Foundations for Geometry
RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCE O
Foundations for Geometry
Section 1A Section 1B Euclidean and Construction Tools Coordinate and Transformation Tools
1-1 Understanding Points, Lines, and Planes 1-5 Using Formulas in Geometry 1-2 Technology Lab Explore Properties Associated with Points Connecting Geometry to Algebra Graphing in the Coordinate 1-2 Measuring and Constructing Segments Plane 1-3 Measuring and Constructing Angles 1-6 Midpoint and Distance in the Coordinate Plane 1-4 Pairs of Angles 1-7 Transformations in the Coordinate Plane 1-7 Technology Lab Explore Transformations
Pacing Guide for 45-Minute Classes
Chapter 1 Countdown to Testing Weeks 1 , 2
DAY 1 DAY 2 DAY 3 DAY 4 DAY 5 1-1 Lesson 1-2 Technology Lab 1-3 Lesson 1-4 Lesson Multi-Step Test Prep 1-2 Lesson Ready to Go On?
DAY 6 DAY 7 DAY 8 DAY 9 DAY 10 1-5 Lesson Connecting Geometry 1-7 Lesson 1-7 Technology Lab Multi-Step Test Prep to Algebra Ready to Go On? 1-6 Lesson
DAY 11 DAY 12 Chapter 1 Review Chapter 1 Test
Pacing Guide for 90-Minute Classes
Chapter 1
DAY 1DAY 2DAY 3DAY 4DAY 5 1-1 Lesson 1-3 Lesson Multi-Step Test Prep Connecting Geometry 1-7 Technology Lab 1-2 Technology Lab 1-4 Lesson Ready to Go On? to Algebra Multi-Step Test Prep 1-2 Lesson 1-5 Lesson 1-6 Lesson Ready to Go On? 1-7 Lesson
DAY 6 Chapter 1 Review Chapter 1 Test
2A Chapter 1
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DIAGNOSE PRESCRIBE
Assess Before Chapter 1 Prior Diagnose readiness for the chapter. Prescribe intervention. Knowledge Are You Ready? SE p. 3 Are You Ready? Intervention Skills 20, 57, 60, 79
Before Every Lesson Diagnose readiness for the lesson. Prescribe intervention. Warm Up TE, every lesson Skills Bank SE pp. S50–S81 Reteach CRB, Ch. 1
During Every Lesson Diagnose understanding of lesson concepts. Prescribe intervention. Check It Out! SE, every example Questioning Strategies TE, every example Think and Discuss SE, every lesson Reading Strategies CRB, every lesson Write About It SE, every lesson Success for ELL pp. 1–14 Journal TE, every lesson
After Every Lesson Diagnose mastery of lesson concepts. Prescribe intervention. Lesson Quiz TE, every lesson Reteach CRB, every lesson Formative Alternative Assessment TE, every lesson Problem Solving CRB, every lesson Assessment Test Prep SE, every lesson Test Prep Doctor TE, every lesson
Test and Practice Generator Homework Help Online
Before Chapter 1 Testing Diagnose mastery of concepts in the chapter. Prescribe intervention.
Ready to Go On? SE pp. 35, 59 Ready to Go On? Intervention pp. 2–14 Multi-Step Test Prep SE pp. 34, 58 Scaffolding Questions TE pp. 34, 58 Section Quizzes AR pp. 5–6
Test and Practice Generator
Before High Stakes Testing Diagnose mastery of benchmark concepts. Prescribe intervention. College Entrance Exam Practice SE p. 65 College Entrance Exam Practice New York Test Prep SE pp. 68–69 New York State Test Prep Workbook
After Chapter 1 Check mastery of chapter concepts. Prescribe intervention. Multiple-Choice Tests (Forms A, B, C) Reteach CRB, every lesson
Summative Free-Response Tests (Forms A, B, C) Lesson Tutorial Videos Chapter 1 Assessment Performance Assessment AR pp. 7–20 Test and Practice Generator Check mastery of benchmark concepts. Prescribe intervention. New York State Regents Exam New York State Test Prep Workbook College Entrance Exams College Entrance Exam Practice
KEY: SE = Student Edition TE = Teacher’s Edition CRB = Chapter Resource Book AR = Assessment Resources Available online Available on CD-ROM 2B
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Supporting the Teacher
Chapter 1 Resource Book Teacher Tools Practice A, B, C Power Presentations® pp. 3–5, 11–13, 19–21, 27–29, 35–37, Complete PowerPoint® presentations for Chapter 1 lessons 43–45, 51–53 ® Lesson Tutorial Videos Reading Strategies %,, Holt authors Ed Burger and Freddie Renfro present tutorials pp. 10, 18, 26, 34, 42, 50, 58 to support the Chapter 1 lessons. Reteach One-Stop Planner® pp. 6–7, 14–15, 22–23, 30–31, 38–39, 46–47, 54–55 Easy access to all Chapter 1 resources and assessments, as well as software for lesson planning, test generation, Problem Solving and puzzle creation pp. 9, 17, 25, 33, 41, 49, 57 IDEA Works!® Challenge Key Chapter 1 resources and assessments modifi ed to address pp. 8, 16, 24, 32, 40, 48, 56 special learning needs Parent Letter pp. 1–2 Lesson Plans ...... pp. 1–7 Solutions Key ...... Chapter 1 Geometry Posters Transparencies TechKeys Lab Resources Lesson Transparencies, Volume 1 ...... Chapter 1 Project Teacher Support Parent Resources • Teaching Tools • Warm Ups • Teaching Transparencies Workbooks • Additional Examples • Lesson Quizzes New York Homework and Practice Workbook Teacher’s Guide ...... pp. 1–7 Alternate Openers: Explorations ...... pp. 1–7 Know-It Notebook Know-It Notebook ...... Chapter 1 • Graphic Organizers Teacher’s Guide ...... Chapter 1 New York Problem Solving Workbook Teacher’s Guide ...... pp. 1–7 Nw York State Test Prep Workbook Teacher’s Guide
Technology Highlights for the Teacher
Power Presentations One-Stop Planner Premier Online Edition Dynamic presentations to engage students. Easy access to Chapter 1 resources and Chapter 1 includes Tutorial Videos, Complete PowerPoint® presentations for assessments. Includes lesson-planning, test- Lesson Activities, Lesson Quizzes, every lesson in Chapter 1. generation, and puzzle-creation software. Homework Help, and Chapter Project.
KEY: SE = Student Edition TE = Teacher’s Edition %,, English Language Learners JG8E@J? Spanish version available Available online Available on CD-ROM
2C Chapter 1
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Reaching All Learners
ENGLISH LANGUAGE Resources for All Learners English Language Learners LEARNERS Geometry Lab Activities ...... Chapter 1 Are You Ready? Vocabulary ...... SE p. 3 Technology Lab Activities ...... Chapter 1 Vocabulary Connections ...... SE p. 4 New York Homework and Practice Workbook ...... pp. 1–7 Lesson Vocabulary ...... SE, every lesson Know-It Notebook ...... Chapter 1 Vocabulary Exercises ...... SE, every exercise set New York Problem Solving Workbook ...... pp. 1–7 Vocabulary Review ...... SE p. 60 English Language Learners ...... TE pp. 5, 7, 14, 22, 23, 39 DEVELOPING LEARNERS Reading Strategies ...... CRB, every lesson Practice A ...... CRB, every lesson Success for English Language Learners ...... pp. 1–14 Reteach ...... CRB, every lesson Multilingual Glossary Inclusion ...... TE pp. 18, 25, 30, 36, 42, 45, 51 ...... TE, every example Questioning Strategies Reaching All Learners Through... Modifi ed Chapter 1 Resources ...... IDEA Works! Inclusion ...... TE pp. 18, 25, 30, 36, 42, 45, 51 Homework Help Online Visual Cues ...... TE pp. 21, 22, 27, 30 ON-LEVEL LEARNERS Kinesthetic Experience ...... TE pp. 18, 26 Practice B ...... CRB, every lesson Concrete Manipulatives ...... TE pp. 14, 51 Multiple Representations ...... TE p. 18 Multiple Representations ...... TE p. 18 Cognitive Strategies ...... TE pp. 29, 44 Cognitive Strategies ...... TE pp. 29, 44 ADVANCED LEARNERS Cooperative Learning ...... TE p. 29 Practice C ...... CRB, every lesson Modeling ...... TE p. 7 Challenge ...... CRB, every lesson Critical Thinking ...... TE p. 14 Test Prep Doctor ...... TE pp. 11, 26, 33, 40 Reading and Writing Math EXTENSION ...... TE p. 5 49, 55, 65, 66, 68 Multi-Step Test Prep EXTENSION ...... TE pp. 34, 58 Common Error Alerts ...... TE pp. 15, 19, 21, 23, Critical Thinking ...... TE p. 14 25, 29, 39, 45, 49 Scaffolding Questions ...... TE pp. 34, 58
Technology Highlights for Reaching All Learners
Lesson Tutorial Videos Multilingual Glossary Online Interactivities Starring Holt authors Ed Burger and Freddie Searchable glossary includes defi nitions Interactive tutorials provide visually engaging Renfro! Live tutorials to support every in English, Spanish, Vietnamese, Chinese, alternative opportunities to learn concepts and lesson in Chapter 1. Hmong, Korean, and 4 other languages. master skills.
KEY: SE = Student Edition TE = Teacher’s Edition CRB = Chapter Resource Book JG8E@J? Spanish version available Available online Available on CD-ROM
2D
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Ongoing Assessment
Assessing Prior Knowledge Daily Assessment Determine whether students have the prerequisite concepts Provide formative assessment for each day of Chapter 1. and skills for success in Chapter 1. Questioning Strategies ...... TE, every example Are You Ready? JG8E@J? ...... SE p. 3 Think and Discuss ...... SE, every lesson Warm Up ...... TE, every lesson Check It Out! Exercises ...... SE, every example Write About It ...... SE, every lesson Test Preparation Journal ...... TE, every lesson Provide review and practice for Chapter 1 and standardized Lesson Quiz ...... TE, every lesson tests. Alternative Assessment ...... TE, every lesson Multi-Step Test Prep ...... SE pp. 34, 58 Modifi ed Lesson Quizzes ...... IDEA Works! Study Guide: Review ...... SE pp. 60–63 Test Tackler ...... SE pp. 66–67 Weekly Assessment New York Test Prep ...... SE pp. 68–69 Provide formative assessment for each week of Chapter 1. College Entrance Exam Practice ...... SE p. 65 Multi-Step Test Prep ...... SE pp. 34, 58 New York State Test Prep Workbook Ready to Go On? ...... SE pp. 35, 59 IDEA Works! Cumulative Assessment ...... SE pp. 68–69 Test and Practice Generator ...... One-Stop Planner Alternative Assessment Assess students’ understanding of Chapter 1 concepts and combined problem-solving skills. Formal Assessment Chapter 1 Project ...... SE p. 2 Provide summative assessment of Chapter 1 mastery. Alternative Assessment ...... TE, every lesson Section Quizzes ...... AR pp. 5–6 Performance Assessment ...... AR pp. 19–20 Chapter 1 Test ...... SE p. 64 Portfolio Assessment ...... AR p. xxxiv Chapter Test (Levels A, B, C) ...... AR pp. 7–18 • Multiple Choice • Free Response Cumulative Test ...... AR pp. 21–24 Test and Practice Generator ...... One-Stop Planner Modifi ed Chapter 1 Test ...... IDEA Works!
Technology Highlights for Ongoing Assessment
Are You Ready? JG8E@J? Ready to Go On? Test and Practice Generator Automatically assess readiness and Automatically assess understanding Use Chapter 1 problem banks to create prescribe intervention for Chapter 1 of and prescribe intervention for assessments and worksheets to print out or prerequisite skills. Sections 1A and 1B. deliver online. Includes dynamic problems.
KEY: SE = Student Edition TE = Teacher’s Edition AR = Assessment Resources JG8E@J? Spanish version available Available online Available on CD-ROM
2E Chapter 1
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Formal Assessment
Three levels (A, B, C) of multiple-choice and free-response chapter tests are available in the Assessment Resources.
