,a2 " (2x) 2 # x2
" 4x2 # x2
" 3x2
a " !3x2"" x!3"
In a 30°-60°-90° right triangle, the hypotenuse is 5 centimeters.Find the lengths of the other two sides of the triangle.If the hypotenuse of a 30°-60°-90° right triangle is 5 centimeters, then the length of theshorter leg is half of 5 or 2.5 centimeters. The length of the longer leg is !3" times the length of the shorter leg, or (2.5)(!3") centimeters.
Find x and y.
1. 2. 3.
4. 5. 6.
7. The perimeter of an equilateral triangle is 32 centimeters. Find the length of an altitudeof the triangle to the nearest tenth of a centimeter.
8. An altitude of an equilateral triangle is 8.3 meters. Find the perimeter of the triangle tothe nearest tenth of a meter.
xy
60#
20
x y60#
12
x y
30#
9!%3
x
y
11
30#
x
y
60#
8
x
y30#
60#12
x
a
N
Q
P
M
2x30#30#
60#
60#
Study Guide and Intervention (continued)
Special Right Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
ExercisesExercises
Example 1Example 1
Example 2Example 2
!MNP is an equilateraltriangle.!MNQ is a 30°-60°-90°right triangle.
Skills PracticeSpecial Right Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
© Glencoe/McGraw-Hill 365 Glencoe Geometry
Less
on
7-3
Find x and y.
1. 2. 3.
4. 5. 6.
For Exercises 7–9, use the figure at the right.
7. If a " 11, find b and c.
8. If b " 15, find a and c.
9. If c " 9, find a and b.
For Exercises 10 and 11, use the figure at the right.
10. The perimeter of the square is 30 inches. Find the length of B"C".
11. Find the length of the diagonal B"D".
12. The perimeter of the equilateral triangle is 60 meters. Find the length of an altitude.
13. !GEC is a 30°-60°-90° triangle with right angle at E, and E"C" is the longer leg. Find the coordinates of G in Quadrant I for E(1, 1) and C(4, 1).
E
FGD 60#
A B
CD 45#
bA
B
C
ac 60#
30#
yx#13
1313
13
y
x60#
16
y
x
45# 8
y
x45#
12
y
x
30#
32
y
x60# 24
© Glencoe/McGraw-Hill 366 Glencoe Geometry
Find x and y.
1. 2. 3.
4. 5. 6.
For Exercises 7–9, use the figure at the right.
7. If a " 4!3", find b and c.
8. If x " 3!3", find a and CD.
9. If a " 4, find CD, b, and y.
10. The perimeter of an equilateral triangle is 39 centimeters. Find the length of an altitudeof the triangle.
11. !MIP is a 30°-60°-90° triangle with right angle at I, and I"P" the longer leg. Find thecoordinates of M in Quadrant I for I(3, 3) and P(12, 3).
12. !TJK is a 45°-45°-90° triangle with right angle at J. Find the coordinates of T inQuadrant II for J(#2, #3) and K(3, #3).
13. BOTANICAL GARDENS One of the displays at a botanical garden is an herb garden planted in the shape of a square. The square measures 6 yards on each side. Visitors can view the herbs from adiagonal pathway through the garden. How long is the pathway?
6 yd 6 yd
6 yd
6 yd
bA
B
C
Da
x
y60#
30#
c
x45#
11
y60#3.5
xy
x#y 28
y
x30#
26y
x2560#
yx
45#9
Practice Special Right Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
Reading to Learn MathematicsSpecial Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
© Glencoe/McGraw-Hill 367 Glencoe Geometry
Less
on
7-3
Pre-Activity How is triangle tiling used in wallpaper design?
Read the introduction to Lesson 7-3 at the top of page 357 in your textbook.• How can you most completely describe the larger triangle and the two
smaller triangles in tile 15?
• How can you most completely describe the larger triangle and the twosmaller triangles in tile 16? (Include angle measures in describing all thetriangles.)
Reading the Lesson1. Supply the correct number or numbers to complete each statement.
a. In a 45°-45°-90° triangle, to find the length of the hypotenuse, multiply the length of a
leg by .
b. In a 30°-60°-90° triangle, to find the length of the hypotenuse, multiply the length of
the shorter leg by .
c. In a 30°-60°-90° triangle, the longer leg is opposite the angle with a measure of .
d. In a 30°-60°-90° triangle, to find the length of the longer leg, multiply the length of
the shorter leg by .
e. In an isosceles right triangle, each leg is opposite an angle with a measure of .
f. In a 30°-60°-90° triangle, to find the length of the shorter leg, divide the length of the
longer leg by .
g. In 30°-60°-90° triangle, to find the length of the longer leg, divide the length of the
hypotenuse by and multiply the result by .
h. To find the length of a side of a square, divide the length of the diagonal by .
2. Indicate whether each statement is always, sometimes, or never true.a. The lengths of the three sides of an isosceles triangle satisfy the Pythagorean
Theorem.b. The lengths of the sides of a 30°-60°-90° triangle form a Pythagorean triple.c. The lengths of all three sides of a 30°-60°-90° triangle are positive integers.
Helping You Remember3. Some students find it easier to remember mathematical concepts in terms of specific
numbers rather than variables. How can you use specific numbers to help you rememberthe relationship between the lengths of the three sides in a 30°-60°-90° triangle?
© Glencoe/McGraw-Hill 368 Glencoe Geometry
Constructing Values of Square RootsThe diagram at the right shows a right isosceles triangle with two legs of length 1 inch. By the Pythagorean Theorem, the length of the hypotenuse is !2" inches. By constructing an adjacent right triangle with legs of !2" inches and 1 inch, you can create a segment of length !3".
By continuing this process as shown below, you can construct a “wheel” of square roots. This wheel is called the “Wheel of Theodorus”after a Greek philosopher who lived about 400 B.C.
Continue constructing the wheel until you make a segment oflength !18".
!%
1
1
1
3!%
2
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
1
1
11
1
1
!%2
!%3
!%5
!%6
!%7
!%8
!"10
!"11 !"12!"13
!"14
!"15
!"17
!"18
!"16 " 4
!%4 " 2
!%9 " 3
Study Guide and InterventionTrigonometry
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
© Glencoe/McGraw-Hill 369 Glencoe Geometry
Less
on
7-4
Trigonometric Ratios The ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine, cosine, and tangent, which are abbreviated sin, cos, and tan,respectively.
sin R " !leg
hyopppotoesnitues#
eR
! cos R " tan R "
" !rt! " !
st! " !
rs!
Find sin A, cos A, and tan A. Express each ratio as a decimal to the nearest thousandth.
sin A " !ohpyppoostietneulseeg
! cos A " !ahdyjpaocteenntulseeg
! tan A " !aopd
pja
ocseintet
lleegg!
" !BAB
C! " !A
ABC! " !
BAC
C!
" !153! " !
11
23! " !1
52!
# 0.385 # 0.923 # 0.417
Find the indicated trigonometric ratio as a fraction and as a decimal. If necessary, round to the nearest ten-thousandth.
1. sin A 2. tan B
3. cos A 4. cos B
5. sin D 6. tan E
7. cos E 8. cos D
16
1620
12
3430
C
B
A D F
E
12
135C
B
A
leg opposite #R!!!leg adjacent to #R
leg adjacent to #R!!!hypotenuse
s
tr
T
S
R
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 370 Glencoe Geometry
Use Trigonometric Ratios In a right triangle, if you know the measures of two sidesor if you know the measures of one side and an acute angle, then you can use trigonometricratios to find the measures of the missing sides or angles of the triangle.
Find x, y, and z. Round each measure to the nearest whole number. 1858#
x # CB y
zA
Study Guide and Intervention (continued)
Trigonometry
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
a. Find x.
x % 58 " 90x " 32
b. Find y.
tan A " !1y8!
tan 58° " !1y8!
y " 18 tan 58°y # 29
c. Find z.
cos A " !1z8!
cos 58° " !1z8!
z cos 58° " 18
z " !cos18
58°!
z # 34
ExercisesExercises
Find x. Round to the nearest tenth.
1. 2.
3. 4.
5. 6.15
64# x1640#
x
4
1x#12
5x#
12 16
x#3228#
x
ExampleExample
Skills PracticeTrigonometry
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
© Glencoe/McGraw-Hill 371 Glencoe Geometry
Less
on
7-4
Use !RST to find sin R, cos R, tan R, sin S, cos S, and tan S.Express each ratio as a fraction and as a decimal to the nearest hundredth.
1. r " 16, s " 30, t " 34 2. r " 10, s " 24, t " 26
Use a calculator to find each value. Round to the nearest ten-thousandth.
3. sin 5 4. tan 23 5. cos 61
6. sin 75.8 7. tan 17.3 8. cos 52.9
Use the figure to find each trigonometric ratio. Express answers as a fraction and as a decimal rounded to thenearest ten-thousandth.
9. tan C 10. sin A 11. cos C
Find the measure of each acute angle to the nearest tenth of a degree.
12. sin B " 0.2985 13. tan A " 0.4168 14. cos R " 0.8443
15. tan C " 0.3894 16. cos B " 0.7329 17. sin A " 0.1176
Find x. Round to the nearest tenth.
18. 19. 20.
19
x
33# UL
S
27
x #8
BA
C27
x #
13
BA
C
41
409B
AC
sR
S
T
rt
© Glencoe/McGraw-Hill 372 Glencoe Geometry
Use !LMN to find sin L, cos L, tan L, sin M, cos M, and tan M.Express each ratio as a fraction and as a decimal to the nearest hundredth.
1. ! " 15, m " 36, n" 39 2. ! " 12, m " 12!3", n " 24
Use a calculator to find each value. Round to the nearest ten-thousandth.
3. sin 92.4 4. tan 27.5 5. cos 64.8
Use the figure to find each trigonometric ratio. Express answers as a fraction and as a decimal rounded to the nearest ten-thousandth.
6. cos A 7. tan B 8. sin A
Find the measure of each acute angle to the nearest tenth of a degree.
9. sin B " 0.7823 10. tan A " 0.2356 11. cos R " 0.6401
Find x. Round to the nearest tenth.
12. 13. 14.
15. GEOGRAPHY Diego used a theodolite to map a region of land for his class in geomorphology. To determine the elevation of a vertical rockformation, he measured the distance from the base of the formation to his position and the angle between the ground and the line of sight to the top of the formation. The distance was 43 meters and the angle was 36 degrees. What is the height of the formation to the nearest meter?
36#43 m
41#x
3229
x #9
23
x #
11
15
5!"105
CA
B
ML
N
Practice Trigonometry
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
Reading to Learn MathematicsTrigonometry
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
© Glencoe/McGraw-Hill 373 Glencoe Geometry
Less
on
7-4
Pre-Activity How can surveyors determine angle measures?
Read the introduction to Lesson 7-4 at the top of page 364 in your textbook.
• Why is it important to determine the relative positions accurately innavigation? (Give two possible reasons.)
• What does calibrated mean?
Reading the Lesson
1. Refer to the figure. Write a ratio using the side lengths in the figure to represent each of the following trigonometric ratios.
A. sin N B. cos N
C. tan N D. tan M
E. sin M F. cos M
2. Assume that you enter each of the expressions in the list on the left into your calculator.Match each of these expressions with a description from the list on the right to tell whatyou are finding when you enter this expression.
P
M N
a. sin 20
b. cos 20
c. sin#1 0.8
d. tan#1 0.8
e. tan 20
f. cos#1 0.8
i. the degree measure of an acute angle whose cosine is 0.8
ii. the ratio of the length of the leg adjacent to the 20° angle to thelength of hypotenuse in a 20°-70°-90° triangle
iii.the degree measure of an acute angle in a right triangle for which the ratio of the length of the opposite leg to the length ofthe adjacent leg is 0.8
iv. the ratio of the length of the leg opposite the 20° angle to thelength of the leg adjacent to it in a 20°-70°-90° triangle
v. the ratio of the length of the leg opposite the 20° angle to thelength of hypotenuse in a 20°-70°-90° triangle
vi. the degree measure of an acute angle in a right triangle for which the ratio of the length of the opposite leg to the length ofthe hypotenuse is 0.8
Helping You Remember
3. How can the co in cosine help you to remember the relationship between the sines andcosines of the two acute angles of a right triangle?
© Glencoe/McGraw-Hill 374 Glencoe Geometry
Sine and Cosine of AnglesThe following diagram can be used to obtain approximate values for the sineand cosine of angles from 0° to 90°. The radius of the circle is 1. So, the sineand cosine values can be read directly from the vertical and horizontal axes.
Find approximate values for sin 40°and cos 40#. Consider the triangle formed by the segment marked 40°, as illustrated by the shaded triangle at right.
sin 40° " !ac! # !
0.164! or 0.64 cos 40° " !
bc! # !
0.177! or 0.77
1. Use the diagram above to complete the chart of values.
2. Compare the sine and cosine of two complementary angles (angles whose sum is 90°). What do you notice?
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
90°
0°
10°
20°
30°
40°
50°
60°
70°80°
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
x° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90°
sin x° 0.64
cos x° 0.77
1
0
40°0.64
c " 1 unit
x °b " cos x ° 0.77 1
a " sin x °
ExampleExample
Study Guide and InterventionAngles of Elevation and Depression
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill 375 Glencoe Geometry
Less
on
7-5
Angles of Elevation Many real-world problems that involve looking up to an object can be described in terms of an angle of elevation, which is the angle between an observer’s line of sight and a horizontal line.
The angle of elevation from point A to the top of a cliff is 34°. If point A is 1000 feet from the base of the cliff,how high is the cliff?Let x " the height of the cliff.
tan 34° " !10x00! tan " !
oapdpjaocseitnet!
1000(tan 34°) " x Multiply each side by 1000.
674.5 " x Use a calculator.
The height of the cliff is about 674.5 feet.
Solve each problem. Round measures of segments to the nearest whole numberand angles to the nearest degree.
1. The angle of elevation from point A to the top of a hill is 49°.If point A is 400 feet from the base of the hill, how high is the hill?
2. Find the angle of elevation of the sun when a 12.5-meter-tall telephone pole casts a 18-meter-long shadow.
3. A ladder leaning against a building makes an angle of 78°with the ground. The foot of the ladder is 5 feet from the building. How long is the ladder?
4. A person whose eyes are 5 feet above the ground is standing on the runway of an airport 100 feet from the control tower.That person observes an air traffic controller at the window of the 132-foot tower. What is the angle of elevation?
?5 ft
100 ft
132 ft
78#5 ft
?
18 m
12.5 m
sun
?
✹
400 ft
?
49#A
?
1000 ft34#A
x
angle ofelevation
line of si
ght
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 376 Glencoe Geometry
Angles of Depression When an observer is looking down, the angle of depression is the angle between the observer’s line of sight and a horizontal line.
The angle of depression from the top of an 80-foot building to point A on the ground is 42°. How far is the foot of the building from point A?Let x " the distance from point A to the foot of the building. Since the horizontal line is parallel to the ground, the angle of depression#DBA is congruent to #BAC.
tan 42° " !8x0! tan " !
oa
pd
pja
ocseitnet!
x(tan 42°) " 80 Multiply each side by x.
x " !tan80
42°! Divide each side by tan 42°.
x # 88.8 Use a calculator.
Point A is about 89 feet from the base of the building.
Solve each problem. Round measures of segments to the nearest whole numberand angles to the nearest degree.
1. The angle of depression from the top of a sheer cliff to point A on the ground is 35°. If point A is 280 feet from the base of the cliff, how tall is the cliff?
2. The angle of depression from a balloon on a 75-foot string to a person on the ground is 36°. How high is the balloon?
3. A ski run is 1000 yards long with a vertical drop of 208 yards. Find the angle of depression from the top of the ski run to the bottom.
4. From the top of a 120-foot-high tower, an air traffic controller observes an airplane on the runway at an angle of depression of 19°. How far from the base of thetower is the airplane?
120 ft
?
19#
208 yd
?
1000 yd
36#
75 ft ?
A
35#
280 ft
?
A C
BD
x42#
angle ofdepression
horizontal
80 ft
Yline of sight
horizontalangle ofdepression
Study Guide and Intervention (continued)
Angles of Elevation and Depression
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
ExercisesExercises
ExampleExample
Skills PracticeAngles of Elevation and Depression
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill 377 Glencoe Geometry
Less
on
7-5
Name the angle of depression or angle of elevation in each figure.
1. 2.
3. 4.
5. MOUNTAIN BIKING On a mountain bike trip along the Gemini Bridges Trail in Moab,Utah, Nabuko stopped on the canyon floor to get a good view of the twin sandstonebridges. Nabuko is standing about 60 meters from the base of the canyon cliff, and thenatural arch bridges are about 100 meters up the canyon wall. If her line of sight is fivefeet above the ground, what is the angle of elevation to the top of the bridges? Round tothe nearest tenth degree.
6. SHADOWS Suppose the sun casts a shadow off a 35-foot building.If the angle of elevation to the sun is 60°, how long is the shadow to the nearest tenth of a foot?
7. BALLOONING From her position in a hot-air balloon, Angie can see her car parked in afield. If the angle of depression is 8° and Angie is 38 meters above the ground, what isthe straight-line distance from Angie to her car? Round to the nearest whole meter.
8. INDIRECT MEASUREMENT Kyle is at the end of a pier 30 feet above the ocean. His eye level is 3 feet above the pier. He is using binoculars to watch a whale surface. If the angle of depression of the whale is 20°, how far is the whale from Kyle’s binoculars? Round to the nearest tenth foot.
whale water level
20#Kyle’s eyes
pier3 ft
30 ft
60#?
35 ft
Z
P
W
R
D
A
C
B
T
W
R
S
F
T
L
S
© Glencoe/McGraw-Hill 378 Glencoe Geometry
Name the angle of depression or angle of elevation in each figure.
1. 2.
3. WATER TOWERS A student can see a water tower from the closest point of the soccerfield at San Lobos High School. The edge of the soccer field is about 110 feet from thewater tower and the water tower stands at a height of 32.5 feet. What is the angle ofelevation if the eye level of the student viewing the tower from the edge of the soccerfield is 6 feet above the ground? Round to the nearest tenth degree.
4. CONSTRUCTION A roofer props a ladder against a wall so that the top of the ladderreaches a 30-foot roof that needs repair. If the angle of elevation from the bottom of theladder to the roof is 55°, how far is the ladder from the base of the wall? Round youranswer to the nearest foot.
5. TOWN ORDINANCES The town of Belmont restricts the height of flagpoles to 25 feet on any property. Lindsay wants to determinewhether her school is in compliance with the regulation. Her eye level is 5.5 feet from the ground and she stands 36 feet from theflagpole. If the angle of elevation is about 25°, what is the height of the flagpole to the nearest tenth foot?
6. GEOGRAPHY Stephan is standing on a mesa at the Painted Desert. The elevation ofthe mesa is about 1380 meters and Stephan’s eye level is 1.8 meters above ground. IfStephan can see a band of multicolored shale at the bottom and the angle of depressionis 29°, about how far is the band of shale from his eyes? Round to the nearest meter.
7. INDIRECT MEASUREMENT Mr. Dominguez is standing on a 40-foot ocean bluff near his home. He can see his two dogs on the beach below. If his line of sight is 6 feet above the ground and the angles of depression to his dogs are 34°and 48°, how far apart are the dogs to the nearest foot?
48# 34#
40 ft
6 ft
Mr. Dominguez
bluff
25#5.5 ft
36 ft
x
R
M
P
L
T
Y
R
Z
Practice Angles of Elevation and Depression
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
Reading to Learn MathematicsAngles of Elevation and Depression
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill 379 Glencoe Geometry
Less
on
7-5
Pre-Activity How do airline pilots use angles of elevation and depression?
Read the introduction to Lesson 7-5 at the top of page 371 in your textbook.
What does the angle measure tell the pilot?
Reading the Lesson
1. Refer to the figure. The two observers are looking at one another. Select the correct choice for each question.
a. What is the line of sight?(i) line RS (ii) line ST (iii) line RT (iv) line TU
b. What is the angle of elevation?(i) #RST (ii) #SRT (iii) #RTS (iv) #UTR
c. What is the angle of depression?(i) #RST (ii) #SRT (iii) #RTS (iv) #UTR
d. How are the angle of elevation and the angle of depression related?(i) They are complementary.(ii) They are congruent.(iii) They are supplementary.(iv) The angle of elevation is larger than the angle of depression.
e. Which postulate or theorem that you learned in Chapter 3 supports your answer forpart c?(i) Corresponding Angles Postulate(ii) Alternate Exterior Angles Theorem(iii) Consecutive Interior Angles Theorem(iv) Alternate Interior Angles Theorem
2. A student says that the angle of elevation from his eye to the top of a flagpole is 135°.What is wrong with the student’s statement?
Helping You Remember
3. A good way to remember something is to explain it to someone else. Suppose a classmatefinds it difficult to distinguish between angles of elevation and angles of depression. Whatare some hints you can give her to help her get it right every time?