A Chapter 1 Test A Chapter 1 Test
C Chapter 1 Test C Chapter 1 Test
MULTIPLE CHOICE FREE RESPONSE MODIFIED FOR IDEA
B Chapter 1 Test B Chapter 1 Test Chapter 1 Test
#IRCLE THE CORRECT ANSWER 'IVEN ,- -0 AND , - AND 0 ARE 5SE THE DIAGRAM FOR %XERCISES n : IS IN THE INTERIOR OF 789 Circle the best answer. 7. Which term describes ЄLMP? COLLINEAR WHICH OF THE FOLLOWING BEST M78: AND M789 5SE THE FIGURE FOR %XERCISES n DESCRIBES THE RELATIONSHIP OF , - M &IND M:89 Use the figure for Exercises 1–4. AND 0 ? ? # 0 M & ,- -0 ? $ ! " # ' - IS THE MIDPOINT OF ,0 1 ? N " $ ! ( - BISECTS ,0 ! # % .AME A PAIR OF ADJACENT ANGLES THAT DO N N , - . " * !LL OF THE ABOVE ./4 FORM A LINEAR PAIR $ % A acute 5SE THE FIGURE FOR %XERCISES AND B obtuse .AME A POINT ON LINE N P 1. What is another name for plane ? 8. What is mЄWXY? 7HAT IS ANOTHER NAME FOR PLANE A plane B C plane ABC ! OR $ ! PLANE !% # PLANE "!$ B plane n 0 1 " PLANE ! $ PLANE "!# .AME A SEGMENT ON LINE M 7 AND AND : , - . ? ? 2. Which segment is on line n? OR AND _ 7HICH SEGMENT IS ON LINE N !% OR %! A DA ? ? _ & !$ ( !# 7HICH TERM DESCRIBES 0-1 ! AND " ARE COMPLEMENTARY ? ? .AME A RAY WITH ENDPOINT # B BC ' "# * "% ! OBTUSE # RIGHT M! X &IND M" 8 9 ??k ??k " STRAIGHT $ ACUTE 3. What is the name of a ray with A 20 C 100 #! OR #" 7HICH IS THE NAME OF A RAY WITH endpoint__ B? __ B 60 7HAT IS M0-. X ‹ › › ENDPOINT ?? ! ?? .AME THREE COLLINEAR POINTS A BC C BA k k _ ! $! # #! & ( ! AND " ARE SUPPLEMENTARY ??k ??k B DB 9. Which angles are adjacent? " "# $ !" ' * M" &IND M! ! " AND # _ 4. Name the intersection of line n and AC . 7HICH ANGLES ARE ADJACENT AND FORM A & IS BETWEEN % AND ' %& X AND .AME THE INTERSECTION OF PLANE AND A point B LINE M LINEAR PAIR %' X &IND &' &IND THE AREA OF A SQUARE WITH S CM B point C & LINE N ( !# ? ' POINT ! * !% X 5. F is between E and G, EF 7, and ? ? CM Є Є ? ,- z 10 AND ,- &IND 10 FG 6. What is the length of EG? A 1 and 2 7HAT IS THE MEASURE OF 24 &IND THE AREA OF A TRIANGLE THAT HAS A A 7 C 26 B Є1 and Є3 2 34 X BASE OF INCHES AND A HEIGHT OF B 13 X ! AND # AND INCHES Є _ _ 10. m A 47 . " AND $ AND M,-0 #LASSIFY ,-0 AS ACUTE 6. LM ഡ QP and LM 12. Find QP. What is the measure of a complement ! # Є RIGHT OR OBTUSE IN of A ? " $ A 6 )F M! X WHAT IS THE MEASURE A 43 OF THE COMPLEMENT OF ! B 12 ACUTE B 133 & ( X ' X * X
B Chapter 1 Test B Chapter 1 Test G1_IDEA_CT01.inddChapter Sec1:1 1 Test 4/10/06 10:14:12 PM (continued) (continued) (continued) )F M" X WHAT IS THE 7HAT IS THE DISTANCE FROM 6 TO 7 &IND THE CIRCUMFERENCE OF A CIRCLE WITH A 5SE THE 0YTHAGOREAN 4HEOREM TO FIND 67 11. If mЄB x , what is the measure of a 17. Use the Distance Formula to find the MEASURE OF THE SUPPLEMENT OF " 6 DIAMETER OF FEET 5SE FOR P supplement of ЄB? length of VW. 6 ! # X A 90 C (180 x) 2 2 d ͙(x2 x1) (y2 y1) " X $ X B (90 x) CM FT Y 6 12. What is the perimeter of a square with 7HAT IS THE PERIMETER OF A SQUARE WHOSE 7 &IND THE COORDINATES OF THE MIDPOINT ? s 8 centimeters? Use P 4s. X SIDE IS CM 7 CM OF '( WITH ENDPOINTS ' AND UNITS A 32 cm 7 & CM ( CM ( & CM ( CM B 64 cm ' CM * CM ' CM * CM )DENTIFY THE TRANSFORMATION 7HAT IS THE AREA OF A TRIANGLE WITH ? / 13. What is the area of a triangle with a base A 5 C 25 7HAT TRANSFORMATION IS SHOWN A HEIGHT OF INCHES AND A BASE OF - IS THE MIDPOINT OF 23 AND of 6 inches and a height of 3 inches? B 9 INCHES - HAS COORDINATES Use A __1bh. 2 2 ! IN # IN A 9 in 18. Use the Pythagorean Theorem, 2 HAS COORDINATES 2 2 2 2 " IN $ IN B 18 in a b c , to find the missing length. &IND THE COORDINATES OF 3 . 0
! CIRCLE HAS A DIAMETER OF FEET 7HAT IS . 0 14. What is the approximate area of a circle ! ROTATION # TRANSLATION ITS APPROXIMATE AREA with a radius of 4 feet? Use 3.14 for . " REFLECTION $ IMAGE CM & FT ( FT A 12.56 ft2 C 50.24 ft2 ' FT * FT 5SE THE $ISTANCE &ORMULA TO FIND !" B 25.12 ft2 'IVEN A POINT IN THE COORDINATE PLANE CM ? THE RULE X Y → X Y Y 'IVEN '( WITH ENDPOINTS ' AND / TRANSLATES THE POINT IN WHICH DIRECTION " 15. What_ are the coordinates of the midpoint A 10 C 48 ( WHAT ARE THE COORDINATES OF ? of GH with endpoints G( 2, 5) and & UNITS TO THE LEFT AND UNITS UP B 14 THE MIDPOINT OF '( REFLECTION H(4, 1)? ' UNITS TO THE LEFT AND UNITS DOWN ! # A ( 6, 4) C ( 3, 2) ( UNITS RIGHT AND UNITS UP 19. What transformation is shown? " $ ! TRIANGLE HAS VERTICES AT ! " B (1, 3) * UNITS TO THE RIGHT AND UNITS DOWN X AND # !FTER A TRANSFORMATION THE 0 0 ? IMAGE OF THE TRIANGLE HAS VERTICES AT _ - IS THE MIDPOINT OF 23 2 HAS ! " AND # 16. M is the midpoint of RS . R has COORDINATES AND - HAS )DENTIFY THE TRANSFORMATION coordinates (2, 5), M has coordinates COORDINATES 7HAT ARE THE (6, 9). Find the coordinates of S. COORDINATES OF 3 ! / . / . A (4.5, 6.5) C (4, 4) & ( TRANSLATION B (10, 13) A translation (slide) ' * B reflection (flip) ^ q OR 7HAT IS THE DISTANCE FROM - 20. Which rule would you use to translate TO . (slide) a figure in the coordinate plane 2 ! UNITS # UNITS units to the right? ^ " q UNITS $ UNITS A (x, y) A (x 2, y) B (x, y) A (x, y 2)
G1_IDEA_CT01.indd Sec1:2 4/10/06 10:14:13 PM
Create and customize Chapter 1 Tests. Instantly generate multiple test versions, answer keys, and practice versions of test items.
2F
gge08te_ny_Che08te_ny_Ch 0011 Interleaf.inddInterleaf.indd 22FF 112/21/062/21/06 112:17:342:17:34 PPMM Foundations for Geometry
SECTION 1A 1A Euclidean and Euclidean and Construction Tools Construction Tools 1-1 Understanding Points, Lines, and Planes On page 34, stu- Lab Explore Properties Associated dents analyze a with Points diagram of an 1-2 Measuring and Constructing archaeological dig by Segments applying definitions, using the dis- 1-3 Measuring and Constructing tance formula, and classifying angles. Angles Exercises designed to prepare stu- 1-4 Pairs of Angles dents for success on the Multi-Step Test Prep can be found on pages 10, 18, 26, and 32. 1B Coordinate and Transformation Tools SECTION 1B 1-5 Using Formulas in Geometry Coordinate and 1-6 Midpoint and Distance in the Transformation Tools Coordinate Plane 1-7 Transformations in the On page 58, stu- Coordinate Plane dents find the area Lab Explore Transformations and perimeter of a patio to determine the total cost of the paving stones used. They use distance, midpoint, and transformations to create the construction plans for the patio. Exercises designed to prepare stu- dents for success on the Multi-Step Test Prep can be found on pages 39, 48, and 54. Picture This! Many geometric concepts and shapes may be used in creating works of art. Unique designs can be made using only points, lines, planes, or circles.
KEYWORD: MG7 ChProj
2 Chapter 1
Picture This
ge07se_c01_0002_0005.inddAbout the 1 Project Project Resources 11/23/06 1:22:27 PM ge07se_c01_0002_0005.indd 3 4/27/06 11:56:23 AM Students begin by using geoboards to All project resources for teachers and stu- explore designs based on line segments. dents are provided online. Then they use a compass and paper folding Materials: to make star designs. Finally, students use • Activity 1: geoboard, colored rubber everything they’ve learned to produce an bands or colored string original piece of string art. • Activity 2: compass, straightedge • Activity 3: straightedge • Activity 4: wooden board, small nails, hammer, colored string or thread
KEYWORD: MG7 ProjectTS 2 Chapter 1
gge08te_ny_c01_0002_0005.indde08te_ny_c01_0002_0005.indd 2 112/21/062/21/06 11:09:50:09:50 PPMM Vocabulary Match each term on the left with a definition on the right. 1. coordinate C A. a mathematical phrase that contains operations, numbers, and/or variables Organizer 2. metric system of measurement E B. the measurement system often used in the United States Objective: Assess students’ 3. expression A C. one of the numbers of an ordered pair that locates a point on a coordinate graph understanding of prerequisite skills. 4. order of operations D D. a list of rules for evaluating expressions E. a decimal system of weights and measures that is used universally in science and commonly throughout the world Prerequisite Skills Measure with Customary and Measure with Customary and Metric Units Metric Units For each object tell which is the better measurement. Combine Like Terms 5. length of an unsharpened pencil 6. the diameter of a quarter 1 3 1 Evaluate Expressions 7__ in. or 9__ in. 7_1 in. 1 m or 2 __ cm 2 _1 cm 2 4 2 2 2 7. length of a soccer field 8. height of a classroom Ordered Pairs 100 yd or 40 yd 100 yd 5 ft or 10 ft 10 ft 9. height of a student’s desk 10. length of a dollar bill 30 in. or 4 ft 30 in. 15.6 cm or 35.5 cm 15.6 cm Assessing Prior Knowledge Combine Like Terms INTERVENTION Simplify each expression. Diagnose and Prescribe 11. -y + 3y - 6y + 12y 8y 12. 63 + 2x - 7 - 4x -2x + 56 Use this page to determine - - - + - - - + + - + 13. 5 9 7x 6x x 14 14. 24 3y y 7 2y 31 whether intervention is necessary or whether enrichment is Evaluate Expressions appropriate. Evaluate each expression for the given value of the variable. 15. x + 3x + 7x for x =-5 -55 16. 5p + 10 for p = 78 400 Resources 17. 2a - 8a for a = 12 -72 18. 3n - 3 for n = 16 45 Are You Ready? Intervention and Enrichment Worksheets Ordered Pairs Þ n Write the ordered pair for each point. Are You Ready? CD-ROM { 19. A (0, 7) 20. B (-5, 4) Ý Are You Ready? Online - - 21. C (6, 3) 22. D ( 8, 2) ä { { n ( - )(- ) { 23. E 3, 5 24. F 6, 4 n
Foundations for Geometry 3
NO YES ge07se_c01_0002_0005.indd 2 12/2/05 10:19:46 AM ge07se_c01_0002_0005.indd 3 INTERVENE Diagnose and Prescribe 4/27/06 11:56:23 AMENRICH
ARE YOU READY? Intervention, Chapter 1 Prerequisite Skill Worksheets CD-ROM Online ARE YOU READY? Enrichment, Chapter 1 Measure with Customary and Metric Units Skill 20 Activity 20 Worksheets Combine Like Terms Skill 57 Activity 57 Diagnose and CD-ROM Evaluate Expressions Skill 60 Activity 60 Prescribe Online Online Ordered Pairs Skill 79 Activity 79
Are You Ready? 3
gge07te_c01_0002_0005.indde07te_c01_0002_0005.indd 3 66/14/06/14/06 11:41:18:41:18 PPMM CHAPTER Study Guide: 1 Preview
Organizer Key Vocabulary/Vocabulario Objective: Help students Previously, you organize the new concepts they angle ángulo • used the order of operations. will learn in Chapter 1. area área • used variables and expressions to represent coordinate plane plano cartesiano @
undefined term término indefinido
Multilingual Glossary Online You will study • applying basic facts about Vocabulary Connections points, lines, planes, KEYWORD: MG7 Glossary segments, and angles. To become familiar with some of the • measuring and constructing vocabulary terms in the chapter, consider segments and angles. the following. You may refer to the chapter, the glossary, or a dictionary if you like. Answers to • using formulas to find Vocabulary Connections distance and the coordinates 1. A definition is a statement that gives the of a midpoint. meaning of a word or phrase. What do 1. An undefined term is a term that you think the phrase undefined term is not defined with a word or • identifying reflections, rotations, and translations. means? phrase. 2. Coordinates are numbers used to describe 2. A coordinate plane is a flat sur- a location. A plane is a flat surface. face that has numbers on it to How can you use these meanings to describe locations. understand the term coordinate plane ? 3. Possible answer: tip of a sharp- 3. A point is often represented by a dot. ened pencil You can use the skills What real-world items could represent learned in this chapter 4. Transformation means “to points? to find distances between change or move a shape.” • 4. Trans- is a prefix that means “across,” cities. as in movement. A form is a shape. • to determine how much How can you use these meanings to material is needed to make understand the term transformation ? a rectangular or triangular object. • in classes such as Biology, when you learn about gene mapping and in physics, when you study angles formed by light waves that bounce off objects.
4 Chapter 1
New York Performance Indicators
ge07se_c01_0002_0005.inddGeometry 4 G.G.66 Find the midpoint of a line segment, given its 12/13/06 12:08:42 PM ge07se_c01_0002_0005.indd 5 12/13/06 12:09:55 PM G.G.17 Construct a bisector of a given angle, using endpoints a straightedge and compass, and justify the G.G.67 Find the length of a line segment, given its construction endpoints G.G.48 Investigate, justify, and apply the Pythagorean Process Theorem and its converse G.PS.5 Choose an effective approach to solve a G.G.54 Define, investigate, justify, and apply problem from a variety of strategies (numeric, isometries in the plane (rotations, reflections, graphic, algebraic) translations, glide reflections) G.CM.11 Understand and use appropriate language, G.G.55 Investigate, justify, and apply the properties representations, and terminology when that remain invariant under translations, describing objects, relationships, mathematical rotations, reflections, and glide reflections solutions, and geometric diagrams G.G.56 Identify specific isometries by observing G.CN.6 Recognize and apply mathematics to situations orientation, numbers of invariant points, in the outside world and/or parallelism
4 Chapter 1 New York Mathematics Performance Indicators within the process strands are covered throughout the book.
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Organizer Reading Strategy: Use Your Book for Success Objective: Help students apply Understanding how your textbook is organized will help you locate and use strategies to understand and retain helpful information. key concepts. @
ENGLISH LANGUAGE Reading Strategy: LEARNERS Use Your Book for The Glossary is found The Index is located at the The Skills Bank is located in in the back of your end of your textbook. If the back of your textbook. Success textbook. Use it when you need to locate the page Look in the Skills Bank for Discuss The index, glossary, margin you need a definition where a particular concept help with math topics that notes, and Skills Bank can provide of an unfamiliar word is explained, use the Index were taught in previous students with a great deal of use- or phrase. to find the corresponding courses, such as the order ful information as they use this page number. of operations. textbook. Have students divide their journal into sections such as research, reflections, constructions, postulates, and so on. Encourage them to use all the resources pro- vided in this book. Extend Ask students to find a term they do not know and look it up in the glossary. Then ask them to enter the definition in their jour- nal. Challenge the students to use Try This information in the book that was not mentioned. For example, look Use your textbook for the following problems. at the table of contents, highlighted 1. Use the index to find the page where right angle is defined. words, words in italics, or words in boldface. 2. What formula does the Know-It Note on the first page of Lesson 1-6 refer to?
3. Use the glossary to find the definition of congruent segments. Answers to Try This 1. p. 21 4. In what part of the textbook can you find help for solving equations? 2. Mdpt. Formula 3. Foundations for Geometry 5 segs. that have the same length 4. Skills Bank
New York Performance Indicators ge07se_c01_0002_0005.indd 4 12/13/06 12:08:42 PM ge07se_c01_0002_0005.inddPerformance 5 LAB LAB12/13/06 12:09:55 PM Indicators 1-1 1-2 1-2 1-3 1-4 1-5 1-6 1-7 1-7 G.G.17 ★ G.G.48 ★ G.G.54 ★ G.G.55 ★ G.G.56 ★ G.G.66 ★ G.G.67 ★ G.PS.5 ★★ G.CM.11 ★★ G.CN.6 ★★ New York Mathematics Performance Indicators are written out completely on pp NY28-NY35. Reading and Writing Math 5
gge08te_ny_c01_0002_0005.indde08te_ny_c01_0002_0005.indd 5 112/21/062/21/06 11:10:24:10:24 PPMM Euclidean and Construction Tools
One-Minute Section Planner
Lesson Lab Resources Materials
Lesson 1-1 Understanding Points, Lines, and Planes Required • Identify, name, and draw points, lines, segments, rays, and planes. straightedge (MK) • Apply basic facts about points, lines, and planes. Optional ✔ ✔ ✔ ✔ □ NY Regents Exam □ SAT-10 □ NAEP □ ACT □ SAT raw spaghetti, index cards, straw, clay, gumdrops, toothpicks
1-2 Technology Lab Explore Properties Associated with Points Technology Lab Activities Required • Use geometry software to measure distances and explore properties 1-2 Lab Recording Sheet geometry software of points on segments. Optional ✔ ✔ ✔ □ NY Regents Exam □ SAT-10 □ NAEP □ ACT □ SAT ruler (MK)
Lesson 1-2 Measuring and Constructing Segments Geometry Lab Activities Required • Use length and midpoint of a segment. 1-2 Geometry Lab compass (MK), straightedge • Construct midpoints and congruent segments. Optional ✔ ✔ ✔ ✔ □ NY Regents Exam □ SAT-10 □ NAEP □ ACT □ SAT road map, masking tape, butcher paper, small plastic disks, meter stick
Lesson 1-3 Measuring and Constructing Angles Geometry Lab Activities Required • Name and classify angles. 1-3 Geometry Lab compass (MK), straightedge, • Measure and construct angles and angle bisectors. protractor (MK), geometry □✔ NY Regents Exam □ SAT-10 □✔ NAEP □✔ ACT □✔ SAT software Optional yarn, index cards, pictures of angles, origami paper, sticky notes, acetate or tracing paper, clock (MK), Mira
Lesson 1-4 Pairs of Angles Required • Identify adjacent, vertical, complementary, and supplementary protractor (MK) angles. • Find measures of pairs of angles. □ NY Regents Exam □ SAT-10 □✔ NAEP □✔ ACT □✔ SAT
MK = Manipulatives Kit
6A Chapter 1
gge08te_ny_c01_0006A_0006B.indde08te_ny_c01_0006A_0006B.indd 66AA 112/21/062/21/06 11:14:11:14:11 PPMM Section Overview
Points, Lines, and Planes Lesson 1-1
Understanding points, lines, planes, and segments is fundamental to the entire geometry course.
• The intersection of two lines is a point. • The intersection of two planes is a line. • If two points lie in a plane, then the line −− they determine lies in the plane. Segment: DC Line: DC Ray: BA Opposite rays: BC , BD
Measuring Segments and Angles Lessons 1-2, 1-3
Distance and angle measure are important in carpentry, engineering, science, and many other areas. Congruent Angles Congruent Segments - Óx Óx nÊ { £ Î Â nÊ ÚÚ ÓÊÊ ÎÚÚ£ÊÊ ÎÚÚ£ÊÊ / Ó Ó −− −− Ç m∠G = m∠ H ⇔ ∠G ∠H AB = ST ⇔ AB ST −− M is the midpoint of AB. Acute Angle Right Angle Obtuse Angle 1 1 AM = _ AB MB = _ AB 2 2 AM = MB
ä Â
Pairs of Angles Lesson 1-4
Properties of angle pairs are used in science and engineering.
Vertical Angles Linear Pair Adjacent Angles
Ê Ã`iÊ>`Ê ÛiÀÌiÝ
Complementary Angles Supplementary Angles
£{£Â Πx£Â ÎÂ
39º + 51º = 90º 39º + 141º = 180º
6B
gge07te_c01_0006A_0006B.indde07te_c01_0006A_0006B.indd 66BB 112/5/052/5/05 110:36:130:36:13 PPMM 1-1 Organizer 1-1 Understanding Points, Pacing: Traditional 1 day Lines, and Planes __1 Block 2 day Objectives: Identify, name, and Objectives Who uses this? draw points, lines, segments, rays, Identify, name, and draw Architects use representations of points, lines, and and planes. points, lines, segments, planes to create models of buildings. Interwoven rays, and planes. Apply basic facts about points, segments were used to model the beams of Apply basic facts about Beijing’s National Stadium for the 2008 Olympics. lines, and planes. points, lines, and planes. The most basic figures in geometry are undefined @ 0 diagrams. Also, G.CN.6. Points that lie in the same plane are coplanar . Otherwise they are noncoplanar. ä £
Also available on transparency EXAMPLE 1 Naming Points, Lines, and Planes Refer to the design in the roof of Beijing’s National Stadium.
A plane may be A Name four coplanar points. named by any three K, L, M, and N all lie in plane R. noncollinear points on that plane. Plane B Name three lines. Q: What did the little acorn say ABC may also be AB , BC , and CA . named BCA, CAB, when it grew up? CBA, ACB, or BAC. A: “Geometry” 1. Use the diagram to name two planes. Possible answer: plane R and plane ABC New York Performance Indicators 6 Chapter 1 Foundations for Geometry
Process G.CM.11 Understand and use appropriate language, representations, 1 Introduce and terminology when describing
objects, relationships, mathematical ge07se_c01_0006_0011.inddEXPLORATION 1 11/23/06 1:23:25 PM solutions, and geometric diagrams Motivate ££ 1`iÀÃÌ>`}Ê*ÌÃ]Ê G.CN.6 Recognize and apply iÃ]Ê>`Ê*>ià Point out different objects in the classroom that 4HE FOLLOWING ARE SOME OF THE TERMS USED AS THE BASIC mathematics to situations in the BUILDING BLOCKS OF GEOMETRY $ESCRIBE A REAL WORLD OBJECT THAT are representations of points, segments, and outside world IS SUGGESTED BY EACH TERM planes, such as the tips of pushpins on the bul- POINT letin board, rulers, and desktops. Discuss with stu- LINE dents what these items have in common. PLANE
4ELL WHETHER EACH OF THE FOLLOWING IS MOST LIKE A POINT A LINE Ask students to give examples of other objects OR A PLANE that have these same characteristics and can be A A DESKTOP B A SPECK OF DUST found in the world around them, such as the loca- C A JET CONTRAIL tions of cities on a map, the lines on a football / Ê Ê - 1-- field, and the bases on a baseball field. %XPLAIN WHAT IS REPRESENTED BY THE ARROWHEAD AT EACH END OF THE LINE IN 0ROBLEM $ISCUSS WHETHER THE SURFACE OF %ARTH CAN BE AN EXAMPLE OF A PLANE Explorations and answers are provided in the Explorations binder. 6 Chapter 1
gge08te_ny_c01_0006_0011.indde08te_ny_c01_0006_0011.indd 6 112/21/062/21/06 112:34:452:34:45 PPMM Segments and Rays
DEFINITION NAME DIAGRAM Additional Examples A segment , or line segment, The two endpoints −− −− is the part of a line consisting AB or BA Example 1 of two points and all points between them.
An endpoint is a point at A capital letter one end of a segment or the C and D starting point of a ray. A ray is a part of a line Its endpoint and any other that starts at an endpoint point on the ray , - A. Name four coplanar points. and extends forever in one RS direction. - , A, B, C, D
Opposite rays are two The common endpoint and B. Name three lines. rays that have a common any other point on each ray Possible answer: AE , BE , CE endpoint and form a line. EF and EG Example 2 Draw and label each of the EXAMPLE 2 Drawing Segments and Rays following. Draw and label each of the following. A. segment with endpoints M 6 and N A a segment with endpoints U and V 1 *+, B opposite rays with a common endpoint Q B. opposite rays with common endpoint T / 2. Draw and label a ray with endpoint M that contains N. M N Example 3 Name a line that passes through A postulate , or axiom, is a statement that is accepted as true without proof. two points. XY Postulates about points, lines, and planes help describe geometric properties.
Postulates Points, Lines, and Planes 89 1-1-1 Through any two points there is exactly one line. Also available on transparency 1-1-2 Through any three noncollinear points there is exactly one plane containing them.