S
T observer attop of building
observeron ground R
U
© Glencoe/McGraw-Hill 380 Glencoe Geometry
Reading MathematicsThe three most common trigonometric ratios are sine, cosine, and tangent. Three other ratios are thecosecant, secant, and cotangent. The chart below shows abbreviations and definitions for all six ratios.Refer to the triangle at the right.
Use the abbreviations to rewrite each statement as an equation.
1. The secant of angle A is equal to 1 divided by the cosine of angle A.
2. The cosecant of angle A is equal to 1 divided by the sine of angle A.
3. The cotangent of angle A is equal to 1 divided by the tangent of angle A.
4. The cosecant of angle A multiplied by the sine of angle A is equal to 1.
5. The secant of angle A multiplied by the cosine of angle A is equal to 1.
6. The cotangent of angle A times the tangent of angle A is equal to 1.
Use the triangle at right. Write each ratio.
7. sec R 8. csc R 9. cot R
10. sec S 11. csc S 12. cot S
13. If sin x° " 0.289, find the value of csc x°.
14. If tan x° " 1.376, find the value of cot x°.
R
T S
ts
r
A
ca
b
B
C
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
Abbreviation Read as: Ratio
sin A the sine of #A " !ac!
cos A the cosine of #A " !bc!
tan A the tangent of #A " !ab!
csc A the cosecant of #A " !ac
!
sec A the secant of #A " !bc
!
cot A the cotangent of #A " !ba!
leg adjacent to #A!!!
leg opposite #A
hypotenuse!!!leg adjacent to #A
hypotenuse!!leg opposite #A
leg opposite #A!!!leg adjacent to #A
leg adjacent to #A!!!hypotenuse
leg opposite #A!!hypotenuse
Study Guide and InterventionThe Law of Sines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
© Glencoe/McGraw-Hill 381 Glencoe Geometry
Less
on
7-6
The Law of Sines In any triangle, there is a special relationship between the angles ofthe triangle and the lengths of the sides opposite the angles.
Law of Sines !sin
aA
! " !sin
bB
! " !sin
cC
!
In !ABC, find b.
!sin
cC
! " !sin
bB
! Law of Sines
!sin
3045°! " !
sinb74°! m#C " 45, c " 30, m#B " 74
b sin 45° " 30 sin 74° Cross multiply.
b " !30
sisnin45
7°4°
! Divide each side by sin 45°.
b # 40.8 Use a calculator.
45#
3074#
b
B
AC
In !DEF, find m"D.
!sin
dD
! " !sin
eE
! Law of Sines
!si
2n8D
! " !sin
2458°!
d " 28, m#E " 58, e " 24
24 sin D " 28 sin 58° Cross multiply.
sin D " !28 s
2in4
58°! Divide each side by 24.
D " sin#1 !28 s
2in4
58°! Use the inverse sine.
D # 81.6° Use a calculator.
58#
24
28
E
FD
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find each measure using the given measures of !ABC. Round angle measures tothe nearest degree and side measures to the nearest tenth.
1. If c " 12, m#A " 80, and m#C " 40, find a.
2. If b " 20, c " 26, and m#C " 52, find m#B.
3. If a " 18, c " 16, and m#A " 84, find m#C.
4. If a " 25, m#A " 72, and m#B " 17, find b.
5. If b " 12, m#A " 89, and m#B" 80, find a.
6. If a " 30, c " 20, and m#A " 60, find m#C.
© Glencoe/McGraw-Hill 382 Glencoe Geometry
Use the Law of Sines to Solve Problems You can use the Law of Sines to solvesome problems that involve triangles.
Law of SinesLet !ABC be any triangle with a, b, and c representing the measures of the sides opposite the angles with measures A, B, and C, respectively. Then !sin
aA! " !sin
bB! " !sin
cC!.
Isosceles !ABC has a base of 24 centimeters and a vertex angle of 68°. Find the perimeter of the triangle.The vertex angle is 68°, so the sum of the measures of the base angles is 112 and m#A " m#C " 56.
!sin
bB
! " !sin
aA
! Law of Sines
!sin
2468°! " !
sina56°! m#B " 68, b " 24, m#A " 56
a sin 68° " 24 sin 56° Cross multiply.
a " !24
sisnin68
5°6°
! Divide each side by sin 68°.
# 21.5 Use a calculator.
The triangle is isosceles, so c " 21.5.The perimeter is 24 % 21.5 % 21.5 or about 67 centimeters.
Draw a triangle to go with each exercise and mark it with the given information.Then solve the problem. Round angle measures to the nearest degree and sidemeasures to the nearest tenth.
1. One side of a triangular garden is 42.0 feet. The angles on each end of this side measure66° and 82°. Find the length of fence needed to enclose the garden.
2. Two radar stations A and B are 32 miles apart. They locate an airplane X at the sametime. The three points form #XAB, which measures 46°, and #XBA, which measures52°. How far is the airplane from each station?
3. A civil engineer wants to determine the distances from points A and B to an inaccessiblepoint C in a river. #BAC measures 67° and #ABC measures 52°. If points A and B are82.0 feet apart, find the distance from C to each point.
4. A ranger tower at point A is 42 kilometers north of a ranger tower at point B. A fire atpoint C is observed from both towers. If #BAC measures 43° and #ABC measures 68°,which ranger tower is closer to the fire? How much closer?
68#
bc a
24
B
CA
Study Guide and Intervention (continued)
The Law of Sines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
ExampleExample
ExercisesExercises
Skills PracticeThe Law of Sines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
© Glencoe/McGraw-Hill 383 Glencoe Geometry
Less
on
7-6
Find each measure using the given measures from !ABC. Round angle measuresto the nearest tenth degree and side measures to the nearest tenth.
1. If m#A " 35, m#B " 48, and b " 28, find a.
2. If m#B " 17, m#C " 46, and c " 18, find b.
3. If m#C " 86, m#A " 51, and a " 38, find c.
4. If a " 17, b " 8, and m#A " 73, find m#B.
5. If c " 38, b " 34, and m#B " 36, find m#C.
6. If a " 12, c " 20, and m#C " 83, find m#A.
7. If m#A " 22, a " 18, and m#B" 104, find b.
Solve each !PQR described below. Round measures to the nearest tenth.
8. p " 27, q " 40, m#P " 33
9. q " 12, r " 11, m#R " 16
10. p " 29, q " 34, m#Q " 111
11. If m#P " 89, p " 16, r " 12
12. If m#Q " 103, m#P " 63, p " 13
13. If m#P " 96, m#R " 82, r " 35
14. If m#R " 49, m#Q " 76, r " 26
15. If m#Q " 31, m#P " 52, p " 20
16. If q " 8, m#Q " 28, m#R " 72
17. If r " 15, p " 21, m#P " 128
© Glencoe/McGraw-Hill 384 Glencoe Geometry
Find each measure using the given measures from !EFG. Round angle measuresto the nearest tenth degree and side measures to the nearest tenth.
1. If m#G " 14, m#E " 67, and e " 14, find g.
2. If e " 12.7, m#E " 42, and m#F " 61, find f.
3. If g " 14, f " 5.8, and m#G " 83, find m#F.
4. If e " 19.1, m#G " 34, and m#E " 56, find g.
5. If f " 9.6, g " 27.4, and m#G " 43, find m#F.
Solve each !STU described below. Round measures to the nearest tenth.
6. m#T " 85, s " 4.3, t " 8.2
7. s " 40, u " 12, m#S " 37
8. m#U " 37, t " 2.3, m#T " 17
9. m#S " 62, m#U " 59, s " 17.8
10. t " 28.4, u " 21.7, m#T " 66
11. m#S " 89, s " 15.3, t " 14
12. m#T " 98, m#U " 74, u " 9.6
13. t " 11.8, m#S " 84, m#T " 47
14. INDIRECT MEASUREMENT To find the distance from the edge of the lake to the tree on the island in the lake, Hannah set up atriangular configuration as shown in the diagram. The distance from location A to location B is 85 meters. The measures of the angles at A and B are 51° and 83°, respectively. What is the distancefrom the edge of the lake at B to the tree on the island at C?
A
C
B
Practice The Law of Sines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
Reading to Learn MathematicsThe Law of Sines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
© Glencoe/McGraw-Hill 385 Glencoe Geometry
Less
on
7-6
Pre-Activity How are triangles used in radio astronomy?
Read the introduction to Lesson 7-6 at the top of page 377 in your textbook.
Why might several antennas be better than one single antenna whenstudying distant objects?
Reading the Lesson
1. Refer to the figure. According to the Law of Sines, which of the following are correct statements?
A. !sinm
M! " !sinn
N! " !sinp
P! B. !sin
Mm
! " !si
Nn n! " !
sinP
p!
C. !co
ms M! " !
cosn
N! " !
cops P! D. !
sinm
M! % !
sinn
N! " !
sinp
P!
E. (sin M)2 % (sin N)2 " (sin P)2 F. !sin
pP
! " !sin
mM
! " !sin
nN
!
2. State whether each of the following statements is true or false. If the statement is false,explain why.
a. The Law of Sines applies to all triangles.
b. The Pythagorean Theorem applies to all triangles.
c. If you are given the length of one side of a triangle and the measures of any twoangles, you can use the Law of Sines to find the lengths of the other two sides.
d. If you know the measures of two angles of a triangle, you should use the Law of Sinesto find the measure of the third angle.
e. A friend tells you that in triangle RST, m#R " 132, r " 24 centimeters, and s " 31centimeters. Can you use the Law of Sines to solve the triangle? Explain.
Helping You Remember
3. Many students remember mathematical equations and formulas better if they can statethem in words. State the Law of Sines in your own words without using variables ormathematical symbols.
P
M Np
mn
© Glencoe/McGraw-Hill 386 Glencoe Geometry
IdentitiesAn identity is an equation that is true for all values of the variable for which both sides are defined. One way to verify an identity is to use a right triangle and the definitions fortrigonometric functions.
Verify that (sin A)2 $ (cos A)2 " 1 is an identity.
(sin A)2 % (cos A)2 " &!ac!'2 % &!
bc!'2
" !a2 %
cb2
! " !cc
2
2! " 1
To check whether an equation may be an identity, you can testseveral values. However, since you cannot test all values, youcannot be certain that the equation is an identity.
Test sin 2x " 2 sin x cos x to see if it could be an identity.
Try x " 20. Use a calculator to evaluate each expression.
sin 2x " sin 40 2 sin x cos x " 2 (sin 20)(cos 20)# 0.643 # 2(0.342)(0.940)
# 0.643
Since the left and right sides seem equal, the equation may be an identity.
Use triangle ABC shown above. Verify that each equation is an identity.
1. !csoins
AA
! " !tan1
A! 2. !tsainn
BB
! " !co1s B!
3. tan B cos B " sin B 4. 1 # (cos B)2 " (sin B)2
Try several values for x to test whether each equation could be an identity.
5. cos 2x " (cos x)2 # (sin x)2 6. cos (90 # x) " sin x
B
A C
ca
b
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
Example 1Example 1
Example 2Example 2
Study Guide and InterventionThe Law of Cosines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
© Glencoe/McGraw-Hill 387 Glencoe Geometry
Less
on
7-7
The Law of Cosines Another relationship between the sides and angles of any triangleis called the Law of Cosines. You can use the Law of Cosines if you know three sides of atriangle or if you know two sides and the included angle of a triangle.
Let !ABC be any triangle with a, b, and c representing the measures of the sides opposite Law of Cosines the angles with measures A, B, and C, respectively. Then the following equations are true.
a2 " b2 % c2 # 2bc cos A b2 " a2 % c2 # 2ac cos B c2 " a2 % b2 # 2ab cos C
In !ABC, find c.c2 " a2 % b2 # 2ab cos C Law of Cosines
c2 " 122 % 102 # 2(12)(10)cos 48° a " 12, b " 10, m#C " 48
c " !122 %" 102 #" 2(12)"(10)co"s 48°" Take the square root of each side.
c # 9.1 Use a calculator.
In !ABC, find m"A.a2 " b2 % c2 # 2bc cos A Law of Cosines
72 " 52 % 82 # 2(5)(8) cos A a " 7, b " 5, c " 8
49 " 25 % 64 # 80 cos A Multiply.
#40 " #80 cos A Subtract 89 from each side.
!12! " cos A Divide each side by #80.
cos#1 !12! " A Use the inverse cosine.
60° " A Use a calculator.
Find each measure using the given measures from !ABC. Round angle measuresto the nearest degree and side measures to the nearest tenth.
1. If b " 14, c " 12, and m#A " 62, find a.
2. If a " 11, b " 10, and c " 12, find m#B.
3. If a " 24, b " 18, and c " 16, find m#C.
4. If a " 20, c " 25, and m#B " 82, find b.
5. If b " 18, c " 28, and m#A " 59, find a.
6. If a " 15, b " 19, and c " 15, find m#C.
58
7 CB
A
48#12 10
c
C
BA
Example 1Example 1
Example 2Example 2
ExercisesExercises
© Glencoe/McGraw-Hill 388 Glencoe Geometry
Use the Law of Cosines to Solve Problems You can use the Law of Cosines tosolve some problems involving triangles.
Let !ABC be any triangle with a, b, and c representing the measures of the sides opposite the Law of Cosines angles with measures A, B, and C, respectively. Then the following equations are true.
a2 " b2 % c2 # 2bc cos A b2 " a2 % c2 # 2ac cos B c2 " a2 % b2 # 2ab cos C
Ms. Jones wants to purchase a piece of land with the shape shown. Find the perimeter of the property.Use the Law of Cosines to find the value of a.
a2 " b2 % c2 # 2bc cos A Law of Cosines
a2 " 3002 % 2002 # 2(300)(200) cos 88° b " 300, c " 200, m#A " 88
a " !130,0"00 #"120,0"00 cos" 88°" Take the square root of each side.
# 354.7 Use a calculator.
Use the Law of Cosines again to find the value of c.
c2 " a2 % b2 # 2ab cos C Law of Cosines
c2 " 354.72 % 3002 # 2(354.7)(300) cos 80° a " 354.7, b " 300, m#C " 80
c " !215,8"12.09" # 21"2,820" cos 8"0°" Take the square root of each side.
# 422.9 Use a calculator.
The perimeter of the land is 300 % 200 % 422.9 % 200 or about 1223 feet.
Draw a figure or diagram to go with each exercise and mark it with the giveninformation. Then solve the problem. Round angle measures to the nearest degreeand side measures to the nearest tenth.
1. A triangular garden has dimensions 54 feet, 48 feet, and 62 feet. Find the angles at eachcorner of the garden.
2. A parallelogram has a 68° angle and sides 8 and 12. Find the lengths of the diagonals.
3. An airplane is sighted from two locations, and its position forms an acute triangle withthem. The distance to the airplane is 20 miles from one location with an angle ofelevation 48.0°, and 40 miles from the other location with an angle of elevation of 21.8°.How far apart are the two locations?
4. A ranger tower at point A is directly north of a ranger tower at point B. A fire at point Cis observed from both towers. The distance from the fire to tower A is 60 miles, and thedistance from the fire to tower B is 50 miles. If m#ACB " 62, find the distance betweenthe towers.
200 ft
300 ft
300 ft
88#
80#ca
Study Guide and Intervention (continued)
The Law of Cosines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
ExampleExample
ExercisesExercises
Skills PracticeThe Law of Cosines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
© Glencoe/McGraw-Hill 389 Glencoe Geometry
Less
on
7-7
In !RST, given the following measures, find the measure of the missing side.
1. r " 5, s " 8, m#T " 39
2. r " 6, t " 11, m#S " 87
3. r " 9, t " 15, m#S " 103
4. s " 12, t " 10, m#R " 58
In !HIJ, given the lengths of the sides, find the measure of the stated angle to thenearest tenth.
5. h " 12, i " 18, j " 7; m#H
6. h " 15, i " 16, j " 22; m#I
7. h " 23, i " 27, j " 29; m#J
8. h " 37, i " 21, j " 30; m#H
Determine whether the Law of Sines or the Law of Cosines should be used first tosolve each triangle. Then solve each triangle. Round angle measures to the nearestdegree and side measures to the nearest tenth.
9. 10.
11. a " 10, b " 14, c "19 12. a " 12, b " 10, m#C " 27
Solve each !RST described below. Round measures to the nearest tenth.
13. r " 12, s " 32, t " 34
14. r " 30, s " 25, m#T " 42
15. r " 15, s " 11, m#R " 67
16. r " 21, s " 28, t " 30
M
L N
!86#
52
24
B
A C
c
66#
33
19
© Glencoe/McGraw-Hill 390 Glencoe Geometry
In !JKL, given the following measures, find the measure of the missing side.
1. j " 1.3, k " 10, m#L " 77
2. j " 9.6, ! " 1.7, m#K " 43
3. j " 11, k " 7, m#L " 63
4. k " 4.7, ! " 5.2, m#J " 112
In !MNQ, given the lengths of the sides, find the measure of the stated angle tothe nearest tenth.
5. m " 17, n " 23, q " 25; m#Q
6. m " 24, n " 28, q " 34; m#M
7. m " 12.9, n " 18, q " 20.5; m#N
8. m " 23, n " 30.1, q " 42; m#Q
Determine whether the Law of Sines or the Law of Cosines should be used first tosolve !ABC. Then sole each triangle. Round angle measures to the nearest degreeand side measure to the nearest tenth.
9. a " 13, b " 18, c " 19 10. a " 6, b " 19, m#C " 38
11. a " 17, b " 22, m#B " 49 12. a " 15.5, b " 18, m#C " 72
Solve each !FGH described below. Round measures to the nearest tenth.
13. m#F " 54, f " 12.5, g " 11
14. f "20, g " 23, m#H " 47
15. f " 15.8, g " 11, h " 14
16. f " 36, h " 30, m#G " 54
17. REAL ESTATE The Esposito family purchased a triangular plot of land on which theyplan to build a barn and corral. The lengths of the sides of the plot are 320 feet, 286 feet,and 305 feet. What are the measures of the angles formed on each side of the property?
Practice The Law of Cosines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
Reading to Learn MathematicsThe Law of Cosines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
© Glencoe/McGraw-Hill 391 Glencoe Geometry
Less
on
7-7
Pre-Activity How are triangles used in building design?
Read the introduction to Lesson 7-7 at the top of page 385 in your textbook.
What could be a disadvantage of a triangular room?
Reading the Lesson1. Refer to the figure. According to the Law of Cosines, which
statements are correct for !DEF?
A. d2 " e2 % f 2 # ef cos D B. e2 " d2 % f 2 # 2df cos E
C. d2 " e2 % f 2 % 2ef cos D D. f 2 " d2 % e2 # 2ef cos F
E. f2 " d2 % e2 # 2de cos F F. d2 " e2 % f 2
G. !sin
dD
! " !sin
eE
! " !sin
fF
! H. d " !e2 % f"2 # 2e"f cos "D"
2. Each of the following describes three given parts of a triangle. In each case, indicatewhether you would use the Law of Sines or the Law of Cosines first in solving a trianglewith those given parts. (In some cases, only one of the two laws would be used in solvingthe triangle.)
a. SSS b. ASA
c. AAS d. SAS
e. SSA
3. Indicate whether each statement is true or false. If the statement is false, explain why.
a. The Law of Cosines applies to right triangles.
b. The Pythagorean Theorem applies to acute triangles.
c. The Law of Cosines is used to find the third side of a triangle when you are given themeasures of two sides and the nonincluded angle.
d. The Law of Cosines can be used to solve a triangle in which the measures of the threesides are 5 centimeters, 8 centimeters, and 15 centimeters.
Helping You Remember4. A good way to remember a new mathematical formula is to relate it to one you already
know. The Law of Cosines looks somewhat like the Pythagorean Theorem. Both formulasmust be true for a right triangle. How can that be?
D
dE
e
F
f
© Glencoe/McGraw-Hill 392 Glencoe Geometry
Spherical TrianglesSpherical trigonometry is an extension of plane trigonometry.Figures are drawn on the surface of a sphere. Arcs of great circles correspond to line segments in the plane. The arcs of three great circles intersecting on a sphere form a spherical triangle. Angles have the same measure as the tangent lines drawn to each great circle at the vertex. Since the sides are arcs, they too can be measured in degrees.
Solve the spherical triangle given a " 72#,b " 105#, and c " 61#.
Use the Law of Cosines.
0.3090 " (–0.2588)(0.4848) % (0.9659)(0.8746) cos Acos A " 0.5143
A " 59°
#0.2588 " (0.3090)(0.4848) % (0.9511)(0.8746) cos Bcos B " #0.4912
B " 119°
0.4848 " (0.3090)(–0.2588) % (0.9511)(0.9659) cos Ccos C " 0.6148
C " 52°
Check by using the Law of Sines.
!ssiinn
75
29
°°! " !
ssiinn
11
01
59
°°! " !
ssiinn
65
12
°°! " 1.1
Solve each spherical triangle.
1. a " 56°, b " 53°, c " 94° 2. a " 110°, b " 33°, c " 97°
3. a " 76°, b " 110°, C " 49° 4. b " 94°, c " 55°, A " 48°
A
C
B
c
ba
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
The sum of the sides of a spherical triangle is less than 360°.The sum of the angles is greater than 180° and less than 540°.The Law of Sines for spherical triangles is as follows.