1-1-3 If two points lie in a plane, then the line containing those points lies in the plane. Reading Math The phrase
“exactly one” ENGLISH means that one exists LANGUAGE LEARNERS EXAMPLE 3 Identifying Points and Lines in a Plane and it is unique. Name a line that passes through two points. There is exactly one line n passing through INTERVENTION G and H. Questioning Strategies EXAMPLE 1 • What are the different ways you 3. Name a plane that contains three noncollinear points. can name the planes represented Possible answer: plane GHF in the stadium roof? 1- 1 Understanding Points, Lines, and Planes 7 • How else could you name the lines?
2 Teach EXAMPLE 2 • What figure is formed by two ge07se_c01_0006_0011.inddGuided 7 Instruction 12/2/05 7:08:00 PMopposite rays? Is there a way to name the opposite rays? Give an Show students how to draw a plane. Through Modeling example. Discuss all the different ways of labeling a Have students work in small groups to use line, segment, ray, and plane. Explain that physical models, such as raw spaghetti and EXAMPLE 3 CD and DC are not the same ray. index cards, to illustrate Postulates 1-1-3, • Are there other ways to name the 1-1-4, and 1-1-5. Use contrasting colors of line determined by the points in Visual Point out to students that paper to emphasize the different planes in the diagram? Explain. the diagrams of planes, lines, and Postulate 1-1-5. rays extend forever, but we can- not draw them that way.
Lesson 1-1 7
gge07te_c01_0006_0011.indde07te_c01_0006_0011.indd 7 66/14/06/14/06 22:06:22:06:22 PPMM Recall that a system of equations is a set of two or more equations containing two or more of the same variables. The coordinates of the solution of the system satisfy all equations in the system. These coordinates also locate the point where Additional Examples all the graphs of the equations in the system intersect. Example 4 An intersection is the set of all points that two or more figures have in common. A. Sketch two lines intersecting The next two postulates describe intersections involving lines and planes. in exactly one point. Postulates Intersection of Lines and Planes
1-1-4 If two lines intersect, then they intersect in exactly one point. B. Sketch a figure that shows a 1-1-5 If two planes intersect, then they intersect in exactly one line. line that lies in a plane.
Use a dashed line to show the hidden parts of any figure that you are drawing. A dashed line will indicate the part of the figure that is not seen.
EXAMPLE 4 Representing Intersections Also available on transparency Sketch a figure that shows each of the following. A A line intersects a plane, but B Two planes intersect in does not lie in the plane. one line. INTERVENTION Ű Questioning Strategies EXAMPLE 4 • Must any two planes intersect? Why or why not? Name planes in the classroom that support your answer. 4. Sketch a figure that shows two lines intersect in one point in a plane, but only one of the lines lies in the plane. • What would happen if planes were extended? Would they then all intersect? • If a line lies in a plane, how many points of intersection do the line G.RP.5, G.CM.11, G.R.1 and the plane have? THINK AND DISCUSS 1. Explain why any two points are collinear. 2. Which postulate explains the fact that two straight roads cannot cross each other more than once? 3. Explain why points and lines may be coplanar even when the plane containing them is not drawn. 4. Name all the possible lines, segments, and rays for the points A and B. Then give the maximum number of planes that can be determined by these points. 5. GET ORGANIZED Copy and complete the graphic organizer 1`ivi`Ê/iÀÃ below. In each box, name, describe, and illustrate one of the undefined terms.
8 Chapter 1 Foundations for Geometry
3 Answers to Think and Discuss Close Possible answers:
ge07se_c01_0006_0011.indd 3 11/23/06 1:23:35 PM and INTERVENTION 1. By Post. 1-1-1, through any 2 pts. there Summarize is a line. Therefore any 2 pts. are Ask students to name the different terms Diagnose Before the Lesson collinear. introduced in this lesson and write the 1-1 Warm Up, TE p. 6 2. Post. 1-1-4 notation for each. Give students the pos- tulates with some missing words and ask 3. Any 3 noncollinear pts. determine a Monitor During the Lesson them to fill in the blanks. Ask students to plane. Check It Out! Exercises, SE pp. 6–8 −− explain the meaning of the word postulate. 4. AB , AB , AB , BA ; 0 planes Questioning Strategies, TE pp. 7–8 to assume or claim as true 5. See p. A2.
Assess After the Lesson 1-1 Lesson Quiz, TE p. 11 Alternative Assessment, TE p. 11
8 Chapter 1
gge08te_ny_c01_0006_0011.indde08te_ny_c01_0006_0011.indd 8 112/21/062/21/06 11:15:56:15:56 PPMM 1-1 1-1 ExercisesExercises Exercises KEYWORD: MG7 1-1
KEYWORD: MG7 Parent GUIDED PRACTICE Assignment Guide Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Give an example from your classroom of three collinear points. Assign Guided Practice exercises Possible answer: the intersection of 2 floor tiles as necessary. 2. Make use of the fact that endpoint is a compound of end and point and name the endpoint of ST . S If you finished Examples 1–2 SEE EXAMPLE 1 Use the figure to name each of the following. Basic 13–17, 33 Average 13–17, 33, 36, 43 p. 6 3. five points A, B, C, D, E Advanced 13–17, 33, 34, 36, 4. Possible answer: AC , BD two lines 43–45
5. two planes Possible answer: ABC and N If you finished Examples 1–4 6. point on BD Possible answer: B, C, or D Basic 13–28, 31–33, 39–42, SEE EXAMPLE 2 Draw and label each of the following. 47–51 p. 7 7. a segment with endpoints M and N MN Average 13–28, 30–33, 35–43, 46–51 8. a ray with endpoint F that passes through G FG Advanced 14–28 even, 29–51 SEE EXAMPLE 3 Use the figure to name each of the following. Possible answer: p. 7 9. a line that contains A and C Homework Quick Check AB Quickly check key concepts. 10. a plane that contains A, D, and C Possible answer: plane ABD Exercises: 14–20 even, 24, 26, SEE EXAMPLE 4 Sketch a figure that shows each of the following. 28, 32 p. 8 11. three coplanar lines that intersect in a common point AB 12. two lines that do not intersect Communicating Math In CD Exercises 3–6, students PRACTICE AND PROBLEM SOLVING sometimes forget to place Independent Practice Use the figure to name each of the following. a symbol above the letters that are For See used to name lines, segments, and Exercises Example 13. three collinear points B, E, A rays. Remind students that two let- 13–15 1 14. four coplanar points Possible answer: ters without a symbol represent 16–17 2 B, C, D, E 15. a plane containing E a distance. 18–19 3 Possible answer: plane ABC 20–21 4 Draw and label each of the following. Extra Practice 16. a line containing X and Y Y Skills Practice p. S4 X Application Practice p. S28 17. a pair of opposite rays that both contain R R
Use the figure to name each of the following. 18. two points and a line that lie in plane T 20. Possible answer: G, J, and 19. two planes that contain R T S Possible answer: planes and S Sketch a figure that shows each of the following. Ű 20. a line that intersects two nonintersecting planes 21. 21. three coplanar lines that intersect in three different points
1- 1 Understanding Points, Lines, and Planes 9
ge07se_c01_0006_0011.indd 9 12/2/05 7:08:05 PM
KEYWORD: MG7 Resources
Lesson 1-1 9
gge07te_c01_0006_0011.indde07te_c01_0006_0011.indd 9 66/14/06/14/06 22:06:38:06:38 PPMM Exercise 22 involves 22. This problem will prepare you for the identifying geometric Multi-Step Test Prep on page 34. Name an shapes at an archae- object at the archaeological site shown that ological site. This exercise prepares is represented by each of the following. students for the Multi-Step Test Prep a. a point on page 34. b. a segment c. a plane Visual For Exercise 36, have students make an organized list of all the combinations of three points that could be chosen and then select 22. Possible Draw each of the following. those that are collinear. answers: 23. plane H containing two lines that intersect at M a. tip of a stake M 24. ST intersecting plane M at R Answers b. string c. grid formed 24. Use the figure to name each of the following. S by string - R 25. the intersection of TV and US U 28. If 2 pts. lie in 6 R / 1 T a plane, then the 26. the intersection of US and plane U −− −− line containing 31. 27. the intersection of TU and UV U those pts. lies in the plane. Write the postulate that justifies each statement. 32. 28. The line connecting two dots on a sheet of paper lies on the same sheet of paper as 30. It is not the dots. 33. A B possible. By Post. 1-1-2, any 29. If two ants are walking in straight lines but in different directions, their paths cannot 3 noncollinear pts. cross more than once. If 2 lines intersect, then they intersect in exactly 1 pt. 34. are contained in 30. Critical Thinking Is it possible to draw three points that are noncoplanar? Explain. a unique plane. S AN 38. Lines may not If the 3 pts. are Tell whether each statement is sometimes, always, or never true. Support your intersect: 0 pts. collinear, they answer with a sketch. of intersection. are contained in 31. If two planes intersect, they intersect in a straight line. A All 3 lines may infinitely many planes. In either 32. If two lines intersect, they intersect at two different points. N intersect in 1 pt. case, the 3 pts. 33. AB is another name for BA . A Two of the lines will be coplanar. may not intersect, 34. If two rays share a common endpoint, then they form a line. S but they might each 35. Post. 1-1-3 35. Art Pointillism is a technique in which tiny dots of intersect a third line. complementary colors are combined to form a picture. Each line may Which postulate ensures that a line connecting two of intersect each of these points also lies in the plane containing the points? the other lines. 36. Probability Three of the labeled points are chosen at random. What is the probability that _1 they are collinear? 4 37. Campers often use a cooking stove with three legs. Which postulate explains why they might prefer this design to a stove that has four legs? Post. 1-1-2 38. Write About It Explain why three coplanar lines may 1-1 PRACTICE A have zero, one, two, or three points of intersection. 1-1 PRACTICE C Support your answer with a sketch.
*À>VÌViÊ York/ADAGP, New Artists’ Rights Society (ARS), © 2004 NY/Detail des Musees Nationaux/Art Resource, Paris/Reunion ,%33/. 1-1 PRACTICE B ££ 5NDERSTANDING 0OINTS ,INES AND 0LANES 10 Chapter 1 Foundations for Geometry 5SE THE FIGURE FOR %XERCISES n 0OSSIBLE ANSWER PLANE "#$ .AME A PLANE ? ? ? ? ,%33/. ,i>`}Ê-ÌÀ>Ìi}ià ,%33/. ,iÌi>V .AME A SEGMENT "$ "# "% OR #% j??k # ££ )DENTIFY 2ELATIONSHIPS1-1 READING STRATEGIES ££ 5NDERSTANDING 0OINTS ,INES AND 0LANES1-1 RETEACH 0OSSIBLE ANSWER %# " .AME A LINE )N GEOMETRY MANY FIGURES CAN BE BROKEN DOWN INTO THREE BASIC BUILDING ! POINT HAS NO SIZE )T IS NAMED USING A CAPITAL LETTER s0 BLOCKS POINTS LINES AND PLANES !LL THE FIGURES BELOW CONTAIN POINTS POINT 0 .AME THREE COLLINEAR POINTS % #ONSIDER THE FOLLOWING POINTS " # AND % &IGURE #HARACTERISTICS $IAGRAM 7ORDS AND 3YMBOLS s ! POINT NAMES A LOCATION AND HAS NO SIZE s 0 j??k .AME THREE NONCOLLINEAR POINTS $ LINE ENDPOINTS LINE !" OR !" s ! LINE IS A STRAIGHT PATH THAT EXTENDS FOREVER IN TWO DIRECTIONS !" EXTENDS FOREVER !" 0OSSIBLE ANSWER POINTS " # AND $ ge07se_c01_0006_0011.indds ! 10PLANE IS A FLAT SURFACE THAT EXTENDS FOREVER . IN TWO DIRECTIONS 4/27/06 11:59:35 AM ge07se_c01_0006_0011.indd 11 12/2/05 7:08:14 PM - ? / .AME THE INTERSECTION OF A LINE AND A SEGMENT NOT ON THE LINE POINT " LINE SEGMENT ENDPOINTS SEGMENT 89 OR 89 ???k ??? k OR SEGMENT 89 .AME A PAIR OF OPPOSITE RAYS "# AND "% 5SE THAT INFORMATION TO THINK ABOUT THE FOLLOWING DEFINITIONS HAS A FINITE LENGTH ???k RAY ENDPOINT s ! SEGMENT IS PART OF A LINE AND CONSISTS OF TWO POINTS AND ' RAY 21 OR 21 ENDPOINT 5SE THE FIGURE FOR %XERCISES n ALL THE POINTS IN BETWEEN & EXTENDS FOREVER 12! RAY IS NAMED STARTING ENDPOINT WITH ITS ENDPOINT M IN ONE DIRECTION .AME THE POINTS THAT DETERMINE PLANE s !N ENDPOINT IS A POINT AT THE END OF A SEGMENT OR A RAY PLANE EXTENDS FOREVER & PLANE &'( OR PLANE s ! RAY IS A PART OF A LINE THAT STARTS AT AN ENDPOINT AND ' POINTS 8 9 AND : IN ALL DIRECTIONS ( EXTENDS FOREVER IN ONE DIRECTION * : .AME THE POINT AT WHICH LINE M INTERSECTS + !NSWER EACH QUESTION $RAW AND LABEL A DIAGRAM FOR EACH FIGURE PLANE POINT : 7 (OW ARE A LINE AND A LINE SEGMENT THE SAME POINT 7 LINE -. .AME TWO LINES IN PLANE THAT INTERSECT LINE M j??k j??k ! LINE SEGMENT IS A SPECIFIC PORTION OF A LINE THAT BEGINS AND ENDS s 7 -. 8: AND 9: 89 (OW ARE A LINE AND A LINE SEGMENT DIFFERENT ? ??k .AME A LINE IN PLANE THAT DOES NOT INTERSECT *+ %& j??k ! LINE GOES ON FOREVER IN BOTH DIRECTIONS WHILE A SEGMENT HAS ENDPOINTS 89 LINE M *+ %& (OW ARE A LINE SEGMENT AND A RAY THE SAME $RAW YOUR ANSWERS IN THE SPACE PROVIDED ! RAY IS A PART OF A LINE THAT STARTS AT THE ENDPOINT AND EXTENDS IN ONE DIRECTION FOREVER .AME EACH FIGURE USING WORDS AND SYMBOLS -ICHELLE +WAN WON A BRONZE MEDAL IN FIGURE SKATING AT THE 3ALT ,AKE (OW ARE A LINE SEGMENT AND A RAY DIFFERENT #ITY 7INTER /LYMPIC 'AMES ! LINE SEGMENT HAS ENDPOINTS A RAY EXTENDS FOREVER IN ONE DIRECTION $ 4 -ICHELLE SKATES STRAIGHT AHEAD FROM POINT , AND STOPS ,- 3 AT POINT - $RAW HER PATH )DENTIFY THE FOLLOWING # " j???k ???k -ICHELLE SKATES STRAIGHT AHEAD FROM POINT , AND CONTINUES RAY s 9 1 THROUGH POINT - .AME A FIGURE THAT REPRESENTS HER PATH , - ! LINE #$ OR #$ RAY 34 OR 34 $RAW HER PATH 0 .AME THE PLANE IN TWO DIFFERENT WAYS -ICHELLE AND HER FRIEND !LEXEI START BACK TO BACK AT POINT , POINT SEGMENT RAY 8 AND SKATE IN OPPOSITE DIRECTIONS -ICHELLE SKATES THROUGH + ,- , POINT - AND !LEXEI SKATES THROUGH POINT + $RAW THEIR PATHS . 7 -
, ? 4 -. * + 3 PLANE ,-. PLANE SEGMENT 78 78 10 Chapter 1 ENDPOINT LINE PLANE
gge07te_c01_0006_0011.indde07te_c01_0006_0011.indd 1100 66/14/06/14/06 11:46:39:46:39 PPMM For help with 39. Which of the following is a set of noncollinear points? Exercise 42, have - students draw three P, R, T P, Q, R * , / points on a sheet of paper and use Q, R, S S, T, U + 1 the tip of a pencil held perpendicu- lar to the paper for the fourth point. 40. What is the greatest number of intersection points four coplanar lines can have? Have a partner hold a sheet of paper 6 2 up in various ways to determine the 4 0 planes.
41. Two flat walls meet in the corner of a classroom. Which postulate best describes Algebra For Exercise 45, this situation? tell students not to mutiply Through any three noncollinear points there is exactly one plane. n(n - 1). This formula If two points lie in a plane, then the line containing them lies in would count a line twice. the plane. If two lines intersect, then they intersect in exactly one point. If two planes intersect, then they intersect in exactly one line.
42. Gridded Response What is the greatest number of planes determined by Journal four noncollinear points? 4 Have students draw and label a point, a line, a plane, a ray, opposite CHALLENGE AND EXTEND rays, 4 coplanar points, and 3 col- linear points. Use the table for Exercises 43–45.
Figure
Number of Points 234 Have students make physical models using straw and clay, or gumdrops Maximum Number 13 and marshmallows with toothpicks, of Segments to illustrate the postulates in this les- 43. What is the maximum number of segments determined by 4 points? 6 son. Have them explain their models. 44. Multi-Step Extend the table. What is the maximum number of segments determined by 10 points? 45 n(n - 1) _ 45. Write a formula for the maximum number of segments determined by n points. 2 1-1 46. Critical Thinking Explain how rescue teams could use two of the postulates from this lesson to locate a distress signal.
SPIRAL REVIEW 47. The combined age of a mother and her twin daughters is 58 years. The mother was 25 years old when the twins were born. Write and solve an equation to find the age of each of the three people. (Previous course) Mother is 36; twins are 11. Name each of the following. Determine whether each set of ordered pairs is a function. (Previous course) 48. (0, 1) , (1, -1) , (5, -1) , (-1, 2) yes49. (3, 8) , (10, 6) , ( 9, 8) , ( 10, -6) no 1. two opposite rays CB and CD 2. a point on BC Find the mean, median, and mode for each set of data. (Previous course) Possible answer: D 50. 0, 6, 1, 3, 5, 2, 7, 10 51. 0.47, 0.44, 0.4, 0.46, 0.44 mean: 4.25; median: 4; mode: none mean: 0.442; median: 0.44; mode: 0.44 3. the intersection of plane and plane 1- 1 Understanding Points, Lines, and Planes 11 Possible answer: BD 4. a plane containing E, D, and B *ÀLiÊ-Û} ,%33/. 1-1 PROBLEM SOLVING ££ 5NDERSTANDING 0OINTS ,INES AND 0LANES plane 5SE THE MAP OF PART OF 3AN !NTONIO FOR %XERCISES AND Answers .AME A POINT THAT APPEARS TO BE COLLINEAR ? WITH %& 7HICH STREETS INTERSECT AT THIS POINT % 4RAVIS 3T ! $ Draw each of the following. 0OINT $ % 4RAVIS 3T AND 46. Rescue teams can use the principles of
.AVARRO 3T 3T -ARYS 3T % (OUSTON 3T ge07se_c01_0006_0011.indd 10 4/27/06 11:59:35 AM ge07se_c01_0006_0011.indd 6 ? " % Post. 1-1-1 and Post. 1-1-4. A distress 11/23/06 1:23:50 PM5. a line intersecting a plane at %XPLAIN WHY POINT ! IS ./4 COLLINEAR WITH "% #OLLEGE 3T 0OINT ! DOES NOT LIE ON THE LINE # & signal is received by 2 rescue teams. one point ?
3AN !NTONIO 2IVER THAT CONTAINS "% .AVARRO 3T By Post. 1-1-1, 2 pts. determine a line. ? 3UPPOSE 56 REPRESENTS THE PENCIL THAT 4WO CYCLISTS START AT THE SAME POINT YOU ARE USING TO DO YOUR HOMEWORK BUT TRAVEL ALONG TWO STRAIGHT STREETS IN So 2 lines are created by the 3 pts., AND PLANE REPRESENTS THE PAPER DIFFERENT DIRECTIONS )F THEY CONTINUE HOW THAT YOU ARE WRITING ON $ESCRIBE THE ?? MANY TIMES WILL THEIR PATHS CROSS AGAIN j k the locations of the rescue teams and RELATIONSHIP BETWEEN 56 AND PLANE %XPLAIN j??k 0OSSIBLE ANSWER 56 INTERSECTS TIMES )F TWO LINES INTERSECT THEN the distress signal. By Post. 1-1-4, the PLANE THEY INTERSECT IN EXACTLY ONE POINT
#HOOSE THE BEST ANSWER intersection of the 2 lines will be the )N A BUILDING PLANES AND 3UPPOSE POINT ' REPRESENTS A DUCK FLYING REPRESENT EACH OF THE THREE FLOORS PLANES OVER A LAKE POINTS ( AND * REPRESENT location of the distress signal. AND REPRESENT THE FRONT AND BACK OF TWO DUCKS SWIMMING ON THE LAKE AND 6. a ray with endpoint P that THE BUILDING PLANES AND REPRESENT THE PLANE REPRESENTS THE LAKE 7HICH IS SIDES 7HICH IS A TRUE STATEMENT A TRUE STATEMENT passes through Q ! 0LANES AND INTERSECT IN A LINE & 4HERE ARE TWO LINES THROUGH ' AND * " 0LANES AND INTERSECT IN A LINE ' 4HE LINE CONTAINING ' AND ( LIES IN # 0LANES AND INTERSECT IN A POINT PLANE $ 0LANES AND INTERSECT IN A POINT ( ' ( AND * ARE NONCOPLANAR * * 4HERE IS EXACTLY ONE PLANE CONTAINING + POINTS ' ( AND *
5SE THE FIGURE FOR %XERCISE + , ! FRAME HOLDING TWO PICTURES SITS ON A TABLE * Also available on transparency 7HICH IS ./4 A TRUE STATEMENT ? ? ! 4HE LINES CONTAINING *0 AND +. INTERSECT PLANE ? ? " 0. AND .- INTERSECT IN A POINT ? # ,- AND . INTERSECT IN A LINE . - ? $ 0 AND .- ARE COPLANAR 0 Lesson 1-1 11
gge08te_ny_c01_0006_0011.indde08te_ny_c01_0006_0011.indd 1111 112/21/062/21/06 11:16:18:16:18 PPMM 1-2 Organizer Explore Properties Use with Lesson 1-2
Pacing: Associated with Points __1 The two endpoints of a segment determine its length. Other points on the Traditional 2 day Block __1 day segment are between the endpoints. Only one of these points is the midpoint 4 of the segment. In this lab, you will use geometry software to measure Objective: Use geometry Use with Lesson 1-2 lengths of segments and explore properties of points on segments. software to measure distances and explore properties of points on NY Performance Indicators segments. G.RP.3, G.R.1 KEYWORD: MG7 Lab1 Materials: geometry software Activity 1 Construct a segment and label its endpoints A and C. @
Resources 3 Measure the distances from A to B and from B to C. Technology Lab Activities Use the Calculate tool to calculate the sum of AB 1-2 Lab Recording Sheet and BC. −− 4 Measure the length of AC . What do you notice Teach about this length compared with the measurements Discuss found in Step 3? Emphasize that there are infinitely −− 5 Drag−− point B along AC . Drag one of the endpoints many points on a line segment of AC. What relationships do you think are true between its endpoints, but only one about the three measurements? of these points is the midpoint. −− Close 6 Construct the midpoint of AC and label it M. −−− −−− Key Concept 7 Measure AM and MC . What relationships−− −−− do you−−− think are true about the lengths of AC, AM, and MC ? When three points A, B, and C are Use the Calculate tool to confirm your findings. + = collinear, then AB BC AC. The −− midpoint of a segment is equidistant 8 How many midpoints of AC exist? from each endpoint. Assessment Journal Have students summarize how they determine, given three col- Try This linear points, if the point between 1. Repeat the activity with a new segment. Drag each of the points in your figure the endpoints is the midpoint. Then (the endpoints, the point on the segment, and the midpoint). Write down any have them support their answer with relationships you observe about the measurements. Check students’ work. a sketch. −− −− −− −− 2. Create a point D not on AC . Measure AD, DC , and AC . Does AD + DC = AC? What do you think has to be true about D for the relationship to always be true? New York Performance No; D must be between A and C. Indicators 12 Chapter 1 Foundations for Geometry
Process G.RP.3 Investigate and evaluate conjectures in mathematical terms, Teacher to Teacher using mathematical strategies to reach a conclusion ge07se_c01_0012.inddTo 1introduce the concept of way through the woods, she is 11/23/06 1:24:18 PM G.R.1 Use physical objects, diagrams, midpoint, I like to tell the fol- as close to the end as to the charts, tables, graphs, symbols, lowing riddle to the class. beginning. After passing the equations, or objects created using midpoint, the hunter is going technology as representations of Q: A hunter walks into the out instead of in. This helps mathematical concepts woods and keeps walking students remember that the in the same direction. At midpoint is not just any point what point does the hunter between the endpoints, but begin to leave the woods? exactly one point in the middle. A: The midpoint. Teresa Salas Corpus Christi, TX Then I can explain to students that once the hunter is half-
12 Chapter 1
gge08te_ny_c01_0012.indde08te_ny_c01_0012.indd 1122 112/21/062/21/06 112:35:442:35:44 PPMM 1-2 Measuring and 1-2 Organizer Constructing Segments Pacing: __1 Traditional 2 day __1 Block 4 day Objectives Why learn this? Objectives: Use length and Use length and midpoint You can measure a segment to calculate the midpoint of a segment. of a segment. distance between two locations. Maps of a Construct midpoints and congruent Construct midpoints and race are used to show the distance between segments. congruent segments. stations on the course. (See Example 4.)