!ssiinn
Aa
! " !ssiinn
Bb
! " !ssiinn
Cc
!
There is also a Law of Cosines for spherical triangles.cos a " cos b cos c % sin b sin c cos Acos b " cos a cos c % sin a sin c cos Bcos c " cos a cos b % sin a sin b cos C
ExampleExample
© Glencoe/McGraw-Hill A2 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Geo
met
ric M
ean
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-1
7-1
©G
lenc
oe/M
cGra
w-Hi
ll35
1G
lenc
oe G
eom
etry
Lesson 7-1
Geo
met
ric
Mea
nT
he g
eom
etri
c m
ean
betw
een
two
num
bers
is t
he s
quar
e ro
ot o
fth
eir
prod
uct.
For
two
posi
tive
num
bers
aan
d b,
the
geom
etri
c m
ean
of a
and
bis
the
po
siti
ve n
umbe
r x
in t
he p
ropo
rtio
n !a x!
"! bx ! .
Cro
ss m
ulti
plyi
ng g
ives
x2
"ab
,so
x"
!ab"
.
Fin
d t
he
geom
etri
c m
ean
bet
wee
n e
ach
pai
r of
nu
mbe
rs.
a.12
an
d 3
Let
xre
pres
ent
the
geom
etri
c m
ean.
!1 x2 !"
! 3x !De
finitio
n of
geo
met
ric m
ean
x2"
36Cr
oss
mul
tiply.
x"
!36"
or 6
Take
the
squa
re ro
ot o
f eac
h sid
e.
b.8
and
4L
et x
repr
esen
t th
e ge
omet
ric
mea
n.
!8 x!"
! 4x !
x2"
32x
"!
32"#
5.7
Exer
cises
Exer
cises
Fin
d t
he
geom
etri
c m
ean
bet
wee
n e
ach
pai
r of
nu
mbe
rs.
1.4
and
44
2.4
and
6!
24"#
4.9
3.6
and
9!
54"#
7.3
4.!1 2!
and
21
5.2!
3"an
d 3!
3"!
18"#
4.2
6.4
and
2510
7.!
3"an
d !
6"18
!1 4!#
2.1
8.10
and
100
!10
00"
#31
.6
9.!1 2!
and
!1 4!$%!1 8!
#0.
410
.an
d $%!1 22 5!
#0.
7
11.4
and
16
812
.3 a
nd 2
4!
72"#
8.5
Th
e ge
omet
ric
mea
n a
nd
on
e ex
trem
e ar
e gi
ven
.Fin
d t
he
oth
er e
xtre
me.
13.!
24"is
the
geo
met
ric
mea
n be
twee
n a
and
b.F
ind
bif
a"
2.12
14.!
12"is
the
geo
met
ric
mea
n be
twee
n a
and
b.F
ind
bif
a"
3.4
Det
erm
ine
wh
eth
er e
ach
sta
tem
ent
is a
lwa
ys,s
omet
imes
,or
nev
ertr
ue.
15.T
he g
eom
etri
c m
ean
of t
wo
posi
tive
num
bers
is g
reat
er t
han
the
aver
age
of t
he t
wo
num
bers
.ne
ver
16.I
f th
e ge
omet
ric
mea
n of
tw
o po
siti
ve n
umbe
rs is
less
tha
n 1,
then
bot
h of
the
num
bers
are
less
tha
n 1.
som
etim
es
3!2"
!5
2!2"
!5
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-Hi
ll35
2G
lenc
oe G
eom
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Alt
itu
de
of
a Tr
ian
gle
In t
he d
iagr
am,!
AB
C$
!A
DB
$!
BD
C.
An
alti
tude
to
the
hypo
tenu
se o
f a
righ
t tr
iang
le f
orm
s tw
o ri
ght
tria
ngle
s.T
he t
wo
tria
ngle
s ar
e si
mila
r an
d ea
ch is
sim
ilar
to t
he
orig
inal
tri
angl
e.C
DB
A
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Geo
met
ric M
ean
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-1
7-1
Use
rig
ht
!A
BC
wit
h
B"D"
⊥A"
C".D
escr
ibe
two
geom
etri
cm
ean
s.
a.!
AD
B&
!B
DC
so ! BA
D D!"
!B CD D!
.
In !
AB
C,t
he a
ltit
ude
is t
he g
eom
etri
cm
ean
betw
een
the
two
segm
ents
of
the
hypo
tenu
se.
b.!
AB
C&
!A
DB
and
!A
BC
&!
BD
C,
so !A A
C B!"
! AADB !
and
! BAC C!
"! DB
C C!.
In !
AB
C,e
ach
leg
is t
he g
eom
etri
cm
ean
betw
een
the
hypo
tenu
se a
nd t
hese
gmen
t of
the
hyp
oten
use
adja
cent
to
that
leg.
Fin
d x
,y,a
nd
z.
! PPQR !
"!P PQ S!
!2 15 5!"
!1 x5 !PR
"25
, PQ
"15
, PS
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oss
mul
tiply.
x"
9Di
vide
each
sid
e by
25.
The
ny
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R#
SP
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#9
"16
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! yz !PR
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, QR
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! 1z 6!y
"16
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400
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ly.z
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Take
the
squa
re ro
ot o
f eac
h sid
e.
zy
x
15
R QP
S25
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d x
,y,a
nd
zto
th
e n
eare
st t
enth
.
1.2.
3.
!3"
#1.
7!
10"#
3.2;
!14"
#3.
7;3;
!72"
#8.
5;!
35"#
5.9
!8"
#2.
8
4.5.
6.
2;3
2;!
8"#
2.8;
3.5;
!8"
#2.
8;!
8"#
2.8
!24"
#4.
9
xz
y
62
x
zy
2
2x
y
1
!%3
!"12
zx
y 81
z
xy
5
2
x
13
Answers (Lesson 7-1)
© Glencoe/McGraw-Hill A3 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Geo
met
ric M
ean
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-1
7-1
©G
lenc
oe/M
cGra
w-Hi
ll35
3G
lenc
oe G
eom
etry
Lesson 7-1
Fin
d t
he
geom
etri
c m
ean
bet
wee
n e
ach
pai
r of
nu
mbe
rs.S
tate
exa
ct a
nsw
ers
and
answ
ers
to t
he
nea
rest
ten
th.
1.2
and
82.
9 an
d 36
3.4
and
7
418
!28"
#5.
3
4.5
and
105.
2!2"
and
5!2"
6.3!
5"an
d 5!
5"
!50"
#7.
1!
20"#
4.5
!75"
#8.
7
Fin
d t
he
mea
sure
of
each
alt
itu
de.
Sta
te e
xact
an
swer
s an
d a
nsw
ers
to t
he
nea
rest
ten
th.
7.8.
!14"
#3.
7!
24"#
4.9
9.10
.
!18"
#4.
26
Fin
d x
and
y.
11.
12.
6;!
108
"#
10.4
!40"
#6.
3;!
56"#
7.5
13.
14.
!60"
#7.
7;!
285
"#
16.9
12.5
;!29"
#5.
425y
x15
4
y
x
10
4
yx
39y
x
RT
S U4.
58
GEH
F
2
9
L
M N
P2
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C
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A2
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ll35
4G
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Fin
d t
he
geom
etri
c m
ean
bet
wee
n e
ach
pai
r of
nu
mbe
rs t
o th
e n
eare
st t
enth
.
1.8
and
122.
3!7"
and
6!7"
3.!4 5!
and
2
!96"
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8!
126
"#
11.2
$%!8 5!#
1.3
Fin
d t
he
mea
sure
of
each
alt
itu
de.
Sta
te e
xact
an
swer
s an
d a
nsw
ers
to t
he
nea
rest
ten
th.
4.5.
!60"
#7.
7!
102
"#
10.1
Fin
d x
,y,a
nd
z.
6.7.
!18
4"
#13
.6;!
248
"#
15.7
;!
114
"#
10.7
;!15
0"
#12
.2;
!71
3"
#26
.7!
475
"#
21.8
8.9.
4.5;
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#3.
6;6.
515
;5; !
300
"#
17.3
10.C
IVIL
EN
GIN
EER
ING
An
airp
ort,
a fa
ctor
y,an
d a
shop
ping
cen
ter
are
at t
he v
erti
ces
of a
righ
t tr
iang
le fo
rmed
by
thre
e hi
ghw
ays.
The
air
port
and
fact
ory
are
6.0
mile
s ap
art.
The
irdi
stan
ces
from
the
sho
ppin
g ce
nter
are
3.6
mile
s an
d 4.
8 m
iles,
resp
ecti
vely
.A s
ervi
ce r
oad
will
be
cons
truc
ted
from
the
sho
ppin
g ce
nter
to
the
high
way
tha
t co
nnec
ts t
he a
irpo
rt a
ndfa
ctor
y.W
hat
is t
he s
hort
est
poss
ible
leng
th f
or t
he s
ervi
ce r
oad?
Rou
nd t
o th
e ne
ares
thu
ndre
dth.
2.88
mi
xy10
z
20x
y 2
3
z
zx
y625
23
z
xy
8
17
6
KLJ
M
125
U
TA
V
Pra
ctic
e (A
vera
ge)
Geo
met
ric M
ean
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-1
7-1
Answers (Lesson 7-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csG
eom
etric
Mea
n
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-1
7-1
©G
lenc
oe/M
cGra
w-Hi
ll35
5G
lenc
oe G
eom
etry
Lesson 7-1
Pre-
Act
ivit
yH
ow c
an t
he
geom
etri
c m
ean
be
use
d t
o vi
ew p
ain
tin
gs?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 7-
1 at
the
top
of
page
342
in y
our
text
book
.
•W
hat
is a
dis
adva
ntag
e of
sta
ndin
g to
o cl
ose
to a
pai
ntin
g?Sa
mpl
e an
swer
:You
don
’t ge
t a g
ood
over
all v
iew.
•W
hat
is a
dis
adva
ntag
e of
sta
ndin
g to
o fa
r fr
om a
pai
ntin
g?Sa
mpl
e an
swer
:You
can
’t se
e al
l the
det
ails
in th
e pa
intin
g.R
ead
ing
th
e Le
sso
n1.
In t
he p
ast,
whe
n yo
u ha
ve s
een
the
wor
d m
ean
in m
athe
mat
ics,
it r
efer
red
to t
heav
erag
eor
ari
thm
etic
mea
nof
the
tw
o nu
mbe
rs.
a.C
ompl
ete
the
follo
win
g by
wri
ting
an
alge
brai
c ex
pres
sion
in e
ach
blan
k.
If a
and
bar
e tw
o po
siti
ve n
umbe
rs,t
hen
the
geom
etri
c m
ean
betw
een
aan
d b
is
and
thei
r ar
ithm
etic
mea
n is
.
b.E
xpla
in in
wor
ds,w
itho
ut u
sing
any
mat
hem
atic
al s
ymbo
ls,t
he d
iffe
renc
e be
twee
nth
e ge
omet
ric
mea
n an
d th
e al
gebr
aic
mea
n.Sa
mpl
e an
swer
:The
geo
met
ricm
ean
betw
een
two
num
bers
is th
e sq
uare
root
of t
heir
prod
uct.
The
arith
met
ic m
ean
of tw
o nu
mbe
rs is
hal
f the
ir su
m.
2.L
et r
and
sbe
tw
o po
siti
ve n
umbe
rs.I
n w
hich
of
the
follo
win
g eq
uati
ons
is z
equa
l to
the
geom
etri
c m
ean
betw
een
ran
d s?
A,C,
D,F
A.
! zs !"
!z r!B
.! zr !
"! zs !
C.
s:z
"z:
rD
.! zr !
"!z s!
E.
!z r!"
!z s!F.
!z s!"
! zr !
3.Su
pply
the
mis
sing
wor
ds o
r ph
rase
s to
com
plet
e th
e st
atem
ent
of e
ach
theo
rem
.
a.T
he m
easu
re o
f the
alt
itud
e dr
awn
from
the
ver
tex
of t
he r
ight
ang
le o
f a r
ight
tri
angl
e
to it
s hy
pote
nuse
is t
he
betw
een
the
mea
sure
s of
the
tw
o
segm
ents
of
the
.
b.If
the
alt
itud
e is
dra
wn
from
the
ver
tex
of t
he
angl
e of
a r
ight
tria
ngle
to
its
hypo
tenu
se,t
hen
the
mea
sure
of
a of
the
tri
angl
e
is t
he
betw
een
the
mea
sure
of t
he h
ypot
enus
e an
d th
e se
gmen
t
of t
he
adja
cent
to
that
leg.
c.If
the
alt
itud
e is
dra
wn
from
the
of
the
rig
ht a
ngle
of
a ri
ght
tria
ngle
to
its
,the
n th
e tw
o tr
iang
les
form
ed a
re
to t
he g
iven
tri
angl
e an
d to
eac
h ot
her.
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al c
once
pt is
to
rela
te it
to
som
ethi
ng y
oual
read
y kn
ow.H
ow c
an t
he m
eani
ng o
f mea
nin
a p
ropo
rtio
n he
lp y
ou t
o re
mem
ber
how
to f
ind
the
geom
etri
c m
ean
betw
een
two
num
bers
?Sa
mpl
e an
swer
:Writ
e a
prop
ortio
n in
whi
ch th
e tw
o m
eans
are
equ
al.T
his
com
mon
mea
n is
the
geom
etric
mea
n be
twee
n th
e tw
o ex
trem
es.
sim
ilar
hypo
tenu
seve
rtex
hypo
tenu
sege
omet
ric m
ean
leg
right
hypo
tenu
sege
omet
ric m
ean
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b!
!ab"
©G
lenc
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cGra
w-Hi
ll35
6G
lenc
oe G
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etry
Mat
hem
atic
s an
d M
usic
Pyt
hago
ras,
a G
reek
phi
loso
pher
who
live
d du
ring
the
six
th c
entu
ry B
.C.,
belie
ved
that
all
natu
re,b
eaut
y,an
d ha
rmon
y co
uld
be e
xpre
ssed
by
who
le-
num
ber
rela
tion
ship
s.M
ost
peop
le r
emem
ber
Pyt
hago
ras
for
his
teac
hing
sab
out
righ
t tr
iang
les.
(The
sum
of
the
squa
res
of t
he le
gs e
qual
s th
e sq
uare
of
the
hypo
tenu
se.)
But
Pyt
hago
ras
also
dis
cove
red
rela
tion
ship
s be
twee
n th
em
usic
al n
otes
of
a sc
ale.
The
se r
elat
ions
hips
can
be
expr
esse
d as
rat
ios.
CD
EF
GA
BC
$
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!4 5!!3 4!
!2 3!!3 5!
! 18 5!!1 2!
Whe
n yo
u pl
ay a
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inge
d in
stru
men
t,T
he C
str
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can
be u
sed
you
prod
uce
diff
eren
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tes
by p
laci
ng
to p
rodu
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by
plac
ing
your
fin
ger
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iffe
rent
pla
ces
on a
str
ing.
a fi
nger
!3 4!of
the
way
Thi
s is
the
res
ult
of c
hang
ing
the
leng
thal
ong
the
stri
ng.
of t
he v
ibra
ting
par
t of
the
str
ing.
Su
pp
ose
a C
str
ing
has
a l
engt
h o
f 16
in
ches
.Wri
te a
nd
sol
ve
pro
por
tion
s to
det
erm
ine
wh
at l
engt
h o
f st
rin
g w
ould
hav
e to
vi
brat
e to
pro
du
ce t
he
rem
ain
ing
not
es o
f th
e sc
ale.
1.D
14!2 9!
in.
2.E
12!4 5!
in.
3.F
12 in
.
4.G
10!2 3!
in.
5.A
9!3 5!in
.6.
B8!
18 5!in
.
7.C
$8
in.
8.C
ompl
ete
to s
how
the
dis
tanc
e be
twee
n fi
nger
pos
itio
ns o
n th
e 16
-inc
h
C s
trin
g fo
r ea
ch n
ote.
For
exam
ple,
C(1
6) #
D&14
!2 9! '"1!
7 9! .
C
D
E
F
G
A
B
C$
9.B
etw
een
two
cons
ecut
ive
mus
ical
not
es,t
here
is e
ithe
r a
who
le s
tep
or
a ha
lf s
tep.
Usi
ng t
he d
ista
nces
you
fou
nd in
Exe
rcis
e 8,
dete
rmin
e w
hat
two
pair
s of
not
es h
ave
a ha
lf s
tep
betw
een
them
.
Ean
d F,
Ban
d C%
8 in
.7! 17 5!
in.
6!2 5!in
.5!1 3!
in.
4 in
.3!1 5!
in.
1!7 9!
in.En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-1
7-1
3 4of
C s
tring
Answers (Lesson 7-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
An
swer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
The
Pyth
agor
ean
Theo
rem
and
Its
Conv
erse
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-2
7-2
©G
lenc
oe/M
cGra
w-Hi
ll35
7G
lenc
oe G
eom
etry
Lesson 7-2
The
Pyth
ago
rean
Th
eore
mIn
a r
ight
tri
angl
e,th
e su
m o
f th
e
squa
res
of t
he m
easu
res
of t
he le
gs e
qual
s th
e sq
uare
of
the
mea
sure
of
the
hypo
tenu
se.
!A
BC
is a
rig
ht t
rian
gle,
so a
2%
b2"
c2.
Pro
ve t
he
Pyt
hag
orea
n T
heo
rem
.W
ith
alti
tude
C"D"
,eac
h le
g a
and
bis
a g
eom
etri
c m
ean
betw
een
hypo
tenu
se c
and
the
segm
ent
of t
he h
ypot
enus
e ad
jace
nt t
o th
at le
g.
! ac !"
!a y!an
d ! bc !
"!b x! ,
so a
2"
cyan
d b2
"cx
.
Add
the
tw
o eq
uati
ons
and
subs
titu
te c
"y
%x
to g
eta2
%b2
"cy
%cx
"c(
y%
x) "
c2.
cy
xa
b
hA
CBD
ca
bA
CB
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
a.F
ind
a.
a2%
b2"
c2Py
thag
orea
n Th
eore
m
a2%
122
"13
2b
"12
, c"
13
a2%
144
"16
9Si
mpl
ify.
a2"
25Su
btra
ct.
a"
5Ta
ke th
e sq
uare
root
of e
ach
side.
a
1213
ACB
b.F
ind
c.
a2%
b2"
c2Py
thag
orea
n Th
eore
m
202
%30
2"
c2a
"20
, b"
30
400
%90
0 "
c2Si
mpl
ify.
1300
"c2
Add.
!13
00"
"c
Take
the
squa
re ro
ot o
f eac
h sid
e.
36.1
#c
Use
a ca
lcula
tor.
c
30
20
ACB
Exer
cises
Exer
cises
Fin
d x
.
1.2.
3.
!18"
#4.
212
60
4.5.
6.
! 13 0!!
1345
"#
36.7
!66
3"
#25
.7x
1128
x
33
16x
5 9
4 9
x6525
x 159
x
33
©G
lenc
oe/M
cGra
w-Hi
ll35
8G
lenc
oe G
eom
etry
Co
nve
rse
of
the
Pyth
ago
rean
Th
eore
mIf
the
sum
of
the
squa
res
of t
he m
easu
res
of t
he t
wo
shor
ter
side
s of
a t
rian
gle
equa
ls t
he s
quar
e of
th
e m
easu
re o
f th
e lo
nges
t si
de,t
hen
the
tria
ngle
is a
rig
ht t
rian
gle.
If t
he t
hree
who
le n
umbe
rs a
,b,a
nd c
sati
sfy
the
equa
tion
a2
%b2
"c2
,the
n th
e nu
mbe
rs a
,b,a
nd c
form
a
If a2
%b2
"c2
, the
n P
yth
agor
ean
tri
ple
.!
ABC
is a
right
tria
ngle
.
Det
erm
ine
wh
eth
er !
PQ
Ris
a r
igh
t tr
ian
gle.
a2%
b2"
c2Py
thag
orea
n Th
eore
m
102
%(1
0!3")
2"
202
a"
10, b
"10
!3",
c"
20
100
%30
0 "
400
Sim
plify
.
400
"40
0✓Ad
d.
The
sum
of
the
squa
res
of t
he t
wo
shor
ter
side
s eq
uals
the
squ
are
of t
he lo
nges
t si
de,s
o th
etr
iang
le is
a r
ight
tri
angl
e.
Det
erm
ine
wh
eth
er e
ach
set
of
mea
sure
s ca
n b
e th
e m
easu
res
of t
he
sid
es o
f a
righ
t tr
ian
gle.
Th
en s
tate
wh
eth
er t
hey
for
m a
Pyt
hag
orea
n t
rip
le.
1.30
,40,
502.
20,3
0,40
3.18
,24,
30
yes;
yes
no;n
oye
s;ye
s
4.6,
8,9
5.!3 7! ,
!4 7! ,!5 7!
6.10
,15,
20
no;n
oye
s;no
no;n
o
7.!
5",!
12",!
13"8.
2,!
8",!
12"9.
9,40
,41
no;n
oye
s;no
yes;
yes
A f
am
ily
of P
yth
agor
ean
tri
ple
s co
nsi
sts
of m
ult
iple
s of
kn
own
tri
ple
s.F
or e
ach
Pyt
hag
orea
n t
rip
le,f
ind
tw
o tr
iple
s in
th
e sa
me
fam
ily.
Sam
ple
answ
ers
are
give
n.10
.3,4
,511
.5,1
2,13
12.7
,24,
25
30,4
0,50
;10
,24,
26;
14,4
8,50
;12
,16,
2015
,36,
3921
,72,
75
10!
%3
2010 Q
R
P
c
ab
A
C
B
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
The
Pyth
agor
ean
Theo
rem
and
Its
Conv
erse
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-2
7-2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 7-2)
© Glencoe/McGraw-Hill A6 Glencoe Geometry
Skil
ls P
ract
ice
The
Pyth
agor
ean
Theo
rem
and
Its
Conv
erse
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-2
7-2
©G
lenc
oe/M
cGra
w-Hi
ll35
9G
lenc
oe G
eom
etry
Lesson 7-2
Fin
d x
.
1.2.
3.
155
!11
68"
#34
.2
4.5.
6.
!46
8.7
"5"
#21
.7!
65"#
8.1
!11
57"
#34
.0
Det
erm
ine
wh
eth
er !
ST
Uis
a r
igh
t tr
ian
gle
for
the
give
n v
erti
ces.
Exp
lain
.
7.S
(5,5
),T
(7,3
),U
(3,2
)8.
S(3
,3),
T(5
,5),
U(6
,0)
no;S
T"
!8",
TU"
!17"
,ye
s;ST
"!
8",TU
"!
26",
US"
!13"
,US
"!
18",
( !8")2
$ ( !
13")2
&( !
17")2
( !8")2
$ ( !
18")2
"( !
26")2
9.S
(4,6
),T
(9,1
),U
(1,3
)10
.S(0
,3),
T(#
2,5)
,U(4
,7)
yes;
ST"
!50"
,TU
"!
68",
yes;
ST"
!8",
TU"
!40"
,US
"!
18",
US"
!32"
,( !
18")2
$( !
50")2
"( !
68")2
( !8")
2$
( !32"
)2"
( !40"
)2
11.S
(#3,
2),T
(2,7
),U
(#1,
1)12
.S(2
,#1)
,T(5
,4),
U(6
,#3)
yes;
ST"
!50"
,TU
"!
45",
no;S
T"
!34"
,TU
"!
50",
US"
!5",
US"
!20"
,( !
45")2
$ ( !
5")2"
( !50"
)2( !
34")2
$ ( !
20")2
&( !
50")2
Det
erm
ine
wh
eth
er e
ach
set
of
mea
sure
s ca
n b
e th
e m
easu
res
of t
he
sid
es o
f a
righ
t tr
ian
gle.
Th
en s
tate
wh
eth
er t
hey
for
m a
Pyt
hag
orea
n t
rip
le.
13.1
2,16
,20
14.1
6,30
,32
15.1
4,48
,50
yes,
yes
no,n
oye
s,ye
s
16.!
2 5! ,!4 5! ,
!6 5!17
.2!
6",5,
718
.2!
2",2!
7",6
no,n
oye
s,no
yes,
nox31
14x
99
8
x12
.5
25
x
1232
x
12
13x 12
9
©G
lenc
oe/M
cGra
w-Hi
ll36
0G
lenc
oe G
eom
etry
Fin
d x
.
1.2.
3.
!69
8"
#26
.4!
715
"#
26.7
!59
5"
#24
.4
4.5.
6.
!16
40"
#40
.5!
60"#
7.7
!13
5"
#11
.6
Det
erm
ine
wh
eth
er !
GH
Iis
a r
igh
t tr
ian
gle
for
the
give
n v
erti
ces.
Exp
lain
.
7.G
(2,7
),H
(3,6
),I(
#4,
#1)
8.G
(#6,
2),H
(1,1
2),I
(#2,
1)
yes;
GH
"!
2",HI
"!
98",
no;G
H"
!14
9"
,HI"
!13
0"
,IG
"!
100
",
IG"
!17"
,( !
2")2$
( !98"
)2"
( !10
0"
)2( !
130
")2
$ ( !
17")2
&( !
149
")2
9.G
(#2,
1),H
(3,#
1),I
(#4,
#4)
10.G
(#2,
4),H
(4,1
),I(
#1,
#9)
yes;
GH
"!
29",H
I"!
58",
yes;
GH
"!
45",H
I" !
125
",
IG"
!29"
,IG
"!
170
",
( !29"
)2$
( !29"
)2"
( !58"
)2( !
45")2
$ ( !
125
")2
"( !
170
")2
Det
erm
ine
wh
eth
er e
ach
set
of
mea
sure
s ca
n b
e th
e m
easu
res
of t
he
sid
es o
f a
righ
t tr
ian
gle.
Th
en s
tate
wh
eth
er t
hey
for
m a
Pyt
hag
orea
n t
rip
le.
11.9
,40,
4112
.7,2
8,29
13.2
4,32
,40
yes,
yes
no,n
oye
s,ye
s
14.!
9 5! ,!1 52 !
,315
.!1 3! ,
,116
.,
,!4 7!
yes,
noye
s,no
yes,
no
17.C
ON
STR
UC
TIO
NT
he b
otto
m e
nd o
f a
ram
p at
a w
areh
ouse
is
10 f
eet
from
the
bas
e of
the
mai
n do
ck a
nd is
11
feet
long
.How
hi
gh is
the
doc
k?ab
out 4
.6 ft
hig
h11
ft?do
ck
ram
p
10 ft
2!3"
!7
!4"
!7
2 !2"
!3
x24
24
42
x16
14
x
34
22
x26
2618
x
3421
x
13
23Pra
ctic
e (A
vera
ge)
The
Pyth
agor
ean
Theo
rem
and
Its
Conv
erse
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-2
7-2
Answers (Lesson 7-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Rea
din
g t
o L
earn
Math
emati
csTh
e Py
thag
orea
n Th
eore
m a
nd It
s Co
nver
se
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-2
7-2
©G
lenc
oe/M
cGra
w-Hi
ll36
1G
lenc
oe G
eom
etry
Lesson 7-2
Pre-
Act
ivit
yH
ow a
re r
igh
t tr
ian
gles
use
d t
o bu
ild
su
spen
sion
bri
dge
s?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 7-
2 at
the
top
of
page
350
in y
our
text
book
.
Do
the
two
righ
t tr
iang
les
show
n in
the
dra
win
g ap
pear
to
be s
imila
r?E
xpla
in y
our
reas
onin
g.Sa
mpl
e an
swer
:No;
thei
r sid
es a
re n
otpr
opor
tiona
l.In
the
smal
ler t
riang
le,t
he lo
nger
leg
is m
ore
than
twic
e th
e le
ngth
of t
he s
hort
er le
g,w
hile
in th
e la
rger
tria
ngle
,th
e lo
nger
leg
is le
ss th
an tw
ice
the
leng
th o
f the
sho
rter
leg.
Rea
din
g t
he
Less
on
1.E
xpla
in in
you
r ow
n w
ords
the
dif
fere
nce
betw
een
how
the
Pyt
hago
rean
The
orem
is u
sed
and
how
the
Con
vers
e of
the
Pyt
hago
rean
The
orem
is u
sed.
Sam
ple
answ
er:T
hePy
thag
orea
n Th
eore
m is
use
d to
find
the
third
sid
e of
a ri
ght t
riang
le if
you
know
the
leng
ths
of a
ny tw
o of
the
side
s.Th
e co
nver
se is
use
d to
tell
whe
ther
a tr
iang
le w
ith th
ree
give
n si
de le
ngth
s is
a ri
ght t
riang
le.
2.R
efer
to
the
figu
re.F
or t
his
figu
re,w
hich
sta
tem
ents
are
tru
e?
A.m
2%
n2"
p2B
.n2
"m
2%
p2B,
E,F,
GC
.m2
"n2
%p2
D.m
2"
p2#
n2
E.p
2"
n2#
m2
F.n2
# p
2"
m2
G.n
"!
m2
%"
p2 "H
.p"
!m
2#
"n2 "
3.Is
the
fol
low
ing
stat
emen
t tr
ue o
r fa
lse?
A P
ytha
gore
an t
ripl
e is
any
gro
up o
f thr
ee n
umbe
rs fo
r w
hich
the
sum
of t
he s
quar
es o
f the
smal
ler
two
num
bers
is e
qual
to
the
squa
re o
f the
larg
est
num
ber.
Exp
lain
you
r re
ason
ing.
Sam
ple
answ
er:T
he s
tate
men
t is
fals
e be
caus
e in
a P
ytha
gore
an tr
iple
,al
l thr
ee n
umbe
rs m
ust b
e w
hole
num
bers
.
4.If
x,y
,and
zfo
rm a
Pyt
hago
rean
tri
ple
and
kis
a p
osit
ive
inte
ger,
whi
ch o
f th
e fo
llow
ing
grou
ps o
f nu
mbe
rs a
re a
lso
Pyt
hago
rean
tri
ples
? B,
DA
.3x,
4y,5
zB
.3x,
3y,3
zC
.#3x
,#3y
,#3z
D.k
x,ky
,kz
Hel
pin
g Y
ou
Rem
emb
er
5.M
any
stud
ents
who
stu
died
geo
met
ry lo
ng a
go r
emem
ber
the
Pyt
hago
rean
The
orem
as
the
equa
tion
a2
%b2
"c2
,but
can
not
tell
you
wha
t th
is e
quat
ion
mea
ns.A
form
ula
is u
sele
ssif
you
don
’t kn
ow w
hat
it m
eans
and
how
to
use
it.H
ow c
ould
you
hel
p so
meo
ne w
ho h
asfo
rgot
ten
the
Pyt
hago
rean
The
orem
rem
embe
r th
e m
eani
ng o
f the
equ
atio
n a2
%b2
"c2
?Sa
mpl
e an
swer
:Dra
w a
righ
t tria
ngle
.Lab
el th
e le
ngth
s of
the
two
legs
as a
and
ban
d th
e le
ngth
of t
he h
ypot
enus
e as
c.
pm
n
©G
lenc
oe/M
cGra
w-Hi
ll36
2G
lenc
oe G
eom
etry
Conv
erse
of a
Rig
ht Tr
iang
le T
heor
emYo
u ha
ve le
arne
d th
at t
he m
easu
re o
f th
e al
titu
de f
rom
the
ver
tex
ofth
e ri
ght
angl
e of
a r
ight
tri
angl
e to
its
hypo
tenu
se is
the
geo
met
ric
mea
n be
twee
n th
e m
easu
res
of t
he t
wo
segm
ents
of
the
hypo
tenu
se.
Is t
he c
onve
rse
of t
his
theo
rem
tru
e? I
n or
der
to f
ind
out,
it w
ill h
elp
to r
ewri
te t
he o
rigi
nal t
heor
em in
if-t
hen
form
as
follo
ws.
If !
AB
Qis
a r
ight
tri
angl
e w
ith
righ
t an
gle
at Q
,the
n
QP
is t
he g
eom
etri
c m
ean
betw
een
AP
and
PB
,whe
re P
is b
etw
een
Aan
d B
and
Q "P "
is p
erpe
ndic
ular
to
A "B "
.
1.W
rite
the
con
vers
e of
the
if-t
hen
form
of
the
theo
rem
.
If Q
Pis
the
geom
etric
mea
n be
twee
n AP
and
PB,w
here
Pis
bet
wee
n A
and
Ban
d Q "
P "#
A "B "
,th
en !
ABQ
is a
righ
t tria
ngle
with
righ
t ang
le
at Q
.
2.Is
the
con
vers
e of
the
ori
gina
l the
orem
tru
e? R
efer
to
the
fig
ure
at t
he r
ight
to
expl
ain
your
ans
wer
.
Yes;
(PQ
)2"
(AP)
(PB)
impl
ies
that
!P AQ P!"
! PP QB !.
Sinc
e bo
th "
APQ
and
"Q
PBar
e rig
ht
angl
es,t
hey
are
cong
ruen
t.Th
eref
ore
!AP
Q&
!Q
PBby
SAS
sim
ilarit
y.So
"
A'
"PQ
Ban
d "
AQP
'"
B.Bu
t the
acu
te
angl
es o
f !AQ
Par
e co
mpl
emen
tary
and
m
"AQ
B"
m"
AQP
$m
"PQ
B.He
nce
m"
AQB
"90
and
!AQ
Bis
a ri
ght t
riang
le
with
righ
t ang
le a
t Q.
You
may
fin
d it
inte
rest
ing
to e
xam
ine
the
othe
r th
eore
ms
inC
hapt
er 7
to
see
whe
ther
the
ir c
onve
rses
are
tru
e or
fal
se.Y
ou w
illne
ed t
o re
stat
e th
e th
eore
ms
care
fully
in o
rder
to
wri
te t
heir
conv
erse
s.
Q
BP
A
Q
BP
A
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-2
7-2
Answers (Lesson 7-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Spec
ial R
ight
Tria
ngle
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-3
7-3
©G
lenc
oe/M
cGra
w-Hi
ll36
3G
lenc
oe G
eom
etry
Lesson 7-3
Pro
per
ties
of
45°-
45°-
90°
Tria
ng
les
The
sid
es o
f a
45°-
45°-
90°
righ
t tr
iang
le h
ave
asp
ecia
l rel
atio
nshi
p.
If t
he
leg
of a
45°
-45°
-90°
righ
t tr
ian
gle
is x
un
its,
show
th
at t
he
hyp
oten
use
is
x!2"
un
its.
Usi
ng t
he P
ytha
gore
an T
heor
em w
ith
a"
b"
x,th
en
c2"
a2%
b2
"x2
%x2
"2x
2
c"
!2x
2"
"x!
2"
x!%
x
x2
45#
45#
In a
45°
-45°
-90°
righ
t tr
ian
gle
the
hyp
oten
use
is
!2"
tim
es
the
leg.
If t
he
hyp
oten
use
is
6 u
nit
s,fi
nd
th
e le
ngt
h o
f ea
ch l
eg.
The
hyp
oten
use
is !
2"ti
mes
the
leg,
sodi
vide
the
leng
th o
f th
e hy
pote
nuse
by
!2".
a" " " "
3 !2"
unit
s
6!2"
!26!
2"! !
2"!2"
6! !
2"
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d x
.
1.2.
3.
8!2"
#11
.33
5!2"
#7.
1
4.5.
6.
9!2"
#12
.718
!2"
#25
.56
7.F
ind
the
peri
met
er o
f a
squa
re w
ith
diag
onal
12
cent
imet
ers.
24!
2"#
33.9
cm
8.F
ind
the
diag
onal
of
a sq
uare
wit
h pe
rim
eter
20
inch
es.
5!2"
#7.
1 in
.
9.F
ind
the
diag
onal
of
a sq
uare
wit
h pe
rim
eter
28
met
ers.
7!2"
#9.
9 m
x3!
%2x
18x
x
18
x10
x
45#
3!%2
x 8
45#
45#
©G
lenc
oe/M
cGra
w-Hi
ll36
4G
lenc
oe G
eom
etry
Pro
per
ties
of
30°-
60°-
90°
Tria
ng
les
The
sid
es o
f a
30°-
60°-
90°
righ
t tr
iang
le a
lso
have
a s
peci
al r
elat
ions
hip.
In a
30°
-60°
-90°
righ
t tr
ian
gle,
show
th
at t
he
hyp
oten
use
is
twic
e th
e sh
orte
r le
g an
d t
he
lon
ger
leg
is !
3"ti
mes
th
e sh
orte
r le
g.
!M
NQ
is a
30°
-60°
-90°
righ
t tr
iang
le,a
nd t
he le
ngth
of
the
hypo
tenu
se M "
N"is
tw
o ti
mes
the
leng
th o
f th
e sh
orte
r si
de N"
Q".
Usi
ng t
he P
ytha
gore
an T
heor
em,
a2"
(2x)
2#
x2
"4x
2#
x2
"3x
2
a"
!3x
2"
"x!
3"
In a
30°
-60°
-90°
righ
t tr
ian
gle,
the
hyp
oten
use
is
5 ce
nti
met
ers.
Fin
d t
he
len
gth
s of
th
e ot
her
tw
o si
des
of
the
tria
ngl
e.If
the
hyp
oten
use
of a
30°
-60°
-90°
righ
t tr
iang
le is
5 c
enti
met
ers,
then
the
leng
th o
f th
esh
orte
r le
g is
hal
f of
5 o
r 2.
5 ce
ntim
eter
s.T
he le
ngth
of
the
long
er le
g is
!3"
tim
es t
he
leng
th o
f th
e sh
orte
r le
g,or
(2.
5)(!
3")ce
ntim
eter
s.
Fin
d x
and
y.
1.2.
3.
1;0.
5!3"
#0.
98!
3"#
13.9
;16
5.5;
5.5!
3"#
9.5
4.5.
6.
9;18
4!3"
#6.
9;8!
3"#
13.9
10!
3"#
17.3
;10
7.T
he p
erim
eter
of
an e
quila
tera
l tri
angl
e is
32
cent
imet
ers.
Fin
d th
e le
ngth
of
an a
ltit
ude
of t
he t
rian
gle
to t
he n
eare
st t
enth
of
a ce
ntim
eter
.9.
2 cm
8.A
n al
titu
de o
f an
equ
ilate
ral t
rian
gle
is 8
.3 m
eter
s.F
ind
the
peri
met
er o
f th
e tr
iang
le t
oth
e ne
ares
t te
nth
of a
met
er.
28.8
m
xy
60#
20
xy
60#
12
xy 30
#
9 !%3
x
y11
30#
x
y
60# 8
x y30
#
60#
1 2
x
a
NQP
M
2x30#
30#
60#
60#
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Spec
ial R
ight
Tria
ngle
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-3
7-3
Exer
cises
Exer
cises
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
!M
NP
is an
equ
ilate
ral
trian
gle.
!M
NQ
is a
30°-6
0°-9
0°rig
ht tr
iang
le.
Answers (Lesson 7-3)
© Glencoe/McGraw-Hill A9 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Spec
ial R
ight
Tria
ngle
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-3
7-3
©G
lenc
oe/M
cGra
w-Hi
ll36
5G
lenc
oe G
eom
etry
Lesson 7-3
Fin
d x
and
y.
1.2.
3.
12,1
2!3"
64,3
2!3"
6!2",
6!2"
4.5.
6.
8,8!
2"8,
8!3"
45,1
3!2"
For
Exe
rcis
es 7
–9,u
se t
he
figu
re a
t th
e ri
ght.
7.If
a"
11,f
ind
ban
d c.
b"
11!
3";c
"22
8.If
b"
15,f
ind
aan
d c.
a"
5!3";
c"
10!
3"
9.If
c"
9,fi
nd a
and
b.
a"
4.5;
b"
4.5!
3"
For
Exe
rcis
es 1
0 an
d 1
1,u
se t
he
figu
re a
t th
e ri
ght.
10.T
he p
erim
eter
of
the
squa
re is
30
inch
es.F
ind
the
leng
th o
f B"C"
.7.
5 in
.
11.F
ind
the
leng
th o
f th
e di
agon
al B"
D".
7.5!
2"in
.or a
bout
10.
61 in
.
12.T
he p
erim
eter
of
the
equi
late
ral t
rian
gle
is 6
0 m
eter
s.F
ind
the
le
ngth
of
an a
ltit
ude.
10!
3"m
or a
bout
17.
32 m
13.!
GE
Cis
a 3
0°-6
0°-9
0°tr
iang
le w
ith
righ
t an
gle
at E
,and
E"C"
is
the
long
er le
g.F
ind
the
coor
dina
tes
of G
in Q
uadr
ant
I fo
r E
(1,1
) an
d C
(4,1
).
( 1,1
$!
3")or
abo
ut (1
,2.7
3)
E
FG
D60
#
AB C
D45
#
bA
B Cac
60#
30#
y x# 13
1313
13
y
x60#
16
y
x
45#
8
y
x45
#
12
y
x
30#
32
y
x60
#24
©G
lenc
oe/M
cGra
w-Hi
ll36
6G
lenc
oe G
eom
etry
Fin
d x
and
y.
1.2.
3.
,25
!3",
5013
,13!
3"
4.5.
6.
45,1
4!2"
3.5!
3",7
;11!
2"
For
Exe
rcis
es 7
–9,u
se t
he
figu
re a
t th
e ri
ght.
7.If
a"
4!3",
find
ban
d c.
b"
12,c
"8!
3"
8.If
x"
3!3",
find
aan
d C
D.
a"
6!3",
CD"
9
9.If
a"
4,fi
nd C
D,b
,and
y.
CD "
2!3",
b"
4!3",
y"
6
10.T
he p
erim
eter
of
an e
quila
tera
l tri
angl
e is
39
cent
imet
ers.
Fin
d th
e le
ngth
of
an a
ltit
ude
of t
he t
rian
gle.
6.5!