Vocabulary A ruler can be used to measure the distance between Geometry Lab coordinate two points. A point corresponds to one and In Geometry Lab Activities distance only one number on the ruler. This number length is called a coordinate . The following @
The distance between any two points is the absolute value of the difference NY Performance of the coordinates. If the coordinates of points A and B are a and b, then the Warm Up Indicators - - distance between A and B is−− a b or b a . The distance between A and B Simplify. G.PS.5 Choose an effective is also called the length of AB, or AB. approach to solve a problem 1. 7 - ( -3) 10 from a variety of strategies (numeric, graphic, algebraic). Ê Ê]>ÊÊL]Ê Ê]LÊÊ>] - - (- ) Also, G.RP.2. > L 2. 1 13 12 3. -7 - 1 8 EXAMPLE 1 Finding the Length of a Segment Solve each equation. { Î Ó £ ä £ ÓÎ{x Find each length. + = - {°x 4. 2x 3 9x 11 2 A DC B EF 5. 3x = 4x - 5 5 DC = 4.5 - 2 EF = -4 - (-1) = 2.5 = -4 + 1 6. How many real numbers are = 2.5 = -3 __1 __3 there between 2 and 4 ? = 3 infinitely many Also available on transparency Find each length. <89 1 1a. XY 3_ Î Ó £ ä £ ÓÎ{xÈ 2 PQ represents a_ 1b. XZ 1 £ÊÊp£ÊÊÊ 4_ Ó number, while PQ 2 represents a geometric figure. Be sure to Congruent segments are segments that have use equality for the same length._ _ In the diagram, PQ = RS, so you +, numbers (PQ = RS) can write PQ RS. This is read as “segment PQ Carpenter 1: Why did you leave your and congruence_ _ for is congruent to segment RS.” Tick marks are *-tape measure in the truck? figures ( PQ RS). used in a figure to show congruent segments. /VÊ>Àà Carpenter 2: Because we’re entering a construction zone. 1- 2 Measuring and Constructing Segments 13 New York Performance 1 Indicators Introduce Process G.PS.5 Choose an effective approach ge07se_c01_0013_0019.inddEXPLORATION 1 Motivate 11/23/06 1:25:08to PM solve a problem from a variety of £Ó i>ÃÕÀ}Ê>`Ê strategies (numeric, graphic, algebraic) ÃÌÀÕVÌ}Ê-i}iÌà Show students a road map. Ask them to locate a 5SE GEOMETRY SOFTWARE OR A RULER FOR THE FOLLOWING G.RP.2 Recognize and verify, where %XPLORATION place that is midway between two other places. ? appropriate, geometric relationships $RAW !# 4HEN PLACE " ON THE SEGMENT BETWEEN ! AND # Ask questions about distance between locations of perpendicularity, parallelism, ? ? ? on a straight road. “Suppose you are driving to a -EASURE !" "# AND !# IN INCHES 4HEN MEASURE EACH congruence, and similarity, using SEGMENT IN CENTIMETERS AND MILLIMETERS 2ECORD THE LENGTHS IN relative’s house. The only place to stop for food is algebraic strategies A TABLE LIKE THE ONE SHOWN __1 5NIT ,ENGTH OF !" ,ENGTH OF "# ,ENGTH OF !# 75 miles from home, and it is of the total trip. )NCH 3 #ENTIMETER How far from the food stop does your relative -ILLIMETER live?” 150 mi / Ê Ê - 1-- $ESCRIBE THE RELATIONSHIP AMONG THE LENGTHS OF !" "# AND !# )S THIS RELATIONSHIP THE SAME NO MATTER WHAT MEASURE MENT UNIT YOU USE Explorations and answers are provided in the $ISCUSS WHAT WOULD HAPPEN IF YOU LOCATED $ BETWEEN ! AND " 7HAT WOULD BE THE RELATIONSHIP AMONG THE LENGTHS OF !$ Explorations binder. $" "# AND !#
Lesson 1-2 13
gge08te_ny_c01_0013_0019.indde08te_ny_c01_0013_0019.indd 1313 112/21/062/21/06 112:37:182:37:18 PPMM You can make a sketch or measure and draw a segment. These may not be exact. A construction is a way of creating a figure that is more precise. One way to make a geometric construction is to use a compass and straightedge. Additional Examples
Example 1 Construction Congruent Segment Find each length. _ A. BC 2 B. AC 5 Construct a segment congruent to AB. Ó ä £Î
Ű Ű Also available on transparency
Draw . Choose a point on and Open the compass to distance AB. Place the point of the compass label it C. at C and make an arc through . INTERVENTION Find the point where the arc and intersect and label it D. Questioning Strategies _ _ CD AB EXAMPLE 1 • To find the length of a segment, why do you subtract the coordi- EXAMPLE 2 Copying a Segment nates instead of adding them? Sketch, draw, _and construct a segment • What equation can you create congruent to MN . using BC, AC, and AB? Step 1 Estimate and sketch._ Estimate_ the length of MN and sketch *+ Construction PQ approximately the same length. Discuss techniques to make Step 2 Measure and draw._
an accurate construction, Use a ruler to measure MN . MN appears_ 89
including using a sharp pencil, lining to be 3.1 cm. Use a ruler and draw XY to have length 3.1 cm. up the compass tip and the pencil tip, and being sure the compass Step 3 Construct and compare. stays the same size. Use a compass_ and straightedge_ to construct ST congruent to MN. _ _ Communicating Math A ruler shows that PQ and XY are _ Have students look up the approximately_ the same length as MN , word between in the dic- but ST is precisely the same length. - / tionary and compare that definition with the mathematical definition. 2. Sketch, draw, and construct_ a Emphasize that the segment congruent to JK . Check students’ work. mathematical definition ENGLISH of between includes LANGUAGE In order for you to say that a point B is between two points A and C, LEARNERS collinearity. all three of the points must lie on the same line, and AB + BC = AC.
Critical Thinking Point Postulate 1-2-2 Segment Addition Postulate out that just as there are infinitely many real num- If B is between A and C, + = bers between any two real numbers, then AB BC AC. there are infinitely many points between any two points on a line.
14 Chapter 1 Foundations for Geometry
2 Teach ge07se_c01_0013_0019.inddGuided 14 Instruction 8/23/05 2:02:43 PM −− Discuss how to read PQ, PQ , and PQ. Ask Through Concrete Manipulatives students to explain the Ruler Postulate Have students use masking tape or draw a = in their own words. Stress that CD DC, long number line, either on their notebook and that distances can never be negative. paper or on a long piece of butcher paper Have students dramatize Postulate 1-2-2. taped to the floor. Give them small plastic Locate three students as A, B, and C on a disks to put on their number lines to rep- line in the classroom. Then ask one of the resent points. They can then use these to students to move off the line to illustrate count the spaces between the points. when one point is not between the other two. Find distances between points by counting floor tiles or paces, or by using a meterstick.
14 Chapter 1
gge07te_c01_0013_0019.indde07te_c01_0013_0019.indd 1144 112/20/052/20/05 99:50:43:50:43 AAMM EXAMPLE 3 Using the Segment Addition Postulate " " Ê ,,", A B is between A and C, AC = 14, and BC = 11.4. Find AB. ,/ AC = AB + BC Seg. Add. Post. When solving equations as in = + 14 AB 11.4 Substitute 14 for AC and 11.4 for BC. Examples 3 and 4, students some- - - −−−−−11.4 −−−−−−−11.4 Subtract 11.4 from both sides. times forget to do the inverse opera- = 2.6 AB Simplify. tion. Review inverse operations if ,-/ÓÝÊ ÊÇ Ón necessary. B S is between R and T. Find RT. {Ý RT = RS + ST Seg. Add. Post. 4x = (2x + 7) + 28 Substitute the given values. 4x = 2x + 35 Simplify. - - Additional Examples −−−−2x −−−−−−− 2x Subtract 2x from both sides. = 2x 35 Simplify. Example 2 2x 35 _ = _ Divide both sides by 2. 2 2 Sketch, draw, and construct−− a
x= _ 35 , or 17.5 segment congruent to MN . 2 Simplify. = RT 4x ( ) = =4 17.5 70 Substitute 17.5 for x. 2 3a. Y is between X and Z, XZ = 3, and XY = 1 __1 . Find YZ. 1_ Check students’ drawings and 3 3 constructions. The segment 3b. E is between D and F. Find DF. ÎÝÊÊ£ £Î 24 should be approximately 1__ 1 in. ÈÝ 4 _ The midpoint M of AB is the point that bisects , _or divides, the segment into Example 3 two congruent segments. If M is the midpoint of AB , then = AM = MB. So if AB = 6, then AM = 3 and MB = 3. A. G is between F and H, FG 6, and FH = 11. Find GH. 5
EXAMPLE 4 Recreation Application B. M is between N and O. Find The map shows the route for a race. NO. 27 You are 365 m from drink station R £Ç ÎÝÊÊx and 2 km from drink station S. " The first-aid station is located XRYS xÝÊ ÊÓ at the midpoint of the two drink 365 m stations. How far are you from 2 km the first-aid station? Example 4 Let your current location be X and The map shows the route for a the location of the first-aid station be Y. race. You are at X, 6000 ft from XR + RS = XS Seg. Add. Post. the first checkpoint C. The second 365 + RS = 2000 Substitute 365 for XR and 2000 for XS. checkpoint D is located at the - - midpoint between C and the end −−−−−−−−365 −−−−−365 Subtract 365 from both sides. RS = 1635 Simplify. of the race Y. The total race is 3 _ = Y is the mdpt. of RS, so RY = __1 RS. miles. How far apart are the 2 RY 817.5 2 checkpoints? 4920 feet XY = XR + RY = 365 + 817.5 = 1182.5 m Substitute 365 for XR and 817.5 for RY. 8 9 You are 1182.5 m from the first-aid station. Also available on transparency 4. What is the distance to a drink station located at the midpoint between your current location and the first-aid station? 591.25 m 1- 2 Measuring and Constructing Segments 15 INTERVENTION Questioning Strategies
EXAMPLE 2 • How are drawing and construct- ge07se_c01_0013_0019.indd 15 8/23/05 2:02:48 PMing different? How are they alike? Which is more accurate?
EXAMPLE 3 • Do you need to know that one of the points is between the other two? Why or why not?
EXAMPLE 4 • If you are given the midpoint and one endpoint of a segment, what three distances do you know?
Lesson 1-2 15
gge07te_c01_0013_0019.indde07te_c01_0013_0019.indd 1155 112/5/052/5/05 110:37:550:37:55 PPMM Construction A segment bisector is any ray, segment, or line that intersects a segment at Give each student a ruler. its midpoint. It divides the segment into two equal parts at its midpoint. These can be found in the Manipulatives Kit (MK). Use paper other than patty paper and have Construction Segment Bisector students bend the line over the ruler. Then fold the paper along the line. Fold the paper end to end to locate the midpoint.
Additional Examples _ Example 5 Draw XY on a sheet of paper. Fold the paper so that Y is on Unfold the paper. The line top of X. represented_ by the crease D is the midpoint of EF, bisects XY. Label the midpoint M. ED = 4x + 6, and DF = 7x - 9. XM = MY Find ED, DF, and EF. 26; 26; 52
Also available on transparency EXAMPLE 5 Using Midpoints to Find Lengths xÝ ÎÝÊ Ê{ −− B is the midpoint of AC , AB = 5x, and BC = 3x + 4. Find AB, BC, and AC. Step 1 Solve for x. INTERVENTION _ AB = BC B is the mdpt. of AC. Questioning Strategies 5x = 3x + 4 Substitute 5x for AB and 3x + 4 for BC. - 3x - 3x Subtract 3x from both sides. EXAMPLE 5 −−−− −−−−−− 2x = 4 Simplify. • Is it possible for x to be a negative _ 2x = _4 Divide both sides by 2. number in this type of problem? 2 2 = Support your answer with an x 2 Simplify. example. Step 2 Find AB, BC, and AC. AB = 5xBC= 3x + 4 AC = AB + BC = 5(2) = 10 = 3(2) + 4 = 10 = 10 + 10 = 20
5. S is the midpoint of RT, RS =-2x, and ST =-3x - 2. Find RS, ST, and RT. RS = 4; ST = 4; RT = 8
THINK AND DISCUSS G.CM.11, G.CN.1 _ 1. Suppose R is the midpoint of ST. Explain how SR and ST are related. 2. GET ORGANIZED Copy and complete the graphic organizer. Make a sketch and write an equation to describe each relationship.