3"in
.or a
bout
11.
26 in
.
11.!
MIP
is a
30°
-60°
-90°
tria
ngle
wit
h ri
ght
angl
e at
I,a
nd I"
P"th
e lo
nger
leg.
Fin
d th
eco
ordi
nate
s of
Min
Qua
dran
t I
for
I(3,
3) a
nd P
(12,
3).
( 3,3
$3!
3")or
abo
ut (3
,8.1
9)
12.!
TJK
is a
45°
-45°
-90°
tria
ngle
wit
h ri
ght
angl
e at
J.F
ind
the
coor
dina
tes
of T
inQ
uadr
ant
II f
or J
(#2,
#3)
and
K(3
,#3)
.
('2,
2)
13.B
OTA
NIC
AL
GA
RD
ENS
One
of
the
disp
lays
at
a bo
tani
cal g
arde
n is
an
herb
gar
den
plan
ted
in t
he s
hape
of
a sq
uare
.The
squ
are
mea
sure
s 6
yard
s on
eac
h si
de.V
isit
ors
can
view
the
her
bs f
rom
adi
agon
al p
athw
ay t
hrou
gh t
he g
arde
n.H
ow lo
ng is
the
pat
hway
?
6!2"
yd o
r abo
ut 8
.48
yd
6 yd
6 yd
6 yd
6 yd
bA
B C
Da
x
y60
#
30#
c
11!
2"!
2
x45
#
11
y60
#3.
5
xy
x#y
28
9!2"
!2
9!2"
!2
y
x30
#
26y
x2560
#
yx
45#
9Pra
ctic
e (A
vera
ge)
Spec
ial R
ight
Tria
ngle
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-3
7-3
Answers (Lesson 7-3)
© Glencoe/McGraw-Hill A10 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csSp
ecia
l Tria
ngle
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-3
7-3
©G
lenc
oe/M
cGra
w-Hi
ll36
7G
lenc
oe G
eom
etry
Lesson 7-3
Pre-
Act
ivit
yH
ow i
s tr
ian
gle
tili
ng
use
d i
n w
allp
aper
des
ign
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 7-
3 at
the
top
of
page
357
in y
our
text
book
.•
How
can
you
mos
t co
mpl
etel
y de
scri
be t
he la
rger
tri
angl
e an
d th
e tw
osm
alle
r tr
iang
les
in t
ile 1
5?Sa
mpl
e an
swer
:The
larg
er tr
iang
le is
an is
osce
les
obtu
se tr
iang
le.T
he tw
o sm
alle
r tria
ngle
s ar
eco
ngru
ent s
cale
ne ri
ght t
riang
les.
•H
ow c
an y
ou m
ost
com
plet
ely
desc
ribe
the
larg
er t
rian
gle
and
the
two
smal
ler
tria
ngle
s in
tile
16?
(In
clud
e an
gle
mea
sure
s in
des
crib
ing
all t
hetr
iang
les.
)Sa
mpl
e an
swer
:The
larg
er tr
iang
le is
equ
ilate
ral,
soea
ch o
f its
ang
le m
easu
res
is 6
0.Th
e tw
o sm
alle
r tria
ngle
sar
e co
ngru
ent r
ight
tria
ngle
s in
whi
ch th
e an
gle
mea
sure
sar
e 30
,60,
and
90.
Rea
din
g t
he
Less
on
1.Su
pply
the
cor
rect
num
ber
or n
umbe
rs t
o co
mpl
ete
each
sta
tem
ent.
a.In
a 4
5°-4
5°-9
0°tr
iang
le,t
o fi
nd t
he le
ngth
of
the
hypo
tenu
se,m
ulti
ply
the
leng
th o
f a
leg
by
.
b.In
a 3
0°-6
0°-9
0°tr
iang
le,t
o fi
nd t
he le
ngth
of
the
hypo
tenu
se,m
ulti
ply
the
leng
th o
f
the
shor
ter
leg
by
.
c.In
a 3
0°-6
0°-9
0°tr
iang
le,t
he lo
nger
leg
is o
ppos
ite
the
angl
e w
ith
a m
easu
re o
f .
d.In
a 3
0°-6
0°-9
0°tr
iang
le,t
o fi
nd t
he le
ngth
of
the
long
er le
g,m
ulti
ply
the
leng
th o
f
the
shor
ter
leg
by
.
e.In
an
isos
cele
s ri
ght
tria
ngle
,eac
h le
g is
opp
osit
e an
ang
le w
ith
a m
easu
re o
f .
f.In
a 3
0°-6
0°-9
0°tr
iang
le,t
o fi
nd t
he le
ngth
of
the
shor
ter
leg,
divi
de t
he le
ngth
of
the
long
er le
g by
.
g.In
30°
-60°
-90°
tria
ngle
,to
find
the
leng
th o
f th
e lo
nger
leg,
divi
de t
he le
ngth
of
the
hypo
tenu
se b
y an
d m
ulti
ply
the
resu
lt b
y .
h.
To f
ind
the
leng
th o
f a
side
of
a sq
uare
,div
ide
the
leng
th o
f th
e di
agon
al b
y .
2.In
dica
te w
heth
er e
ach
stat
emen
t is
alw
ays,
som
etim
es,o
r ne
ver
true
.a.
The
leng
ths
of t
he t
hree
sid
es o
f an
isos
cele
s tr
iang
le s
atis
fy t
he P
ytha
gore
anT
heor
em.
som
etim
esb.
The
leng
ths
of t
he s
ides
of
a 30
°-60
°-90
°tr
iang
le f
orm
a P
ytha
gore
an t
ripl
e.ne
ver
c.T
he le
ngth
s of
all
thre
e si
des
of a
30°
-60°
-90°
tria
ngle
are
pos
itiv
e in
tege
rs.
neve
r
Hel
pin
g Y
ou
Rem
emb
er3.
Som
e st
uden
ts f
ind
it e
asie
r to
rem
embe
r m
athe
mat
ical
con
cept
s in
ter
ms
of s
peci
fic
num
bers
rat
her
than
var
iabl
es.H
ow c
an y
ou u
se s
peci
fic
num
bers
to
help
you
rem
embe
rth
e re
lati
onsh
ip b
etw
een
the
leng
ths
of t
he t
hree
sid
es in
a 3
0°-6
0°-9
0°tr
iang
le?
Sam
ple
answ
er:D
raw
a 3
0#-6
0#-9
0#tri
angl
e.La
bel t
he le
ngth
of t
hesh
orte
r leg
as
1.Th
en th
e le
ngth
of t
he h
ypot
enus
e is
2,a
nd th
e le
ngth
of
the
long
er le
g is
!3".
Just
rem
embe
r:1,
2,!
3".
!2"
!3"
2
!3"
45!
3"
602
!2"
©G
lenc
oe/M
cGra
w-Hi
ll36
8G
lenc
oe G
eom
etry
Cons
truct
ing
Valu
es o
f Squ
are
Root
sT
he d
iagr
am a
t th
e ri
ght
show
s a
righ
t is
osce
les
tria
ngle
wit
h tw
o le
gs o
f le
ngth
1 in
ch.B
y th
e P
ytha
gore
an T
heor
em,t
he le
ngth
of
the
hyp
oten
use
is !
2"in
ches
.By
cons
truc
ting
an
adja
cent
rig
ht
tria
ngle
wit
h le
gs o
f !2"
inch
es a
nd 1
inch
,you
can
cre
ate
a se
gmen
t of
leng
th !
3".
By
cont
inui
ng t
his
proc
ess
as s
how
n be
low
,you
can
con
stru
ct a
“w
heel
”of
squ
are
root
s.T
his
whe
el is
cal
led
the
“Whe
el o
f The
odor
us”
afte
r a
Gre
ek p
hilo
soph
er w
ho li
ved
abou
t 40
0 B.C
.
Con
tinu
e co
nstr
ucti
ng t
he w
heel
unt
il yo
u m
ake
a se
gmen
t of
leng
th !
18".
!%
1
1
1
3!
%
2
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-3
7-3
1
1
11
1
1!%2
!%3
!%5
!%6
!%7 !%8
!"10
!"11
!"12
!"13
!"14
!"15
!"17!"18
!"16
" 4
!%4 "
2
!%9 "
3
Answers (Lesson 7-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
An
swer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Trig
onom
etry
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-4
7-4
©G
lenc
oe/M
cGra
w-Hi
ll36
9G
lenc
oe G
eom
etry
Lesson 7-4
Trig
on
om
etri
c R
atio
sT
he r
atio
of
the
leng
ths
of t
wo
side
s of
a r
ight
tr
iang
le is
cal
led
a tr
igon
omet
ric
rati
o.T
he t
hree
mos
t co
mm
on r
atio
s ar
e si
ne,
cosi
ne,
and
tan
gen
t,w
hich
are
abb
revi
ated
sin
,cos
,and
tan
,re
spec
tive
ly.
sin
R"!le
g hyop pp oto es nit ue s# eR
!co
s R
"ta
n R
"
"!r t!
"!s t!
"!r s!
Fin
d s
in A
,cos
A,a
nd
tan
A.E
xpre
ss e
ach
rat
io a
s
a d
ecim
al t
o th
e n
eare
st t
hou
san
dth
.
sin
A"!o hp yp po os ti et ne ul se eg
!co
s A
"!a hd yj pa oc te en nt ul se eg
!ta
n A
"! aop dp jao cs ei nte t
l le eg g!
"!B A
BC !"
! AABC !
"!B A
CC !
"! 15 3!
"!1 12 3!
"! 15 2!
#0.
385
#0.
923
#0.
417
Fin
d t
he
ind
icat
ed t
rigo
nom
etri
c ra
tio
as a
fra
ctio
n
and
as
a d
ecim
al.I
f n
eces
sary
,rou
nd
to
the
nea
rest
te
n-t
hou
san
dth
.
1.si
n A
2.ta
n B
!1 15 7!;0
.882
4! 18 5!
;0.5
333
3.co
s A
4.co
s B
! 18 7!;0
.470
6!1 15 7!
;0.8
824
5.si
n D
6.ta
n E
!4 5! ;0.
8!3 4! ;
0.75
7.co
s E
8.co
s D
!4 5! ;0.
8!3 5! ;
0.6
16
1620
12
3430 CB
AD
FE
12135 CB
A
leg
oppo
site
#R
!!
!le
g ad
jace
nt t
o #
Rle
g ad
jace
nt t
o #
R!
!!
hypo
tenu
se
str TS
R
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-Hi
ll37
0G
lenc
oe G
eom
etry
Use
Tri
go
no
met
ric
Rat
ios
In a
rig
ht t
rian
gle,
if y
ou k
now
the
mea
sure
s of
tw
o si
des
or if
you
kno
w t
he m
easu
res
of o
ne s
ide
and
an a
cute
ang
le,t
hen
you
can
use
trig
onom
etri
cra
tios
to
find
the
mea
sure
s of
the
mis
sing
sid
es o
r an
gles
of
the
tria
ngle
.
Fin
d x
,y,a
nd
z.R
oun
d e
ach
mea
sure
to
the
nea
rest
w
hol
e n
um
ber.
1858
#
x#C
Byz
A
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Trig
onom
etry
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-4
7-4
a.F
ind
x.
x%
58"
90x
"32
b.F
ind
y.
tan
A"
! 1y 8!
tan
58°"
! 1y 8!
y"
18 t
an 5
8°y
#29
c.F
ind
z.
cos
A"
!1 z8 !
cos
58°
"!1 z8 !
zco
s 58
°"
18
z"
! cos18
58°
!
z#
34
Exer
cises
Exer
cises
Fin
d x
.Rou
nd
to
the
nea
rest
ten
th.
1.2.
17.0
48.6
3.4.
22.6
76.0
5.6.
24.9
34.2
1564
#x
1640
#
x
4
1x#
12
5x#
1216 x#
3228
#x
Exam
ple
Exam
ple
Answers (Lesson 7-4)
© Glencoe/McGraw-Hill A12 Glencoe Geometry
Skil
ls P
ract
ice
Trig
onom
etry
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-4
7-4
©G
lenc
oe/M
cGra
w-Hi
ll37
1G
lenc
oe G
eom
etry
Lesson 7-4
Use
!R
ST
to f
ind
sin
R,c
os R
,tan
R,s
in S
,cos
S,a
nd
tan
S.
Exp
ress
eac
h r
atio
as
a fr
acti
on a
nd
as
a d
ecim
al t
o th
e n
eare
st h
un
dre
dth
.
1.r
"16
,s"
30,t
"34
2.r
"10
,s"
24,t
"26
sin
R"
!1 36 4!#
0.47
;si
n R
"!1 20 6!
#0.
38;
cos
R"
!3 30 4!#
0.88
;co
s R
"!2 24 6!
#0.
92;
tan
R"
!1 36 0!#
0.53
;ta
n R
"!1 20 4!
#0.
42;
sin
S"
!3 30 4!#
0.88
;si
n S
"!2 24 6!
#0.
92;
cos
S"
!1 36 4!#
0.47
;co
s S
"!1 20 6!
#0.
38;
tan
S"
!3 10 6!#
1.88
tan
S"
!2 14 0!"
2.4
Use
a c
alcu
lato
r to
fin
d e
ach
val
ue.
Rou
nd
to
the
nea
rest
ten
-th
ousa
nd
th.
3.si
n 5
0.08
724.
tan
230.
4245
5.co
s 61
0.48
48
6.si
n 75
.80.
9694
7.ta
n 17
.30.
3115
8.co
s 52
.90.
6032
Use
th
e fi
gure
to
fin
d e
ach
tri
gon
omet
ric
rati
o.E
xpre
ss
answ
ers
as a
fra
ctio
n a
nd
as
a d
ecim
al r
oun
ded
to
the
nea
rest
ten
-th
ousa
nd
th.
9.ta
n C
10.s
in A
11.c
os C
! 49 0!#
0.22
50!4 40 1!
#0.
9756
!4 40 1!#
0.97
56
Fin
d t
he
mea
sure
of
each
acu
te a
ngl
e to
th
e n
eare
st t
enth
of
a d
egre
e.
12.s
in B
"0.
2985
17.4
13.t
an A
"0.
4168
22.6
14.c
os R
"0.
8443
32.4
15.t
an C
"0.
3894
21.3
16.c
os B
"0.
7329
42.9
17.s
in A
"0.
1176
6.8
Fin
d x
.Rou
nd
to
the
nea
rest
ten
th.
18.
19.
20.
28.8
73.5
15.9
19
x
33#
UL
S
27
x#8
BAC
27
x#13 B
A
C
41
409
B
AC
sR
S Trt
©G
lenc
oe/M
cGra
w-Hi
ll37
2G
lenc
oe G
eom
etry
Use
!L
MN
to f
ind
sin
L,c
os L
,tan
L,s
in M
,cos
M,a
nd
tan
M.
Exp
ress
eac
h r
atio
as
a fr
acti
on a
nd
as
a d
ecim
al t
o th
e n
eare
st h
un
dre
dth
.
1.!
"15
,m"
36,n
"39
2.!
"12
,m"
12!
3",n
"24
sin
L"
!1 35 9!#
0.38
;si
n L
"!1 22 4!
"0.
50;
cos
L"
!3 36 9!#
0.92
;co
s L
"#
0.87
;
tan
L"
!1 35 6!#
0.42
;ta
n L
"#
0.58
;
sin
M"
!3 36 9!#
0.92
;si
n M
"#
0.87
;
cos
M"
!1 35 9!#
0.38
;co
s M
"!1 22 4!
"0.
50;
tan
M"
!3 16 5!"
2.4
tan
M"
#1.
73
Use
a c
alcu
lato
r to
fin
d e
ach
val
ue.
Rou
nd
to
the
nea
rest
ten
-th
ousa
nd
th.
3.si
n 92
.40.
9991
4.ta
n 27
.50.
5206
5.co
s 64
.80.
4258
Use
th
e fi
gure
to
fin
d e
ach
tri
gon
omet
ric
rati
o.E
xpre
ss
answ
ers
as a
fra
ctio
n a
nd
as
a d
ecim
al r
oun
ded
to
the
nea
rest
ten
-th
ousa
nd
th.
6.co
s A
7.ta
n B
8.si
n A
#0.
9487
!3 1!"
3.00
00#
0.31
62
Fin
d t
he
mea
sure
of
each
acu
te a
ngl
e to
th
e n
eare
st t
enth
of
a d
egre
e.
9.si
n B
"0.
7823
51.5
10.t
an A
"0.
2356
13.3
11.c
os R
"0.
6401
50.2
Fin
d x
.Rou
nd
to
the
nea
rest
ten
th.
12.
64.4
13.
18.1
14.
24.2
15.G
EOG
RA
PHY
Die
go u
sed
a th
eodo
lite
to m
ap a
reg
ion
of la
nd f
or h
is
clas
s in
geo
mor
phol
ogy.
To d
eter
min
e th
e el
evat
ion
of a
ver
tica
l roc
kfo
rmat
ion,
he m
easu
red
the
dist
ance
fro
m t
he b
ase
of t
he f
orm
atio
n to
hi
s po
siti
on a
nd t
he a
ngle
bet
wee
n th
e gr
ound
and
the
line
of
sigh
t to
th
e to
p of
the
for
mat
ion.
The
dis
tanc
e w
as 4
3 m
eter
s an
d th
e an
gle
was
36
deg
rees
.Wha
t is
the
hei
ght
of t
he f
orm
atio
n to
the
nea
rest
met
er?
31 m
36# 43
m
41#
x
3229
x#9
23
x#11
!10"
!10
3!10"
!10
15
5 !"10
5 CA
B
12!
3"!
12
12!
3"!
2412! 12
!3"
12!
3"!
24
ML
N
Pra
ctic
e (A
vera
ge)
Trig
onom
etry
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-4
7-4
Answers (Lesson 7-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
An
swer
s
Rea
din
g t
o L
earn
Math
emati
csTr
igon
omet
ry
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-4
7-4
©G
lenc
oe/M
cGra
w-Hi
ll37
3G
lenc
oe G
eom
etry
Lesson 7-4
Pre-
Act
ivit
yH
ow c
an s
urv
eyor
s d
eter
min
e an
gle
mea
sure
s?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 7-
4 at
the
top
of
page
364
in y
our
text
book
.
•W
hy is
it im
port
ant
to d
eter
min
e th
e re
lati
ve p
osit
ions
acc
urat
ely
inna
viga
tion
? (G
ive
two
poss
ible
rea
sons
.)Sa
mpl
e an
swer
s:(1
) To
avoi
d co
llisi
ons
betw
een
ship
s,an
d (2
) to
prev
ent s
hips
from
losi
ng th
eir b
earin
gs a
nd g
ettin
g lo
st a
t sea
.•
Wha
t do
es c
alib
rate
dm
ean?
Sam
ple
answ
er:m
arke
d pr
ecis
ely
tope
rmit
accu
rate
mea
sure
men
ts to
be
mad
eR
ead
ing
th
e Le
sso
n
1.R
efer
to
the
figu
re.W
rite
a r
atio
usi
ng t
he s
ide
leng
ths
in t
he
figu
re t
o re
pres
ent
each
of
the
follo
win
g tr
igon
omet
ric
rati
os.
A.s
in N
! MMNP !
B.c
os N
! MNP N!
C.t
an N
!M NPP !
D.t
an M
! MNPP!
E.s
in M
! MNP N!F.
cos
M! MM
NP !
2.A
ssum
e th
at y
ou e
nter
eac
h of
the
exp
ress
ions
in t
he li
st o
n th
e le
ft in
to y
our
calc
ulat
or.
Mat
ch e
ach
of t
hese
exp
ress
ions
wit
h a
desc
ript
ion
from
the
list
on
the
righ
t to
tel
l wha
tyo
u ar
e fi
ndin
g w
hen
you
ente
r th
is e
xpre
ssio
n.
P
MN
a.si
n 20
vb.
cos
20ii
c.si
n#1
0.8
vid.
tan#
10.
8iii
e.ta
n 20
ivf.
cos#
10.
8i
i.th
e de
gree
mea
sure
of
an a
cute
ang
le w
hose
cos
ine
is 0
.8
ii.
the
rati
o of
the
leng
th o
f th
e le
g ad
jace
nt t
o th
e 20
°an
gle
to t
hele
ngth
of
hypo
tenu
se in
a 2
0°-7
0°-9
0°tr
iang
le
iii.
the
degr
ee m
easu
re o
f an
acu
te a
ngle
in a
rig
ht t
rian
gle
for
whi
ch t
he r
atio
of
the
leng
th o
f th
e op
posi
te le
g to
the
leng
th o
fth
e ad
jace
nt le
g is
0.8
iv.t
he r
atio
of t
he le
ngth
of t
he le
g op
posi
te t
he 2
0°an
gle
to t
hele
ngth
of t
he le
g ad
jace
nt t
o it
in a
20°
-70°
-90°
tria
ngle
v.th
e ra
tio
of t
he le
ngth
of
the
leg
oppo
site
the
20°
angl
e to
the
leng
th o
f hy
pote
nuse
in a
20°
-70°
-90°
tria
ngle
vi.t
he d
egre
e m
easu
re o
f an
acu
te a
ngle
in a
rig
ht t
rian
gle
for
whi
ch t
he r
atio
of
the
leng
th o
f th
e op
posi
te le
g to
the
leng
th o
fth
e hy
pote
nuse
is 0
.8
Hel
pin
g Y
ou
Rem
emb
er
3.H
ow c
an t
he c
oin
cos
ine
help
you
to
rem
embe
r th
e re
lati
onsh
ip b
etw
een
the
sine
s an
dco
sine
s of
the
tw
o ac
ute
angl
es o
f a
righ
t tr
iang
le?
Sam
ple
answ
er:T
he c
oin
cos
ine
com
es fr
om c
ompl
emen
t,as
inco
mpl
emen
tary
angl
es.T
he c
osin
e of
an
acut
e an
gle
is e
qual
to th
e si
neof
its
com
plem
ent.
©G
lenc
oe/M
cGra
w-Hi
ll37
4G
lenc
oe G
eom
etry
Sine
and
Cos
ine
of A
ngle
sT
he f
ollo
win
g di
agra
m c
an b
e us
ed t
o ob
tain
app
roxi
mat
e va
lues
for
the
sin
ean
d co
sine
of
angl
es f
rom
0°
to 9
0°.T
he r
adiu
s of
the
cir
cle
is 1
.So,
the
sine
and
cosi
ne v
alue
s ca
n be
rea
d di
rect
ly f
rom
the
ver
tica
l and
hor
izon
tal a
xes.
Fin
d a
pp
roxi
mat
e va
lues
for
sin
40°
and
cos
40#
.Con
sid
er t
he
tria
ngl
e fo
rmed
by
the
segm
ent
mar
ked
40°
,as
illu
stra
ted
by
the
shad
ed
tria
ngl
e at
rig
ht.
sin
40°
"!a c!
#!0.
164 !or
0.6
4co
s 40
°"
!b c!#
!0.177 !
or 0
.77
1.U
se t
he d
iagr
am a
bove
to
com
plet
e th
e ch
art
of v
alue
s.
2.C
ompa
re t
he s
ine
and
cosi
ne o
f tw
o co
mpl
emen
tary
ang
les
(ang
les
who
se
sum
is 9
0°).
Wha
t do
you
not
ice?
The
sine
of a
n an
gle
is e
qual
to th
e co
sine
of t
he c
ompl
emen
t of
the
angl
e.
00.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.91
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
90°
0°10°
20°
30°
40°
50°
60°
70°
80°
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-4
7-4 x°
0°10
°20
°30
°40
°50
°60
°70
°80
°90
°
sin x
°0
0.17
0.34
0.5
0.64
0.77
0.87
0.94
0.98
1co
s x°
10.
980.
940.
870.
770.
640.
50.
340.
170
1 0
40°
0.64
c "
1 u
nit
x°b
" c
os x
°0.
771
a "
sin
x°
Exam
ple
Exam
ple
Answers (Lesson 7-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Angl
es o
f Ele
vatio
n an
d De
pres
sion
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-5
7-5
©G
lenc
oe/M
cGra
w-Hi
ll37
5G
lenc
oe G
eom
etry
Lesson 7-5
An
gle
s o
f El
evat
ion
Man
y re
al-w
orld
pro
blem
s th
at in
volv
e lo
okin
g up
to
an o
bjec
t ca
n be
des
crib
ed in
ter
ms
of a
n an
gle
of
elev
atio
n,w
hich
is t
he a
ngle
bet
wee
n an
obs
erve
r’s li
ne o
f si
ght
and
a ho
rizo
ntal
line
.
Th
e an
gle
of e
leva
tion
fro
m p
oin
t A
to t
he
top
of
a c
liff
is
34°.
If p
oin
t A
is 1
000
feet
fro
m t
he
base
of
the
clif
f,h
ow h
igh
is
the
clif
f?L
et x
"th
e he
ight
of
the
clif
f.
tan
34°
"! 10
x 00!ta
n "
!o ap dp jao cs eit ne t!
1000
(tan
34°
)"
xM
ultip
ly ea
ch s
ide
by 1
000.
674.
5"
xUs
e a
calcu
lato
r.
The
hei
ght
of t
he c
liff
is a
bout
674
.5 f
eet.
Sol
ve e
ach
pro
blem
.Rou
nd
mea
sure
s of
seg
men
ts t
o th
e n
eare
st w
hol
e n
um
ber
and
an
gles
to
the
nea
rest
deg
ree.
1.T
he a
ngle
of
elev
atio
n fr
om p
oint
Ato
the
top
of
a hi
ll is
49°
.If
poi
nt A
is 4
00 f
eet
from
the
bas
e of
the
hill
,how
hig
h is
th
e hi
ll?
460
ft
2.F
ind
the
angl
e of
ele
vati
on o
f th
e su
n w
hen
a 12
.5-m
eter
-tal
l te
leph
one
pole
cas
ts a
18-
met
er-l
ong
shad
ow.
35°
3.A
ladd
er le
anin
g ag
ains
t a
build
ing
mak
es a
n an
gle
of 7
8°w
ith
the
grou
nd.T
he f
oot
of t
he la
dder
is 5
fee
t fr
om t
he
build
ing.
How
long
is t
he la
dder
?
24 ft
4.A
per
son
who
se e
yes
are
5 fe
et a
bove
the
gro
und
is s
tand
ing
on t
he r
unw
ay o
f an
air
port
100
fee
t fr
om t
he c
ontr
ol t
ower
.T
hat
pers
on o
bser
ves
an a
ir t
raff
ic c
ontr
olle
r at
the
win
dow
of
the
132
-foo
t to
wer
.Wha
t is
the
ang
le o
f el
evat
ion?
52°
?5
ft10
0 ft
132
ft
78#
5 ft
?
18 m
12.5
msun
?
✹
400
ft
?
49#
A
?
1000
ft34
#A
x
angl
e of
elev
atio
n
line o
f sigh
t
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-Hi
ll37
6G
lenc
oe G
eom
etry
An
gle
s o
f D
epre
ssio
nW
hen
an o
bser
ver
is lo
okin
g do
wn,
the
angl
e of
dep
ress
ion
is t
he a
ngle
bet
wee
n th
e ob
serv
er’s
line
of
sigh
t an
d a
hori
zont
al li
ne.
Th
e an
gle
of d
epre
ssio
n f
rom
th
e to
p o
f an
80
-foo
t bu
ild
ing
to p
oin
t A
on t
he
grou
nd
is
42°.
How
far
is
th
e fo
ot o
f th
e bu
ild
ing
from
poi
nt
A?
Let
x"
the
dist
ance
fro
m p
oint
Ato
the
foo
t of
the
bui
ldin
g.Si
nce
the
hori
zont
al li
ne is
par
alle
l to
the
grou
nd,t
he a
ngle
of
depr
essi
on#
DB
Ais
con
grue
nt t
o #
BA
C.
tan
42°
"!8 x0 !
tan
"!o ap dp jao cs eit ne t
!
x(ta
n 42
°)"
80M
ultip
ly ea
ch s
ide
by x
.
x"
! tan80
42°
!Di
vide
each
sid
e by
tan
42°.
x#
88.8
Use
a ca
lcula
tor.
Poin
t A
is a
bout
89
feet
fro
m t
he b
ase
of t
he b
uild
ing.
Sol
ve e
ach
pro
blem
.Rou
nd
mea
sure
s of
seg
men
ts t
o th
e n
eare
st w
hol
e n
um
ber
and
an
gles
to
the
nea
rest
deg
ree.
1.T
he a
ngle
of
depr
essi
on f
rom
the
top
of
a sh
eer
clif
f to
po
int
Aon
the
gro
und
is 3
5°.I
f po
int
Ais
280
fee
t fr
om
the
base
of
the
clif
f,ho
w t
all i
s th
e cl
iff?
196
ft
2.T
he a
ngle
of
depr
essi
on f
rom
a b
allo
on o
n a
75-f
oot
stri
ng t
o a
pers
on o
n th
e gr
ound
is 3
6°.H
ow h
igh
is
the
ballo
on?
44 ft
3.A
ski
run
is 1
000
yard
s lo
ng w
ith
a ve
rtic
al d
rop
of
208
yard
s.F
ind
the
angl
e of
dep
ress
ion
from
the
top
of
the
ski
run
to
the
bott
om.
12°
4.F
rom
the
top
of
a 12
0-fo
ot-h
igh
tow
er,a
n ai
r tr
affi
c co
ntro
ller
obse
rves
an
airp
lane
on
the
runw
ay a
t an
an
gle
of d
epre
ssio
n of
19°
.How
far
fro
m t
he b
ase
of t
heto
wer
is t
he a
irpl
ane?
349
ft
120
ft
?
19#
208
yd
?
1000
yd
36#
75 ft
?
A
35#
280
ft
?
ACB
D
x42
#
angl
e of
depr
essi
on
horiz
onta
l
80 ftY
line o
f sigh
t
horiz
onta
lan
gle
ofde
pres
sion
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Angl
es o
f Ele
vatio
n an
d De
pres
sion
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-5
7-5
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 7-5)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Angl
es o
f Ele
vatio
n an
d De
pres
sion
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-5
7-5
©G
lenc
oe/M
cGra
w-Hi
ll37
7G
lenc
oe G
eom
etry
Lesson 7-5
Nam
e th
e an
gle
of d
epre
ssio
n o
r an
gle
of e
leva
tion
in
eac
h f
igu
re.
1.2.
"FL
S;"
TSL
"RT
W;"
SWT
3.4.
"DC
B;"
ABC
"W
ZP;"
RPZ
5.M
OU
NTA
IN B
IKIN
GO
n a
mou
ntai
n bi
ke t
rip
alon
g th
e G
emin
i Bri
dges
Tra
il in
Moa
b,U
tah,
Nab
uko
stop
ped
on t
he c
anyo
n fl
oor
to g
et a
goo
d vi
ew o
f th
e tw
in s
ands
tone
brid
ges.
Nab
uko
is s
tand
ing
abou
t 60
met
ers
from
the
bas
e of
the
can
yon
clif
f,an
d th
ena
tura
l arc
h br
idge
s ar
e ab
out
100
met
ers
up t
he c
anyo
n w
all.
If h
er li
ne o
f si
ght
is f
ive
feet
abo
ve t
he g
roun
d,w
hat
is t
he a
ngle
of
elev
atio
n to
the
top
of
the
brid
ges?
Rou
nd t
oth
e ne
ares
t te
nth
degr
ee.
abou
t 57.
7#
6.SH
AD
OW
SSu
ppos
e th
e su
n ca
sts
a sh
adow
off
a 3
5-fo
ot b
uild
ing.
If t
he a
ngle
of
elev
atio
n to
the
sun
is 6
0°,h
ow lo
ng is
the
sha
dow
to
the
nea
rest
ten
th o
f a
foot
?ab
out 2
0.2
ft
7.B
ALL
OO
NIN
GF
rom
her
pos
itio
n in
a h
ot-a
ir b
allo
on,A
ngie
can
see
her
car
par
ked
in a
fiel
d.If
the
ang
le o
f de
pres
sion
is 8
°an
d A
ngie
is 3
8 m
eter
s ab
ove
the
grou
nd,w
hat
isth
e st
raig
ht-l
ine
dist
ance
fro
m A
ngie
to
her
car?
Rou
nd t
o th
e ne
ares
t w
hole
met
er.
abou
t 273
m
8.IN
DIR
ECT
MEA
SUR
EMEN
TK
yle
is a
t th
e en
d of
a p
ier
30 f
eet
abov
e th
e oc
ean.
His
eye
leve
l is
3 fe
et a
bove
the
pie
r.H
e is
usi
ng b
inoc
ular
s to
w
atch
a w
hale
sur
face
.If
the
angl
e of
dep
ress
ion
of t
he w
hale
is 2
0°,h
ow f
ar is
the
wha
le f
rom
K
yle’
s bi
nocu
lars
? R
ound
to
the
near
est
tent
h fo
ot.
abou
t 96.
5 ft
wha
lew
ater
leve
l
20#
Kyle
’s ey
es
pier
3 ft
30 ft
60# ?
35 ft
Z
PW
R
D
AC
B
T
WR
S
F
T
L
S
©G
lenc
oe/M
cGra
w-Hi
ll37
8G
lenc
oe G
eom
etry
Nam
e th
e an
gle
of d
epre
ssio
n o
r an
gle
of e
leva
tion
in
eac
h f
igu
re.
1.2.
"TR
Z;"
YZR
"PR
M;"
LMR
3.W
ATE
R T
OW
ERS
A s
tude
nt c
an s
ee a
wat
er t
ower
fro
m t
he c
lose
st p
oint
of
the
socc
erfi
eld
at S
an L
obos
Hig
h Sc
hool
.The
edg
e of
the
soc
cer
fiel
d is
abo
ut 1
10 f
eet
from
the
wat
er t
ower
and
the
wat
er t
ower
sta
nds
at a
hei
ght
of 3
2.5
feet
.Wha
t is
the
ang
le o
fel
evat
ion
if t
he e
ye le
vel o
f th
e st
uden
t vi
ewin
g th
e to
wer
fro
m t
he e
dge
of t
he s
occe
rfi
eld
is 6
fee
t ab
ove
the
grou
nd?
Rou
nd t
o th
e ne
ares
t te
nth
degr
ee.
abou
t 13.
5#
4.C
ON
STR
UC
TIO
NA
roo
fer
prop
s a
ladd
er a
gain
st a
wal
l so
that
the
top
of
the
ladd
erre
ache
s a
30-f
oot
roof
tha
t ne
eds
repa
ir.I
f th
e an
gle
of e
leva
tion
fro
m t
he b
otto
m o
f th
ela
dder
to
the
roof
is 5
5°,h
ow f
ar is
the
ladd
er f
rom
the
bas
e of
the
wal
l? R
ound
you
ran
swer
to
the
near
est
foot
.
abou
t 21
ft
5.TO
WN
OR
DIN
AN
CES
The
tow
n of
Bel
mon
t re
stri
cts
the
heig
ht
of f
lagp
oles
to
25 f
eet
on a
ny p
rope
rty.
Lin
dsay
wan
ts t
o de
term
ine
whe
ther
her
sch
ool i
s in
com
plia
nce
wit
h th
e re
gula
tion
.Her
eye
le
vel i
s 5.
5 fe
et f
rom
the
gro
und
and
she
stan
ds 3
6 fe
et f
rom
the
flag
pole
.If
the
angl
e of
ele
vati
on is
abo
ut 2
5°,w
hat
is t
he h
eigh
t of
the
fla
gpol
e to
the
nea
rest
ten
th f
oot?
abou
t 22.
3 ft
6.G
EOG
RA
PHY
Step
han
is s
tand
ing
on a
mes
a at
the
Pai
nted
Des
ert.
The
ele
vati
on o
fth
e m
esa
is a
bout
138
0 m
eter
s an
d St
epha
n’s
eye
leve
l is
1.8
met
ers
abov
e gr
ound
.If
Step
han
can
see
a ba
nd o
f m
ulti
colo
red
shal
e at
the
bot
tom
and
the
ang
le o
f de
pres
sion
is 2
9°,a
bout
how
far
is t
he b
and
of s
hale
fro
m h
is e
yes?
Rou
nd t
o th
e ne
ares
t m
eter
.
abou
t 285
0 m
7.IN
DIR
ECT
MEA
SUR
EMEN
TM
r.D
omin
guez
is s
tand
ing
on a
40-
foot
oce
an b
luff
nea
r hi
s ho
me.
He
can
see
his
two
dogs
on
the
beac
h be
low
.If
his
line
of s
ight
is 6
fee
t ab
ove
the
grou
nd a
nd t
he a
ngle
s of
dep
ress
ion
to h
is d
ogs
are
34°
and
48°,
how
far
apa
rt a
re t
he d
ogs
to t
he n
eare
st f
oot?
abou
t 27
ft48
#34
#
40 ft
6 ft
Mr.
Dom
ingu
ez
bluf
f
25#
5.5
ft36
ft
x
R
M
P
L
T
YR
Z
Pra
ctic
e (A
vera
ge)
Angl
es o
f Ele
vatio
n an
d De
pres
sion
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-5
7-5
Answers (Lesson 7-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csAn
gles
of E
leva
tion
and
Depr
essi
on
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-5
7-5
©G
lenc
oe/M
cGra
w-Hi
ll37
9G
lenc
oe G
eom
etry
Lesson 7-5
Pre-
Act
ivit
yH
ow d
o ai
rlin
e p
ilot
s u
se a
ngl
es o
f el
evat
ion
an
d d
epre
ssio
n?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 7-
5 at
the
top
of
page
371
in y
our
text
book
.
Wha
t do
es t
he a
ngle
mea
sure
tel
l the
pilo
t?Sa
mpl
e an
swer
:how
stee
p he
r asc
ent m
ust b
e to
cle
ar th
e pe
ak
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.T
he t
wo
obse
rver
s ar
e lo
okin
g at
on
e an
othe
r.Se
lect
the
cor
rect
cho
ice
for
each
que
stio
n.
a.W
hat
is t
he li
ne o
f si
ght?
iii(i
) lin
e R
S(i
i) li
ne S
T(i
ii) li
ne R
T(i
v) li
ne T
U
b.W
hat
is t
he a
ngle
of
elev
atio
n?ii
(i) #
RS
T(i
i) #
SR
T(i
ii) #
RT
S(i
v) #
UT
R
c.W
hat
is t
he a
ngle
of
depr
essi
on?
iv(i
) #R
ST
(ii)
#S
RT
(iii)
#R
TS
(iv)
#U
TR
d.H
ow a
re t
he a
ngle
of
elev
atio
n an
d th
e an
gle
of d
epre
ssio
n re
late
d?ii
(i)
The
y ar
e co
mpl
emen
tary
.(i
i)T
hey
are
cong
ruen
t.(i
ii)T
hey
are
supp
lem
enta
ry.
(iv)
The
ang
le o
f el
evat
ion
is la
rger
tha
n th
e an
gle
of d
epre
ssio
n.
e.W
hich
pos
tula
te o
r th
eore
m t
hat
you
lear
ned
in C
hapt
er 3
sup
port
s yo
ur a
nsw
er f
orpa
rt c
?iv
(i)
Cor
resp
ondi
ng A
ngle
s Po
stul
ate
(ii)
Alt
erna
te E
xter
ior
Ang
les
The
orem
(iii)
Con
secu
tive
Int
erio
r A
ngle
s T
heor
em(i
v)A
lter
nate
Int
erio
r A
ngle
s T
heor
em
2.A
stu
dent
say
s th
at t
he a
ngle
of
elev
atio
n fr
om h
is e
ye t
o th
e to
p of
a f
lagp
ole
is 1
35°.
Wha
t is
wro
ng w
ith
the
stud
ent’s
sta
tem
ent?
An a
ngle
of e
leva
tion
cann
ot b
e ob
tuse
.
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
som
ethi
ng is
to
expl
ain
it t
o so
meo
ne e
lse.
Supp
ose
a cl
assm
ate
finds
it d
iffic
ult
to d
isti
ngui
sh b
etw
een
angl
es o
f ele
vati
on a
nd a
ngle
s of
dep
ress
ion.
Wha
tar
e so
me
hint
s yo
u ca
n gi
ve h
er t
o he
lp h
er g
et it
rig
ht e
very
tim
e?Sa
mpl
e an
swer
s:(1
) The
ang
le o
f dep
ress
ion
and
the
angl
e of
ele
vatio
n ar
e bo
th m
easu
red
betw
een
the
horiz
onta
l and
the
line
of s
ight
.(2)
The
ang
le o
f dep
ress
ion
is a
lway
s co
ngru
ent t
o th
e an
gle
of e
leva
tion
in th
e sa
me
diag
ram
.(3
) Ass
ocia
te th
e w
ord
elev
atio
nw
ith th
e w
ord
upan
d th
e w
ord
depr
essi
onw
ith th
e w
ord
dow
n.
STob
serv
er a
tto
p of
bui
ldin
g
obse
rver
on g
roun
dR
U
©G
lenc
oe/M
cGra
w-Hi
ll38
0G
lenc
oe G
eom
etry
Read
ing
Mat
hem
atic
sT
he t
hree
mos
t co
mm
on t
rigo
nom
etri
c ra
tios
are
si
ne,
cosi
ne,
and
tan
gen
t.T
hree
oth
er r
atio
s ar
e th
eco
seca
nt,
seca
nt,
and
cota
nge
nt.
The
cha
rt b
elow
sh
ows
abbr
evia
tion
s an
d de
fini
tion
s fo
r al
l six
rat
ios.
Ref
er t
o th
e tr
iang
le a
t th
e ri
ght.
Use
th
e ab
brev
iati
ons
to r
ewri
te e
ach
sta
tem
ent
as a
n e
quat
ion
.
1.T
he s
ecan
t of
ang
le A
is e
qual
to
1 di
vide
d by
the
cos
ine
of a
ngle
A.
sec
A"
! co1 s
A!