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16 Chapter 1 Foundations for Geometry
Answers to Think and Discuss 3 −− Close 1. Since R is the mdpt. of ST , you know = = + ge07se_c01_0013_0019.indd 4 SR RT. Also, ST SR RT. By subst., 11/23/06 1:25:20 PM Summarize and INTERVENTION ST = SR + SR = 2SR. So ST is twice SR. Review the Segment Addition Postulate, Diagnose Before the Lesson 2. See p. A2. emphasizing that betweenness involves 1-2 Warm Up, TE p. 13 collinearity. Remind students that a seg- ment has exactly one midpoint. Review the Monitor During the Lesson steps used to construct a segment congru- Check It Out! Exercises, SE pp. 13–16 ent to a given segment. Questioning Strategies, TE pp. 14–16
Assess After the Lesson 1-2 Lesson Quiz, TE p. 19 Alternative Assessment, TE p. 19
16 Chapter 1
gge08te_ny_c01_0013_0019.indde08te_ny_c01_0013_0019.indd 1616 112/21/062/21/06 11:18:01:18:01 PPMM 1-2 1-2 ExercisesExercises Exercises KEYWORD: MG7 1-2
KEYWORD: MG7 Parent GUIDED PRACTICE Assignment Guide Vocabulary Apply_ the vocabulary from_ this lesson to answer each question. Assign Guided Practice exercises 1. Line bisects−− XY at −−M and divides XY into two equal parts. Name a pair of congruent segments. XM and MY as necessary. 2. __?__ is the amount of space between two points on a line. It is always expressed as a If you finished Examples 1–3 nonnegative number. (distance or midpoint) distance Basic 11–15, 20, 24, 26 SEE EXAMPLE 1 Find each length. Average 11–15, 20, 24, 26, p. 13 3. AB 3.54. BC 2.5 { Î Ó £ ä £ÓÎ{x 28–30, 35, 36, 43 Ó°x _ ΰx Advanced 11–15, 20, 24, 26,
SEE EXAMPLE 2 5. Sketch, draw, and construct a segment congruent to RS. Check students’ work. 28–30, 35, 36, 42–45 p. 14 , - If you finished Examples 1–5 SEE EXAMPLE 3 6. B is between A and C, AC = 15.8, and AB = 9.9. Find BC. 5.9 Basic 11–20, 22, 27, 28, 31, 34, 35, 37–40, 46–53 p. 15 £Ç ÎÞ * 7. Find MP. Average 11–28, 30–32, 34–41,
29 xÞÊ Ê 43–53 SEE EXAMPLE 4 8. Travel If a picnic area is located at the midpoint Advanced 12–14, 16–20, 22–30 p. 15 between Sacramento and Oakland, find the distance 2OSEVILLE even, 31, 32, 34–53 to the picnic area from the road sign. 66.5 mi 3ACRAMENTO _ /AKLAND Homework Quick Check SEE EXAMPLE 5 9. Multi-Step K is the midpoint of JL , JL = 4x - 2, Quickly check key concepts. p. 16 and JK = 7. Find x, KL, and JL. x = 4; KL = 7; JL = 14 _ Exercises: 12–14, 16, 18, 20, 10. = = - DE = EF = 1; DF = 2 E bisects DF , DE 2y, and EF 8y 3. Find DE, EF, and DF. 22, 31
PRACTICE AND PROBLEM SOLVING Algebra For Exercise 17, Independent Practice Find each length. remind students that they For See _11 _1 Exercises Example 11. DB 5 12. CD 3 È { Ó ä Ó have to substitute the 12 4 xÊÊp£ pÓ 11–12 1 { Î value of x into the expressions 13 2 + - 13. Sketch, draw, _and construct a segment twice 2x 4 and 3x 1 to find DE, EF, 14–15 3 the length of AB . Check students’ work. and DF. 16 4 14. D is between C and E, CE = 17.1, and DE = 8. Find CD. 9.1 17–18 5 Ó°xÝ Ý , Extra Practice 15. Find MN. 5 Skills Practice p. S4 xÝÊÊÎ Application Practice p. S28
16. Sports During a football game, a quarterback
standing at the 9-yard line passes the ball to a %.$:/.% receiver at the 24-yard line. The receiver then %.$:/.% runs with the ball halfway to the 50-yard line. How many total yards (passing plus running) did the team gain on the play? 28 yd _ 17. Multi-Step E is the midpoint of DF , DE = 2x + 4, and EF = 3x - 1. Find DE, EF, and DF. DE = EF = 14; DF = 28 _ 18. Q bisects PR, PQ = 3y, and PR = 42. Find y and QR. y = 7 ; QR = 21
1- 2 Measuring and Constructing Segments 17
ge07se_c01_0013_0019.indd 17 8/23/05 2:03:00 PM
KEYWORD: MG7 Resources
Lesson 1-2 17
gge07te_c01_0013_0019.indde07te_c01_0013_0019.indd 1177 112/5/052/5/05 110:38:020:38:02 PPMM Exercise 19 involves 19. This problem will prepare you for the Multi-Step Test finding distances Prep on page 34. Archaeologists at Valley Forge were eager between points. This to find what remained of the winter camp that soldiers led exercise prepares students for the by George Washington called home for several months. Multi-Step Test Prep on page 34. The diagram represents_ one of the restored log cabins. −− a. How is C related to AE? C is the mdpt. of AE . Kinesthetic To dem- b. If AC = 7 ft, EF = 2 (AC) + 2, and AB = 2(EF) - 16, onstrate the difference what are AB and EF? 16 between equal length and congruence in Exercise 24, have students place one hand over the Use the diagram for Exercises 20–23. other to show how they match for 2 _1 congruence. Then, for equality, have 20. GD = 4_ . Find GH. 9 _ _3 _3 them show 5 fingers on each hand 21. CD DF, E bisects DF, and CD = 14.2. Find EF. 7.1 for 5 = 5. 22. GH = 4x - 1, and DH = 8. Find x. 4.25 _ _ Inclusion If students are 23. GH bisects CF, CF = 2y - 2, and CD = 3y - 11. Find CD. 4 using a ruler for a straight- edge in Exercise 35, be Tell whether each statement is sometimes, always, or never true. Support each of sure that they are not using the your answers with a sketch. marks on the ruler to measure dis- 24. Two segments that have the same length must be congruent. A; tances, and that they show construc- _ 25. If M is between A and B, then M bisects AB. S; tion arcs. AMBAMB 26. If Y is between X and Z, then X, Y, and Z are collinear. A XYZ _ Multiple 27. AM MB is 27. /////ERROR ANALYSIS///// Below are two statements about the midpoint of AB. Representations For incorrect. The Which is incorrect? Explain the error. Exercise 40, encourage statement should students to draw and label a dia- be−− written−− as gram with all the given information AM MB , not ÊÃÊÌ iÊ`«Ì°ÊvÊ ° ÊÃÊÌ iÊ`«Ì°ÊvÊ ° / iÀivÀiÊ ÊɂÊ ° / iÀivÀiÊ Ê Ê ° before attempting to solve the as 2 distances problem. that are . 28. Carpentry A carpenter has a wooden dowel that is 72 cm long. She wants to cut it into two pieces so that one piece is 5 times as long as the other. What are the lengths of the two pieces? 60 cm; 12 cm 6.5; -1.5 29. The coordinate of M is 2.5, and MN = 4. What are the possible coordinates for N? 30. Draw three collinear points where E is between D and F. Then write an equation 30. DEF using these points and the Segment Addition Postulate. Possible−− −− answer:−− DE + EF = DF Suppose S is between R and T. Use the Segment Addition Postulate to solve for each variable. 31. RS = 7y - 4 32. RS = 3x + 1 33. RS = 2z + 6 ST = y + 5 ST = _ 1x + 3 ST = 4z - 3 2 RT = 28 3.375 RT = 18 4 RT = 5z + 12 9 34. Write About It In the diagram, B is not
between A and C. Explain. B is not between A and C, because A, B, and C are not collinear. 35. Construction Use a compass and straightedge to construct a segment 1-2 PRACTICE A whose length is AB + CD. 1-2 PRACTICE C Check students’ constructions.
*À>VÌViÊ ,%33/. 1-2 PRACTICE B £Ó -EASURING AND #ONSTRUCTING 3EGMENTS 18 Chapter 1 Foundations for Geometry $RAW YOUR ANSWER IN THE SPACE PROVIDED ? ? 5SE A COMPASS AND STRAIGHTEDGE TO CONSTRUCT 89 CONGRUENT TO 56
56 ,%33/. ,i>`}Ê-ÌÀ>Ìi}ià ,%33/. ,iÌi>V £Ó 5NDERSTANDING 6OCABULARY1-2 READING STRATEGIES £Ó -EASURING AND #ONSTRUCTING 3EGMENTS1-2 RETEACH 8 9 #ONSIDER THE FOLLOWING TERMS 4HE DISTANCE BETWEEN ANY TWO POINTS IS THE LENGTH OF THE SEGMENT THAT CONNECTS THEM CONGRUENT SEGMENTSSEGMENTS OF 4HESE SEGMENTS ARE %& '(* THE SAME LENGTH NOT CONGRUENT &IND THE COORDINATE OF EACH POINT " * " % $ # ' CENTIMETERS CM $ ! & ? # ( 4HE DISTANCE BETWEEN % AND * IS %* THE LENGTH OF %* 4O FIND THE DISTANCE SUBTRACT ge07se_c01_0013_0019.indd 18 THE NUMBERS CORRESPONDING TO THE POINTS AND THEN TAKE THE ABSOLUTE VALUE 8/23/05 2:03:05 PM ge07se_c01_0013_0019.indd 19 8/23/05 2:03:11 PM $ # % MIDPOINTPOINT AT THE EXACT CENTER OF A SEGMENT %* : : &IND EACH LENGTH MIDPOINT :: BISECTDIVIDING INTO TWO CONGRUENT PARTS CM "% $" %# SEGMENT BISECTORANY RAY SEGMENT OR LINE THAT INTERSECTS A SEGMENT AT ITS MIDPOINT 5SE THE FIGURE ABOVE TO FIND EACH LENGTH %' %& &( &OR %XERCISES n ( IS BETWEEN ) AND * SEGMENT SEGMENT SEGMENT BISECTED BISECTED BISECTED () AND (* &IND )* BY RAY BY SEGMENT BY LINE CM CM CM *) AND )( &IND (* ? 5SE THE FIGURE FOR %XERCISES n ( IS THE MIDPOINT OF )* AND )( &IND (* ! " X ? ? .AME THE CONGRUENT SEGMENTS 0 1 2 /N 02 1 IS BETWEEN 0 AND 2 )F 02 ( IS THE MIDPOINT OF )* AND )* &IND )( IN THIS FIGURE % 01 12 02 WE CAN FIND 12
# $ 01 12 02 &IND THE MEASUREMENTS ? ? ? ? ? ? ? ? ? ? X !" #$ "# !$ !# "$ $% "% #% AND !% X +-X , X X ? 12 .AME THE MIDPOINT OF "# 0OINT % ? Y Z &IND ,- ? *+ , !"# .AME THE SEGMENT BISECTOR OF "# !$ ! POLE VAULTER USES A FOOT LONG POLE 3HE GRIPS THE POLE SO THAT THE SEGMENT BELOW HER LEFT HAND IS TWICE THE LENGTH OF THE SEGMENT ABOVE HER LEFT HAND (ER RIGHT 5SE THE FIGURE FOR %XERCISES n &IND *+ &IND "# HAND GRIPS THE POLE FEET ABOVE HER LEFT HAND (OW FAR .AME THE CONGRUENT SEGMENTS 8 N AA UP THE POLE IS HER RIGHT HAND FT IN THIS FIGURE 346 78 9 N
: 9 ? ? ? ? 0 &IND 36 &IND 89 89 8: :0 09 Y ? $%& 345 .AME THE MIDPOINT OF :9 0OINT 0 X Y ? ? 80 &IND $& &IND 34 18 Chapter 1 .AME THE SEGMENT BISECTOR OF :9
gge07te_c01_0013_0019.indde07te_c01_0013_0019.indd 1188 112/5/052/5/05 110:38:060:38:06 PPMM " " Ê ,,", 36. Q is between P and R. S is between Q and R, and R_ is between Q and T. PT = 34, ,/ QR = 8, and PQ = SQ = SR. What is the length of RT? 9 10 18 22 In Exercise 36, students might inter- _ _ pret the word between to mean at = 37. C is the midpoint of AD _. B is the midpoint of AC . BC 12. the midpoint. Remind them that What is the length of AD ? between does not imply midway, 12 24 36 48 and suggest that they draw a dia- _ _ gram to help them sort out the 38. Which expression correctly states that XY is congruent to VW ? _ _ _ _ information. XY VW XY VW XY = VW XY = VW