2.T
he c
osec
ant
of a
ngle
A is
equ
al t
o 1
divi
ded
by t
he s
ine
of a
ngle
A.
csc
A"
! sin1
A!
3.T
he c
otan
gent
of
angl
e A
is e
qual
to
1 di
vide
d by
the
tan
gent
of
angl
e A
.co
t A"
! tan1
A!
4.T
he c
osec
ant
of a
ngle
A m
ulti
plie
d by
the
sin
e of
ang
le A
is e
qual
to
1.cs
c A
sin
A"
1
5.T
he s
ecan
t of
ang
le A
mul
tipl
ied
by t
he c
osin
e of
ang
le A
is e
qual
to
1.se
c A
cos
A"
1
6.T
he c
otan
gent
of
angl
e A
tim
es t
he t
ange
nt o
f an
gle
Ais
equ
al t
o 1.
cot A
tan
A"
1
Use
th
e tr
ian
gle
at r
igh
t.W
rite
eac
h r
atio
.
7.se
c R
! st !8.
csc
R! rt !
9.co
t R
!s r!
10.s
ec S
! rt !11
.cs
c S
! st !12
.co
t S
! sr !
13.I
f si
n x°
"0.
289,
find
the
val
ue o
f cs
c x°.
#3.
46
14.I
f ta
n x°
"1.
376,
find
the
val
ue o
f co
t x°.
#0.
727
R TS
ts
r
A
ca
b
B C
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-5
7-5
Abbr
evia
tion
Read
as:
Ratio
sin A
the
sine
of #
A"
!a c!
cos
Ath
e co
sine
of #
A"
!b c!
tan
Ath
e ta
ngen
t of #
A"
!a b!
csc
Ath
e co
seca
nt o
f #A
"! ac !
sec
Ath
e se
cant
of #
A"
! bc !
cot A
the
cota
ngen
t of #
A"
!b a!le
gad
jace
nt to
#A
!!
!le
gop
posit
e#
A
hypo
tenu
se!
!!
leg
adja
cent
to#
A
hypo
tenu
se!
!le
gop
posit
e#
A
leg
oppo
site
#A
!!
!le
gad
jace
nt to
#A
leg
adja
cent
to#
A!
!!
hypo
tenu
se
leg
oppo
site
#A
!!
hypo
tenu
se
Answers (Lesson 7-5)
© Glencoe/McGraw-Hill A17 Glencoe Geometry
An
swer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
The
Law
of S
ines
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-6
7-6
©G
lenc
oe/M
cGra
w-Hi
ll38
1G
lenc
oe G
eom
etry
Lesson 7-6
The
Law
of
Sin
esIn
any
tri
angl
e,th
ere
is a
spe
cial
rel
atio
nshi
p be
twee
n th
e an
gles
of
the
tria
ngle
and
the
leng
ths
of t
he s
ides
opp
osit
e th
e an
gles
.
Law
of S
ines
!sinaA !
"!sin
bB !"
!sincC !
In !
AB
C,f
ind
b.
!sin c
C !"
!sin b
B !La
w of
Sin
es
!sin 30
45°
!"
!sin b74
°!
m#
C"
45, c
"30
, m#
B"
74
bsi
n 45
°"
30 s
in 7
4°Cr
oss
mul
tiply.
b"
!30si
s nin 457 °4°
!Di
vide
each
sid
e by
sin
45°
.
b#
40.8
Use
a ca
lcula
tor.
45#
3074
#
bB
AC
In !
DE
F,f
ind
m"
D.
!sin d
D !"
!sin e
E !La
w of
Sin
es
!si2n 8D !
"!si
n 2458
°!
d"
28, m
#E
"58
, e
"24
24 s
in D
"28
sin
58°
Cros
s m
ultip
ly.
sin
D"
!28s 2in 4
58°
!Di
vide
each
sid
e by
24.
D"
sin#
1 !28
s 2in 458
°!
Use
the
inve
rse
sine.
D#
81.6
°Us
e a
calcu
lato
r.
58#
24
28
E
FD
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d e
ach
mea
sure
usi
ng
the
give
n m
easu
res
of !
AB
C.R
oun
d a
ngl
e m
easu
res
toth
e n
eare
st d
egre
e an
d s
ide
mea
sure
s to
th
e n
eare
st t
enth
.
1.If
c"
12,m
#A
"80
,and
m#
C"
40,f
ind
a.18
.4
2.If
b"
20,c
"26
,and
m#
C"
52,f
ind
m#
B.
37
3.If
a"
18,c
"16
,and
m#
A"
84,f
ind
m#
C.
62
4.If
a"
25,m
#A
"72
,and
m#
B"
17,f
ind
b.7.
7
5.If
b"
12,m
#A
"89
,and
m#
B"
80,f
ind
a.
12.2
6.If
a"
30,c
"20
,and
m#
A"
60,f
ind
m#
C.
35
©G
lenc
oe/M
cGra
w-Hi
ll38
2G
lenc
oe G
eom
etry
Use
th
e La
w o
f Si
nes
to
So
lve
Pro
ble
ms
You
can
use
the
Law
of
Sin
esto
sol
veso
me
prob
lem
s th
at in
volv
e tr
iang
les.
Law
of S
ines
Let !
ABC
be a
ny tr
iang
le w
ith a
, b, a
nd c
repr
esen
ting
the
mea
sure
s of
the
sides
opp
osite
th
e an
gles
with
mea
sure
s A,
B, a
nd C
, res
pect
ively.
The
n !sin
aA !"
!sinbB !
"!sin
cC !.
Isos
cele
s !
AB
Ch
as a
bas
e of
24
cen
tim
eter
s an
d a
ve
rtex
an
gle
of 6
8°.F
ind
th
e p
erim
eter
of
the
tria
ngl
e.T
he v
erte
x an
gle
is 6
8°,s
o th
e su
m o
f th
e m
easu
res
of t
he b
ase
angl
es is
11
2 an
d m
#A
"m
#C
"56
.
!sin b
B !"
!sin a
A !La
w of
Sin
es
!sin 24
68°
!"
!sin a56
°!
m#
B"
68, b
"24
, m#
A"
56
asi
n 68
°"
24 s
in 5
6°Cr
oss
mul
tiply.
a"
!24si
s nin 685 °6°
!Di
vide
each
sid
e by
sin
68°
.
#21
.5Us
e a
calcu
lato
r.
The
tri
angl
e is
isos
cele
s,so
c"
21.5
.T
he p
erim
eter
is 2
4 %
21.5
%21
.5 o
r ab
out
67 c
enti
met
ers.
Dra
w a
tri
angl
e to
go
wit
h e
ach
exe
rcis
e an
d m
ark
it
wit
h t
he
give
n i
nfo
rmat
ion
.T
hen
sol
ve t
he
pro
blem
.Rou
nd
an
gle
mea
sure
s to
th
e n
eare
st d
egre
e an
d s
ide
mea
sure
s to
th
e n
eare
st t
enth
.
1.O
ne s
ide
of a
tri
angu
lar
gard
en is
42.
0 fe
et.T
he a
ngle
s on
eac
h en
d of
thi
s si
de m
easu
re66
°an
d 82
°.F
ind
the
leng
th o
f fe
nce
need
ed t
o en
clos
e th
e ga
rden
.19
2.9
ft
2.T
wo
rada
r st
atio
ns A
and
Bar
e 32
mile
s ap
art.
The
y lo
cate
an
airp
lane
Xat
the
sam
eti
me.
The
thr
ee p
oint
s fo
rm #
XA
B,w
hich
mea
sure
s 46
°,an
d #
XB
A,w
hich
mea
sure
s52
°.H
ow f
ar is
the
air
plan
e fr
om e
ach
stat
ion?
25.5
mi f
rom
A;2
3.2
mi f
rom
B
3.A
civ
il en
gine
er w
ants
to
dete
rmin
e th
e di
stan
ces
from
poi
nts
Aan
d B
to a
n in
acce
ssib
lepo
int
Cin
a r
iver
.#B
AC
mea
sure
s 67
°an
d #
AB
Cm
easu
res
52°.
If p
oint
s A
and
Bar
e82
.0 f
eet
apar
t,fi
nd t
he d
ista
nce
from
Cto
eac
h po
int.
86.3
ft to
poi
nt B
;73.
9 ft
to p
oint
A
4.A
ran
ger
tow
er a
t po
int
Ais
42
kilo
met
ers
nort
h of
a r
ange
r to
wer
at
poin
t B
.A fi
re a
tpo
int
Cis
obs
erve
d fr
om b
oth
tow
ers.
If #
BA
Cm
easu
res
43°
and
#A
BC
mea
sure
s 68
°,w
hich
ran
ger
tow
er is
clo
ser
to t
he f
ire?
How
muc
h cl
oser
?To
wer
Bis
11
km c
lose
r tha
n To
wer
A.
68# b
ca
24B
CA
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
The
Law
of S
ines
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-6
7-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 7-6)
© Glencoe/McGraw-Hill A18 Glencoe Geometry
Skil
ls P
ract
ice
The
Law
of S
ines
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-6
7-6
©G
lenc
oe/M
cGra
w-Hi
ll38
3G
lenc
oe G
eom
etry
Lesson 7-6
Fin
d e
ach
mea
sure
usi
ng
the
give
n m
easu
res
from
!A
BC
.Rou
nd
an
gle
mea
sure
sto
th
e n
eare
st t
enth
deg
ree
and
sid
e m
easu
res
to t
he
nea
rest
ten
th.
1.If
m#
A"
35,m
#B
"48
,and
b"
28,f
ind
a.21
.6
2.If
m#
B"
17,m
#C
"46
,and
c"
18,f
ind
b.7.
3
3.If
m#
C"
86,m
#A
"51
,and
a"
38,f
ind
c.48
.8
4.If
a"
17,b
"8,
and
m#
A"
73,f
ind
m#
B.
26.7
5.If
c"
38,b
"34
,and
m#
B"
36,f
ind
m#
C.
41.1
or 1
38.9
6.If
a"
12,c
"20
,and
m#
C"
83,f
ind
m#
A.
36.6
7.If
m#
A"
22,a
"18
,and
m#
B"
104,
find
b.
46.6
Sol
ve e
ach
!P
QR
des
crib
ed b
elow
.Rou
nd
mea
sure
s to
th
e n
eare
st t
enth
.
8.p
"27
,q"
40,m
#P
"33
m"
Q#
53.8
,m"
R#
93.2
,r#
49.5
;or
m"
Q#
126.
2,m
"R
#20
.8,r
#17
.69.
q"
12,r
"11
,m#
R"
16m
"P
#14
6.5,
m"
Q#
17.5
,p#
22.0
;or
m"
P#
1.5,
m"
Q#
162.
5,p
#1.
010
.p"
29,q
"34
,m#
Q"
111
m"
P#
52.8
,m"
R#
16.2
,r#
10.2
11.I
f m#
P"
89,p
"16
,r"
12m
"Q
#42
.4,m
"R
#48
.6,q
#10
.8
12.I
f m#
Q"
103,
m#
P"
63,p
"13
m"
R#
14,q
#14
.2,r
#3.
5
13.I
f m#
P"
96,m
#R
"82
,r"
35m
"Q
#2,
p#
35.2
,q#
1.2
14.I
f m#
R"
49,m
#Q
"76
,r"
26m
"P
#55
,p#
28.2
,q#
33.4
15.I
f m#
Q"
31,m
#P
"52
,p"
20m
"R
#97
,q#
13.1
,r#
25.2
16.I
f q"
8,m
#Q
"28
,m#
R"
72m
"P
#80
,p#
16.8
,r#
16.2
17.I
f r"
15,p
"21
,m#
P"
128
m"
Q#
17.7
,m"
R#
34.3
,q#
8.1
©G
lenc
oe/M
cGra
w-Hi
ll38
4G
lenc
oe G
eom
etry
Fin
d e
ach
mea
sure
usi
ng
the
give
n m
easu
res
from
!E
FG
.Rou
nd
an
gle
mea
sure
sto
th
e n
eare
st t
enth
deg
ree
and
sid
e m
easu
res
to t
he
nea
rest
ten
th.
1.If
m#
G"
14,m
#E
"67
,and
e"
14,f
ind
g.3.
7
2.If
e"
12.7
,m#
E"
42,a
nd m
#F
"61
,fin
d f.
16.6
3.If
g"
14,f
"5.
8,an
d m
#G
"83
,fin
d m
#F
.24
.3
4.If
e"
19.1
,m#
G"
34,a
nd m
#E
"56
,fin
d g.
12.9
5.If
f"
9.6,
g"
27.4
,and
m#
G"
43,f
ind
m#
F.
13.8
Sol
ve e
ach
!S
TU
des
crib
ed b
elow
.Rou
nd
mea
sure
s to
th
e n
eare
st t
enth
.
6.m
#T
"85
,s"
4.3,
t"
8.2
m"
S#
31.5
,m"
U#
63.5
,u#
7.4
7.s
"40
,u"
12,m
#S
"37
m"
T#
132.
6,m
"U
#10
.4,t
#48
.9
8.m
#U
"37
,t"
2.3,
m#
T"
17m
"S
#12
6,s
#6.
4,u
#4.
7
9.m
#S
"62
,m#
U"
59,s
"17
.8m
"T
#59
,t#
17.3
,u#
17.3
10.t
"28
.4,u
"21
.7,m
#T
"66
m"
S#
69.7
,m"
U#
44.3
,s#
29.2
11.m
#S
"89
,s"
15.3
,t"
14m
"T
#66
.2,m
"U
#24
.8,u
#6.
4
12.m
#T
"98
,m#
U"
74,u
"9.
6m
"S
#8,
s#
1.4,
t#9.
9
13.t
"11
.8,m
#S
"84
,m#
T"
47m
"U
"49
,s#
16.0
,u#
12.2
14.I
ND
IREC
T M
EASU
REM
ENT
To f
ind
the
dist
ance
fro
m t
he e
dge
of t
he la
ke t
o th
e tr
ee o
n th
e is
land
in t
he la
ke,H
anna
h se
t up
atr
iang
ular
con
figu
rati
on a
s sh
own
in t
he d
iagr
am.T
he d
ista
nce
from
loca
tion
Ato
loca
tion
Bis
85
met
ers.
The
mea
sure
s of
the
an
gles
at
Aan
d B
are
51°
and
83°,
resp
ecti
vely
.Wha
t is
the
dis
tanc
efr
om t
he e
dge
of t
he la
ke a
t B
to t
he t
ree
on t
he is
land
at
C?
abou
t 91.
8 m
A
C
B
Pra
ctic
e (A
vera
ge)
The
Law
of S
ines
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-6
7-6
Answers (Lesson 7-6)
© Glencoe/McGraw-Hill A19 Glencoe Geometry
An
swer
s
Rea
din
g t
o L
earn
Math
emati
csTh
e La
w o
f Sin
es
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-6
7-6
©G
lenc
oe/M
cGra
w-Hi
ll38
5G
lenc
oe G
eom
etry
Lesson 7-6
Pre-
Act
ivit
yH
ow a
re t
rian
gles
use
d i
n r
adio
ast
ron
omy?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 7-
6 at
the
top
of
page
377
in y
our
text
book
.
Why
mig
ht s
ever
al a
nten
nas
be b
ette
r th
an o
ne s
ingl
e an
tenn
a w
hen
stud
ying
dis
tant
obj
ects
?Sa
mpl
e an
swer
:Obs
ervi
ng a
n ob
ject
from
onl
y on
e po
sitio
n of
ten
does
not
pro
vide
eno
ugh
info
rmat
ion
to c
alcu
late
thin
gs s
uch
as th
e di
stan
ce fr
om th
eob
serv
er to
the
obje
ct.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.A
ccor
ding
to
the
Law
of
Sine
s,w
hich
of
the
fo
llow
ing
are
corr
ect
stat
emen
ts?
A,F
A.!
sinm
M!"
! sinn
N!"
! sinp
P!B
.!si
n Mm !
"!si
Nnn
!"
!sin P
p!
C.!
coms
M !"
!cos n
N !"
!cops
P !D
.!si
n mM !
%!si
n nN !
"!si
n pP !
E.(
sin
M)2
%(s
in N
)2"
(sin
P)2
F.!si
n pP !
"!si
n mM !
"!si
n nN !
2.St
ate
whe
ther
eac
h of
the
fol
low
ing
stat
emen
ts is
tru
eor
fal
se.I
f th
e st
atem
ent
is f
alse
,ex
plai
n w
hy.
a.T
he L
aw o
f Si
nes
appl
ies
to a
ll tr
iang
les.
true
b.T
he P
ytha
gore
an T
heor
em a
pplie
s to
all
tria
ngle
s.Fa
lse;
sam
ple
answ
er:I
ton
ly a
pplie
s to
righ
t tria
ngle
s.c.
If y
ou a
re g
iven
the
leng
th o
f on
e si
de o
f a
tria
ngle
and
the
mea
sure
s of
any
tw
oan
gles
,you
can
use
the
Law
of
Sine
s to
fin
d th
e le
ngth
s of
the
oth
er t
wo
side
s.tru
ed.
If y
ou k
now
the
mea
sure
s of
tw
o an
gles
of
a tr
iang
le,y
ou s
houl
d us
e th
e L
aw o
f Si
nes
to f
ind
the
mea
sure
of
the
thir
d an
gle.
Fals
e;sa
mpl
e an
swer
:You
sho
uld
use
the
Angl
e Su
m T
heor
em.
e.A
fri
end
tells
you
tha
t in
tri
angl
e R
ST
,m#
R"
132,
r"
24 c
enti
met
ers,
and
s"
31ce
ntim
eter
s.C
an y
ou u
se t
he L
aw o
f Si
nes
to s
olve
the
tri
angl
e? E
xpla
in.
No;
sam
ple
answ
er:I
n an
y tri
angl
e,th
e lo
nges
t sid
e is
opp
osite
the
larg
est
angl
e.Be
caus
e a
trian
gle
can
have
onl
y on
e ob
tuse
ang
le,"
Rm
ust b
eth
e la
rges
t ang
le,b
ut s
(r,
so it
is im
poss
ible
to h
ave
a tri
angl
e w
ithth
e gi
ven
mea
sure
s.
Hel
pin
g Y
ou
Rem
emb
er
3.M
any
stud
ents
rem
embe
r m
athe
mat
ical
equ
atio
ns a
nd f
orm
ulas
bet
ter
if t
hey
can
stat
eth
em in
wor
ds.S
tate
the
Law
of
Sine
s in
you
r ow
n w
ords
wit
hout
usi
ng v
aria
bles
or
mat
hem
atic
al s
ymbo
ls.
Sam
ple
answ
er:I
n an
y tri
angl
e,th
e ra
tio o
f the
sin
e of
an
angl
e to
the
leng
th o
f the
opp
osite
sid
e is
the
sam
e fo
r all
thre
e an
gles
.
P
MN
p
mn
©G
lenc
oe/M
cGra
w-Hi
ll38
6G
lenc
oe G
eom
etry
Iden
titie
sA
n id
enti
tyis
an
equa
tion
tha
t is
tru
e fo
r al
l val
ues
of t
he
vari
able
for
whi
ch b
oth
side
s ar
e de
fine
d.O
ne w
ay t
o ve
rify
an
iden
tity
is t
o us
e a
righ
t tr
iang
le a
nd t
he d
efin
itio
ns f
ortr
igon
omet
ric
func
tion
s.
Ver
ify
that
(si
n A
)2$
(cos
A)2
"1
is a
n i
den
tity
.
(sin
A)2
%(c
os A
)2"
&!a c! '2%
&!b c! '2
"!a2
% cb2
!"
!c c2 2!"
1
To c
heck
whe
ther
an
equa
tion
may
be a
n id
enti
ty,y
ou c
an t
est
seve
ral v
alue
s.H
owev
er,s
ince
you
can
not
test
all
valu
es,y
ouca
nnot
be
cert
ain
that
the
equ
atio
n is
an
iden
tity
.
Test
sin
2x
"2
sin
xco
s x
to s
ee i
f it
cou
ld b
e an
id
enti
ty.
Try
x"
20.U
se a
cal
cula
tor
to e
valu
ate
each
exp
ress
ion.
sin
2x"
sin
402
sin
xco
s x
"2
(sin
20)
(cos
20)
#0.
643
#2(
0.34
2)(0
.940
)#
0.64
3
Sinc
e th
e le
ft a
nd r
ight
sid
es s
eem
equ
al,t
he e
quat
ion
may
be
an id
enti
ty.
Use
tri
angl
e A
BC
show
n a
bove
.Ver
ify
that
eac
h e
quat
ion
is
an i
den
tity
.
1.!c so ins
AA !"
! tan1
A!
2.!t sa inn
BB!
"! co
1 sB!
!c so ins AA!
"!b c!
)!a c!
"!b a!
"! ta
n1A!
!t sa innBB !
"!b a!
)!b c!
"!c a!
"! co
1 sB
!
3.ta
n B
cos
B"
sin
B4.
1#
(cos
B)2
"(s
in B
)2
tan
B co
s B
"!b a!
*!a c!
"!b c!
"si
n B
1(co
s B)
2"
1'
(!a c! )2
"!c c2 2!
'!a c2 2!
"!c2
c' 2a2
!"
!b c22 !or
(sin
B)2
Try
sev
eral
val
ues
for
x t
o te
st w
het
her
eac
h e
quat
ion
cou
ld b
e an
id
enti
ty.
5.co
s 2x
"(c
os x
)2#
(sin
x)2
6.co
s (9
0#
x)"
sin
x
Yes;
see
stud
ents
’wor
k.Ye
s;se
e st
uden
ts’w
ork.
B
AC
ca
b
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-6
7-6
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Answers (Lesson 7-6)
© Glencoe/McGraw-Hill A20 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
The
Law
of C
osin
es
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-7
7-7
©G
lenc
oe/M
cGra
w-Hi
ll38
7G
lenc
oe G
eom
etry
Lesson 7-7
The
Law
of
Co
sin
esA
noth
er r
elat
ions
hip
betw
een
the
side
s an
d an
gles
of
any
tria
ngle
is c
alle
d th
e L
aw o
f C
osin
es.Y
ou c
an u
se t
he L
aw o
f C
osin
es if
you
kno
w t
hree
sid
es o
f a
tria
ngle
or
if y
ou k
now
tw
o si
des
and
the
incl
uded
ang
le o
f a
tria
ngle
.
Let !
ABC
be a
ny tr
iang
le w
ith a
, b, a
nd c
repr
esen
ting
the
mea
sure
s of
the
sides
opp
osite
La
w o
f Cos
ines
the
angl
es w
ith m
easu
res
A, B
, and
C, r
espe
ctive
ly. T
hen
the
follo
wing
equ
atio
ns a
re tr
ue.
a2"
b2%
c2#
2bc
cos
Ab2
"a2
%c2
#2a
cco
s B
c2"
a2%
b2#
2ab
cos
C
In !
AB
C,f
ind
c.
c2"
a2%
b2#
2ab
cos
CLa
w of
Cos
ines
c2"
122
%10
2#
2(12
)(10
)cos
48°
a"
12, b
"10
, m#
C"
48
c"
!12
2%
"10
2#
"2(
12)
"(1
0)co
"s
48°
"Ta
ke th
e sq
uare
root
of e
ach
side.
c#
9.1
Use
a ca
lcula
tor.
In !
AB
C,f
ind
m"
A.
a2"
b2%
c2#
2bc
cos
ALa
w of
Cos
ines
72"
52%
82#
2(5)
(8)
cos
Aa
"7,
b"
5, c
"8
49 "
25 %
64 #
80 c
os A
Mul
tiply.
#40
"#
80 c
os A
Subt
ract
89
from
eac
h sid
e.
!1 2!"
cos
ADi
vide
each
sid
e by
#80
.
cos#
1!1 2!
"A
Use
the
inve
rse
cosin
e.
60°
"A
Use
a ca
lcula
tor.
Fin
d e
ach
mea
sure
usi
ng
the
give
n m
easu
res
from
!A
BC
.Rou
nd
an
gle
mea
sure
sto
th
e n
eare
st d
egre
e an
d s
ide
mea
sure
s to
th
e n
eare
st t
enth
.
1.If
b"
14,c
"12
,and
m#
A"
62,f
ind
a.13
.5
2.If
a"
11,b
"10
,and
c"
12,f
ind
m#
B.
51
3.If
a"
24,b
"18
,and
c"
16,f
ind
m#
C.
42
4.If
a"
20,c
"25
,and
m#
B"
82,f
ind
b.29
.8
5.If
b"
18,c
"28
,and
m#
A"
59,f
ind
a.24
.3
6.If
a"
15,b
"19
,and
c"
15,f
ind
m#
C.
51
58
7C
B
A
48#
1210
c
C
BA
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll38
8G
lenc
oe G
eom
etry
Use
th
e La
w o
f C
osi
nes
to
So
lve
Pro
ble
ms
You
can
use
the
Law
of
Cos
ines
toso
lve
som
e pr
oble
ms
invo
lvin
g tr
iang
les.
Let !
ABC
be a
ny tr
iang
le w
ith a
, b, a
nd c
repr
esen
ting
the
mea
sure
s of
the
sides
opp
osite
the
Law
of C
osin
esan
gles
with
mea
sure
s A,
B, a
nd C
, res
pect
ively.
The
n th
e fo
llowi
ng e
quat
ions
are
true
.a2
"b2
%c2
#2b
cco
s A
b2"
a2%
c2#
2ac
cos
Bc2
"a2
%b2
#2a
bco
s C
Ms.
Jon
es w
ants
to
pu
rch
ase
a p
iece
of
lan
d w
ith
th
e sh
ape
show
n.F
ind
th
e p
erim
eter
of
the
pro
per
ty.
Use
the
Law
of
Cos
ines
to
find
the
val
ue o
f a.
a2"
b2%
c2#
2bc
cos
ALa
w of
Cos
ines
a2"
3002
%20
02#
2(30
0)(2
00)
cos
88°
b"
300,
c"
200,
m#
A"
88
a"
!13
0,0
"00
#"
120,
0"
00 c
os"
88°
"Ta
ke th
e sq
uare
root
of e
ach
side.
#35
4.7
Use
a ca
lcula
tor.
Use
the
Law
of
Cos
ines
aga
in t
o fi
nd t
he v
alue
of c
.
c2"
a2%
b2#
2ab
cos
CLa
w of
Cos
ines
c2"
354.
72%
3002
#2(
354.
7)(3
00)
cos
80°
a"
354.
7, b
"30
0, m
#C
"80
c"
!21
5,8
"12
.09
"#
21"
2,82
0"
cos
8"
0°"Ta
ke th
e sq
uare
root
of e
ach
side.
#42
2.9
Use
a ca
lcula
tor.
The
per
imet
er o
f th
e la
nd is
300
%20
0 %
422.
9 %
200
or a
bout
122
3 fe
et.
Dra
w a
fig
ure
or
dia
gram
to
go w
ith
eac
h e
xerc
ise
and
mar
k i
t w
ith
th
e gi
ven
info
rmat
ion
.Th
en s
olve
th
e p
robl
em.R
oun
d a
ngl
e m
easu
res
to t
he
nea
rest
deg
ree
and
sid
e m
easu
res
to t
he
nea
rest
ten
th.
1.A
tri
angu
lar
gard
en h
as d
imen
sion
s 54
fee
t,48
fee
t,an
d 62
fee
t.F
ind
the
angl
es a
t ea
chco
rner
of
the
gard
en.
75°;
48°;
57°
2.A
par
alle
logr
am h
as a
68°
angl
e an
d si
des
8 an
d 12
.Fin
d th
e le
ngth
s of
the
dia
gona
ls.
11.7
;16.
73.
An
airp
lane
is s
ight
ed f
rom
tw
o lo
cati
ons,
and
its
posi
tion
for
ms
an a
cute
tri
angl
e w
ith
them
.The
dis
tanc
e to
the
air
plan
e is
20
mile
s fr
om o
ne lo
cati
on w
ith
an a
ngle
of
elev
atio
n 48
.0°,
and
40 m
iles
from
the
oth
er lo
cati
on w
ith
an a
ngle
of
elev
atio
n of
21.
8°.
How
far
apa
rt a
re t
he t
wo
loca
tion
s?50
.5 m
i4.
A r
ange
r to
wer
at
poin
t A
is d
irec
tly
nort
h of
a r
ange
r to
wer
at
poin
t B
.A fi
re a
t po
int
Cis
obs
erve
d fr
om b
oth
tow
ers.
The
dis
tanc
e fr
om t
he f
ire
to t
ower
Ais
60
mile
s,an
d th
edi
stan
ce f
rom
the
fir
e to
tow
er B
is 5
0 m
iles.
If m
#A
CB
"62
,fin
d th
e di
stan
ce b
etw
een
the
tow
ers.
57.3
mi
200
ft
300
ft
300
ft
88#80
#c
a
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
The
Law
of C
osin
es
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-7
7-7
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 7-7)
© Glencoe/McGraw-Hill A21 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
The
Law
of C
osin
es
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-7
7-7
©G
lenc
oe/M
cGra
w-Hi
ll38
9G
lenc
oe G
eom
etry
Lesson 7-7
In !
RS
T,g
iven
th
e fo
llow
ing
mea
sure
s,fi
nd
th
e m
easu
re o
f th
e m
issi
ng
sid
e.
1.r
"5,
s"
8,m
#T
"39
t#5.
2
2.r
"6,
t"
11,m
#S
"87
s#
12.3
3.r
"9,
t"
15,m
#S
"10
3s
#19
.2
4.s
"12
,t"
10,m
#R
"58
r#10
.8
In !
HIJ
,giv
en t
he
len
gth
s of
th
e si
des
,fin
d t
he
mea
sure
of
the
stat
ed a
ngl
e to
th
en
eare
st t
enth
.
5.h
"12
,i"
18,j
"7;
m#
H24
.7
6.h
"15
,i"
16,j
"22
;m#
I46
.7
7.h
"23
,i"
27,j
"29
;m#
J70
.4
8.h
"37
,i"
21,j
"30
;m#
H91
.3
Det
erm
ine
wh
eth
er t
he
Law
of
Sin
esor
th
e L
aw o
f C
osin
essh
ould
be
use
d f
irst
to
solv
e ea
ch t
rian
gle.
Th
en s
olve
eac
h t
rian
gle.
Rou
nd
an
gle
mea
sure
s to
th
e n
eare
std
egre
e an
d s
ide
mea
sure
s to
th
e n
eare
st t
enth
.
9.10
.
Cosi
nes;
m"
A #
34;
Sine
s;m
"L
#67
;m
"B
#80
;c#
30.7
m"
N#
27;!
#47
.8
11.a
"10
,b"
14,c
"19
12.a
"12
,b"
10,m
#C
"27
Cosi
nes;
m"
A#
31;
Cosi
nes;
m"
A#
97;
m"
B#
46;m
"C
#10
3m
"B
#56
;c#
5.5
Sol
ve e
ach
!R
ST
des
crib
ed b
elow
.Rou
nd
mea
sure
s to
th
e n
eare
st t
enth
.
13.r
"12
,s"
32,t
"34
m"
R#
20.7
,m"
S#
70.2
,m"
T#
89.1
14.r
"30
,s"
25,m
#T
"42
m"
R#
82.2
,m"
S#
55.7
,t#
20.3
15.r
"15
,s"
11,m
#R
"67
m"
S#
42.5
,m"
T#
70.5
,t#
15.4
16.r
"21
,s"
28,t
"30
m"
R#
42.3
,m"
S#
63.8
,m"
T#
74.0
M
LN
!86
#
52
24
B
AC
c
66#
33
19
©G
lenc
oe/M
cGra
w-Hi
ll39
0G
lenc
oe G
eom
etry
In !
JK
L,g
iven
th
e fo
llow
ing
mea
sure
s,fi
nd
th
e m
easu
re o
f th
e m
issi
ng
sid
e.
1.j"
1.3,
k"
10,m
#L
"77
!#
9.8
2.j"
9.6,
!"
1.7,
m#
K"
43k
#8.
43.
j"11
,k"
7,m
#L
"63
!#
10.0
4.k
"4.
7,!
"5.
2,m
#J
"11
2j#
8.2
In !
MN
Q,g
iven
th
e le
ngt
hs
of t
he
sid
es,f
ind
th
e m
easu
re o
f th
e st
ated
an
gle
toth
e n
eare
st t
enth
.
5.m
"17
,n"
23,q
"25
;m#
Q75
.76.
m"
24,n
"28
,q"
34;m
#M
44.2
7.m
"12
.9,n
"18
,q"
20.5
;m#
N60
.28.
m"
23,n
"30
.1,q
"42
;m#
Q10
3.7
Det
erm
ine
wh
eth
er t
he
Law
of
Sin
es o
r th
e L
aw o
f C
osin
es s
hou
ld b
e u
sed
fir
st t
oso
lve
!A
BC
.Th
en s
ole
each
tri
angl
e.R
oun
d a
ngl
e m
easu
res
to t
he
nea
rest
deg
ree
and
sid
e m
easu
re t
o th
e n
eare
st t
enth
.
9.a
"13
,b"
18,c
"19
10.a
"6,
b"
19,m
#C
"38
Cosi
nes;
m"
A#
41;
Cosi
nes;
m"
A#
15;
m"
B#
65;m
"C
#74
m"
B#
127;
c#
14.7
11.a
"17
,b"
22,m
#B
"49
12.a
"15
.5,b
"18
,m#
C"
72
Sine
s;m
"A
#36
;Co
sine
s;m
"A
#48
;m
"C
#95
;c#
29.0
m"
B#
60;c
#19
.8
Sol
ve e
ach
!F
GH
des
crib
ed b
elow
.Rou
nd
mea
sure
s to
th
e n
eare
st t
enth
.
13.m
#F
"54
,f"
12.5
,g"
11m
"G
#45
.4,m
"H
#80
.6,h
#15
.214
.f"
20,g
"23
,m#
H"
47m
"F
#57
.4,m
"G
#75
.6,h
#17
.415
.f"
15.8
,g"
11,h
"14
m"
F#
77.4
,m"
G#
42.8
,m"
H#
59.8
16.f
"36
,h"
30,m
#G
"54
m"
F#
73.1
,m"
H#
52.9
,g#
30.4
17.R
EAL
ESTA
TET
he E
spos
ito
fam
ily p
urch
ased
a t
rian
gula
r pl
ot o
f la
nd o
n w
hich
the
ypl
an t
o bu
ild a
bar
n an
d co
rral
.The
leng
ths
of t
he s
ides
of
the
plot
are
320
fee
t,28
6 fe
et,
and
305
feet
.Wha
t ar
e th
e m
easu
res
of t
he a
ngle
s fo
rmed
on
each
sid
e of
the
pro
pert
y?
65.5
,54.
4,60
.1
Pra
ctic
e (A
vera
ge)
The
Law
of C
osin
es
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-7
7-7
Answers (Lesson 7-7)
© Glencoe/McGraw-Hill A22 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csTh
e La
w o
f Cos
ines
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-7
7-7
©G
lenc
oe/M
cGra
w-Hi
ll39
1G
lenc
oe G
eom
etry
Lesson 7-7
Pre-
Act
ivit
yH
ow a
re t
rian
gles
use
d i
n b
uil
din
g d
esig
n?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 7-
7 at
the
top
of
page
385
in y
our
text
book
.
Wha
t co
uld
be a
dis
adva
ntag
e of
a t
rian
gula
r ro
om?
Sam
ple
answ
er:
Furn
iture
will
not
fit i
n th
e co
rner
s.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.A
ccor
ding
to
the
Law
of
Cos
ines
,whi
ch
stat
emen
ts a
re c
orre
ct f
or !
DE
F?
B,E,
HA
.d2
"e2
%f2
#ef
cos
DB
.e2
"d
2%
f2#
2df
cos
E
C.d
2"
e2%
f2%
2ef
cos
DD
.f2
"d2
%e2
#2e
fco
s F
E.f
2"
d2%
e2#
2de
cos
FF.
d2
"e2
%f2
G.!
sin d
D !"
!sin e
E !"
!sin f
F !H
.d"
!e2
%f
"2
#2e
"f
cos
"D"
2.E
ach
of t
he f
ollo
win
g de
scri
bes
thre
e gi
ven
part
s of
a t
rian
gle.
In e
ach
case
,ind
icat
ew
heth
er y
ou w
ould
use
the
Law
of
Sine
s or
the
Law
of
Cos
ines
fir
st in
sol
ving
a t
rian
gle
wit
h th
ose
give
n pa
rts.
(In
som
e ca
ses,
only
one
of
the
two
law
s w
ould
be
used
in s
olvi
ngth
e tr
iang
le.)
a.SS
S La
w o
f Cos
ines
b.A
SA L
aw o
f Sin
esc.
AA
S La
w o
f Sin
esd.
SAS
Law
of C
osin
ese.
SSA
Law
of S
ines
3.In
dica
te w
heth
er e
ach
stat
emen
t is
tru
eor
fal
se.I
f th
e st
atem
ent
is f
alse
,exp
lain
why
.
a.T
he L
aw o
f C
osin
es a
pplie
s to
rig
ht t
rian
gles
. tru
eb.
The
Pyt
hago
rean
The
orem
app
lies
to a
cute
tri
angl
es.F
alse
;sam
ple
answ
er:
It on
ly a
pplie
s to
righ
t tria
ngle
s.c.
The
Law
of
Cos
ines
is u
sed
to f
ind
the
thir
d si
de o
f a
tria
ngle
whe
n yo
u ar
e gi
ven
the
mea
sure
s of
tw
o si
des
and
the
noni
nclu
ded
angl
e.Fa
lse;
sam
ple
answ
er:I
t is
used
whe
n yo
u ar
e gi
ven
the
mea
sure
s of
two
side
s an
d th
e in
clud
edan
gle.
d.T
he L
aw o
f C
osin
es c
an b
e us
ed t
o so
lve
a tr
iang
le in
whi
ch t
he m
easu
res
of t
he t
hree
side
s ar
e 5
cent
imet
ers,
8 ce
ntim
eter
s,an
d 15
cen
tim
eter
s.Fa
lse;
sam
ple
answ
er:5
$8
+15
,so,
by th
e Tr
iang
le In
equa
lity
Theo
rem
,no
such
trian
gle
exis
ts.
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al f
orm
ula
is t
o re
late
it t
o on
e yo
u al
read
ykn
ow.T
he L
aw o
f C
osin
es lo
oks
som
ewha
t lik
e th
e P
ytha
gore
an T
heor
em.B
oth
form
ulas
mus
t be
tru
e fo
r a
righ
t tr
iang
le.H
ow c
an t
hat
be?
cos
90 "
0,so
in a
righ
ttri
angl
e,w
here
the
incl
uded
ang
le is
the
right
ang
le,t
he L
aw o
f Cos
ines
beco
mes
the
Pyth
agor
ean
Theo
rem
.
D
dE
e F
f
©G
lenc
oe/M
cGra
w-Hi
ll39
2G
lenc
oe G
eom
etry
Sphe
rical
Tria
ngle
sSp
heri
cal t
rigo
nom
etry
is a
n ex
tens
ion
of p
lane
tri
gono
met
ry.
Fig
ures
are
dra
wn
on t
he s
urfa
ce o
f a
sphe
re.
Arc
s of
gre
at
circ
les
corr
espo
nd t
o lin
e se
gmen
ts in
the
pla
ne.
The
arc
s of
th
ree
grea
t ci
rcle
s in
ters
ecti
ng o
n a
sphe
re f
orm
a s
pher
ical
tr
iang
le.
Ang
les
have
the
sam
e m
easu
re a
s th
e ta
ngen
t lin
es
draw
n to
eac
h gr
eat
circ
le a
t th
e ve
rtex
.Si
nce
the
side
s ar
e ar
cs,t
hey
too
can
be m
easu
red
in d
egre
es.
Sol
ve t
he
sph
eric
al t
rian
gle
give
n a
"72
#,b
"10
5#,a
nd
c"
61#.
Use
the
Law
of
Cos
ines
.
0.30
90"
(–0.
2588
)(0.
4848
)%(0
.965
9)(0
.874
6) c
os A
cos
A"
0.51
43A
"59
°
#0.
2588
" (0
.309
0)(0
.484
8)%
(0.9
511)
(0.8
746)
cos
Bco
s B
"#
0.49
12B
"11
9°
0.48
48"
(0.3
090)
(–0.
2588
)%(0
.951
1)(0
.965
9) c
os C
cos
C"
0.61
48C
"52
°
Che
ck b
y us
ing
the
Law
of
Sine
s.
!s si in n7 52 9° °!
"!s si in n
1 10 15 9° °!
"!s si in n
6 51 2° °!
"1.
1
Sol
ve e
ach
sp
her
ical
tri
angl
e.
1.a
"56
°,b
"53
°,c
"94
°2.
a"
110°
,b"
33°,
c"
97°
A"
41#,
B"
39#,
C"
128#
A"
116#
,B"
31#,
C"
71#
3.a
"76
°,b
"11
0°,C
"49
°4.
b"
94°,
c"
55°,
A"
48°
A"
59#,
B"
124#
,c"
59#
a"
60#,
B"
121#
,C"
45#
A
C
B
c
ba
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
7-7
7-7
The
sum
of t
he s
ides
of a
sph
erica
l tria
ngle
is le
ss th
an 3
60°.
The
sum
of t
he a
ngle
s is
grea
ter t
han
180°
and
less
than
540
°.Th
e La
w of
Sin
es fo
r sph
erica
l tria
ngle
s is
as fo
llows
.
! ss ii nnAa !
"! ss ii nn
Bb !"
! ss ii nn Cc!
Ther
e is
also
a L
aw o
f Cos
ines
for s
pher
ical t
riang
les.
cos
a"
cos
bco
s c
%sin
bsin
cco
s A
cos
b"
cos
aco
s c
%sin
asin
cco
s B
cos
c"
cos
aco
s b
%sin
asin
bco
s C
Exam
ple
Exam
ple
Answers (Lesson 7-7)
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