39. A, B, C, D, and E are collinear points. AE = 34, BD = 16, _ and AB = BC = CD. What is the length of CE? Journal 10 16 18 24 Have students define the midpoint of a segment in their own words CHALLENGE AND EXTEND and illustrate what it means for 40. HJ is twice JK. J is between H and K. If HJ = 4x and HK = 78, find JK. 26 one point to be between two other points. 41. A, D, N, and X are collinear points. D is between N and A. NA + AX = NX. Draw a diagram that represents this information. Sports XA D N Sports Use the following information for Exercises 42 and 43. The table shows regulation distances between hurdles in women’s and men’s races. Have students create and solve a In both the women’s and men’s events, the race consists of a straight track with real-world problem involving alge- 10 equally spaced hurdles. braic expressions that illustrate Distance from Distance Distance from distance and midpoint. Ask them to Distance of Start to First Between Last Hurdle include a diagram and explain their Event Race Hurdle Hurdles to Finish problem. Joanna Hayes, of the Women’s 100 m 13.00 m 8.50 m United States, clears a hurdle on her way to Men’s 110 m 13.72 m 9.14 m winning the gold medal in the women’s 100 m 42. Find the distance from the last hurdle to the finish line for the women’s race. 10.5 m 1-2 hurdles during the 2004 Olympic Games. 43. Find the distance from the last hurdle to the finish line for the men’s race. 14.02 m 44. Critical Thinking Given that J, K, and L are collinear and that K is between 1. M is between N and O, J and L, is it possible that JK = JL? If so, draw an example. If not, explain. MO = 15, and MN = 7.6. JK cannot be equal to JL because JK + KL = JL and KL ≠ 0. Find NO. 22.6 −− SPIRAL REVIEW 2. S is the midpoint of TV, TS = Evaluate each expression. (Previous course) 4x - 7, and SV = 5x - 15. 45. 20 - 8 1246. -9 + 23 14 47. - 4 - 27 -23 Find TS, SV, and TV. 25, 25, 50 Simplify each expression. (Previous course) 3. Sketch, draw, and construct a 48. 8a - 3 (4 + a) - 10 5a - 22 49. x + 2 (5 - 2x) - (4 + 5x) -8x + 6 −− segment congruent to CD. Use the figure to name each of the following. (Lesson 1-1) 50. two lines that contain B AB , CB −− −− 51. two segments containing D AD, BD Check students’ constructions. 52. three collinear points A, B, D −− −− 4. LH bisects GK at M. GM = 2x 53. a ray with endpoint C CB + 6, and GK = 24. Find x. 3 1- 2 Measuring and Constructing Segments 19
*ÀLiÊ-Û} >i}i ,%33/. 1-2 PROBLEM SOLVING ,%33/. 1-2 CHALLENGE £Ó -EASURING AND #ONSTRUCTING 3EGMENTS £Ó #ENTER OF -ASS &OR %XERCISES AND USE THE FIGURE )T SHOWS THE TOP VIEW OF A STAGE THAT HAS 4HE CENTER OF MASS ALSO CALLED THE CENTER OF GRAVITY IS THE POINT WHERE THE WEIGHT OF THREE TRAP DOORS AN OBJECT IS FOCUSED )T IS THE POINT OF BALANCE OF AN OBJECT 1 STAGE )N A TRIANGLE THE CENTER OF MASS IS CALLED THE CENTROID 4HE CENTROID IS FOUND BY DRAWING THE THREE MEDIANS OF X FT X FT X THE TRIANGLE ! MEDIAN OF A TRIANGLE IS A SEGMENT THAT 4 JOINS A VERTEX AND THE MIDPOINT OF THE SIDE OPPOSITE !" ge07se_c01_0013_0019.indd 18 8/23/05 2:03:05 PM ge07se_c01_0013_0019.indd 7 TRAP THE VERTEX 4HE CENTROID OF THE TRIANGLE IS THE INTERSECTION 11/23/06 1:25:34 PM DOOR 2 3 POINT OF THE MEDIANS FT 24 is a median of N123 -ATERIALS NEEDED RULER 4HE TOTAL LENGTH OF THE STAGE IS FEET !N ACTOR STARTS AT POINT ! WALKS ACROSS 5SE A RULER TO DRAW A TRIANGLE WITH VERTICES * + AND , $RAW AND LABEL THE )F THE TRAP DOORS ARE CENTERED ACROSS THE THE STAGE AND THEN STOPS AT POINT " THREE MIDPOINTS ! " AND # $RAW THE THREE MEDIANS ,ABEL THE CENTROID - STAGE WHAT IS THE DISTANCE FROM THE LEFT BEFORE DISAPPEARING THROUGH THE TRAP DOOR 5. Tell whether the statement SIDE OF THE STAGE TO THE FIRST TRAP DOOR (OW FAR DID HE WALK ACROSS THE STAGE * ?? FT FT below is sometimes, always, !NNA IS FEET HIGH ON A ROCK CLIMBING *AMILLA HAS A PIECE OF RIBBON THAT IS # ! or never true. Support your WALL 3HE DESCENDS TO THE FOOT MARK CENTIMETERS LONG &OR HER SCRAPBOOK RESTS AND THEN CLIMBS DOWN UNTIL SHE SHE CUTS IT INTO TWO PIECES SO THAT ONE - REACHES HER FRIEND WHO IS FEET FROM PIECE IS TIMES AS LONG AS THE OTHER answer with a sketch. THE GROUND (OW MANY FEET HAS !NNA 7HAT ARE THE LENGTHS OF THE PIECES ,+ −− DESCENDED " CM AND CM If M is the midpoint of KL, FT 5SE A RULER TO FIND THE LENGTH OF 5SE YOUR RESULTS FROM %XERCISE TO MAKE #HOOSE THE BEST ANSWER EACH MEDIAN 4HEN FIND THE DISTANCE A CONJECTURE ABOUT HOW THE DISTANCE FROM FROM EACH VERTEX TO THE CENTROID A VERTEX TO THE CENTROID IS RELATED TO THE then M, K, and L are collinear. *ORDAN WANTS TO ADJUST THE SHELVES IN HIS BOOKCASE SO THAT X 2OUND YOUR ANSWERS TO THE NEAREST DISTANCE FROM THAT VERTEX TO THE MIDPOINT OF TOP SHELF THERE IS TWICE AS MUCH SPACE ON THE BOTTOM SHELF AS ON THE TENTH OF A CENTIMETER THE OPPOSITE SIDE TOP SHELF AND ONE AND A HALF TIMES MORE SPACE ON THE A X ?? MIDDLE SHELF AS ON THE TOP SHELF )F THE TOTAL HEIGHT OF THE 4HE CENTROID IS OF THE DISTANCE MIDDLE SHELF 3EE STUDENTS WORK BOOKCASE IS METER HOW MUCH SPACE IS BETWEEN THE TOP AND MIDDLE SHELVES FROM EACH VERTEX TO THE MIDPOINT ! M # M X " M $ M BOTTOM SHELF OF THE OPPOSITE SIDE 0OINTS 7 8 AND 9 ARE THE MIDPOINTS OF )N A ROWING RACE THE DISTANCE BETWEEN /N A SUBWAY ROUTE STATION # IS LOCATED ? ? ? % THE TEAMS IN FIRST AND SECOND PLACE IS AT THE MIDPOINT BETWEEN STATIONS ! %& &' AND '% RESPECTIVELY 5SE THE METERS 4HE DISTANCE BETWEEN THE TEAMS AND $ 3TATION " IS LOCATED AT THE TRIANGLE TO VERIFY THE CONJECTURE THAT YOU Also available on transparency IN SECOND AND THIRD PLACE IS ONE THIRD THAT MIDPOINT BETWEEN STATIONS ! AND # )F MADE IN %XERCISE 9 7 DISTANCE (OW MUCH FARTHER AHEAD IS THE THE DISTANCE BETWEEN STATIONS ! AND $ %. ?? %8 &. ?? &9 TEAM IN FIRST PLACE THAN THE TEAM IN THIRD IS KILOMETERS WHAT IS THE DISTANCE . & M ( M BETWEEN STATIONS " AND $ ?? '. '7 '& ' M * M ! KM # KM 8 " KM $ KM Lesson 1-2 19
gge08te_ny_c01_0013_0019.indde08te_ny_c01_0013_0019.indd 1919 112/21/062/21/06 11:18:19:18:19 PPMM 1-3 Organizer 1-3 Measuring and Pacing: Traditional 1 day Constructing Angles __1 Block 2 day Objectives: Name and classify Objectives Who uses this? angles. Name and classify angles. Surveyors use angles to help them Measure and construct angles and Measure and construct measure and map the earth’s surface. angles and angle (See Exercise 27.) angle bisectors. bisectors. A transit is a tool for measuring angles. Geometry Lab Vocabulary It consists of a telescope that swivels horizontally angle In Geometry Lab Activities and vertically. Using a transit, a surveyor vertex interior of an angle can measure the angle formed by his or @
given angle, using a straightedge 1. Draw AB and AC , where A, and compass, and justify the You cannot name an angle just by its vertex if the point is the vertex of more B, and C are noncollinear. construction. Also, G.PS.5, G.RP.2. than one angle. In this case, you must use all three points to name the angle, Check students’ drawings and the middle point is always the vertex.
2. Draw opposite rays DE and DF . Check students’ EXAMPLE 1 Naming Angles + drawings A surveyor recorded the angles formed by a transit (point T) and three distant , Solve each equation. points, Q, R, and S. Name three of / £ Ó 3. 2x + 3 + x – 4 + 3x - 5 = the angles. 180 31 ∠QTR, ∠QTS, and ∠RTS - 4. 5x + 2 = 8x - 10 4
Also available on transparency 1. Write the different ways you can name the angles in the diagram. ∠RTQ, ∠T, ∠STR, ∠1, ∠2
The measure of an angle is usually given in degrees. Since there are 360° in ___1 a circle, one degree is 360 of a circle. When you use a protractor to measure angles, you are applying the following postulate. Q: What do you call people who are Postulate 1-3-1 Protractor Postulate in favor of tractors? ‹___› ‹___› A: Protractors! Given AB and a point O on AB, all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180. New York Performance Indicators 20 Chapter 1 Foundations for Geometry
Geometry G.G.17 Construct a bisector of a given angle, using a straightedge and 1 Introduce compass, and justify the construction Process ge07se_c01_0020_0027.inddEXPLORATION 1 Motivate 11/23/06 1:26:04 PM G.PS.5 Choose an effective approach £Î i>ÃÕÀ}Ê>`Ê to solve a problem from a variety of ÃÌÀÕVÌ}Ê}ià Point out examples of different types of angles in 5SE RECTANGULAR PIECES OF PAPER AND A PROTRACTOR FOR THIS strategies (numeric, graphic, algebraic) %XPLORATION the classroom, such as the corner of a piece of &OLD THE PAPER AND MAKE A &OLD DOWN paper for a right angle, and open scissors to show G.RP.2 Recognize and verify, where CREASE 4HEN UNFOLD THE PAPER THE TOP OF THE PAPER SO THE an acute angle or an obtuse angle. Have students appropriate, geometric relationships EDGE ALIGNS WITH THE get in groups of three and use a long strand of of perpendicularity, parallelism, CREASE congruence, and similarity, using yarn and three index cards, each labeled with a algebraic strategies &OLD UP THE 5NFOLD THE different letter, to model for the class the different BOTTOM OF PAPER AND THE PAPER LABEL THE angles that can be formed. SO THE EDGE ANGLES AS ALIGNS WITH SHOW THE CREASE Explorations and answers are provided in the -EASURE !"# !"$ AND $"# 2ECORD THE RESULTS 2EPEAT 3TEPS THROUGH WITH A NEW PIECE OF PAPER Explorations binder.
/ Ê Ê - 1-- $ESCRIBE WHAT YOU NOTICE ABOUT THE MEASURE OF !"# $ISCUSS THE RELATIONSHIP AMONG THE MEASURES OF !"# 20 Chapter 1 !"$ AND $"#
gge08te_ny_c01_0020_0027.indde08te_ny_c01_0020_0027.indd 2020 112/21/062/21/06 112:38:252:38:25 PPMM " " Ê ,,", Using a Protractor ,/ Most protractors have two sets / of numbers around the edge. Remind students to place the center When I measure an angle and mark of the protractor (MK) on the need to know which number to vertex and to align one side of the use, I first ask myself whether the angle with the 0º mark. Some pro- angle is acute, right, or obtuse. tractors have their zero line on the ∠ , For example, RST looks like it - bottom edge, while others have it is obtuse, so I know its measure higher. must be 110°, not 70°. José Muñoz Lincoln High School
Additional Examples
1 You can use the Protractor Postulate Example
to help you classify angles by their A surveyor recorded the angles measure. The measure of an angle is the formed by a transit (point A) and absolute value of the difference of the ` three distant points, B, C, and D. real numbers that the rays correspond V with on a protractor. If OC corresponds Name three of the angles. with c and OD corresponds with d, " m∠DOC = d - c or c - d .
Types of Angles
Acute Angle Right Angle Obtuse Angle Straight Angle Possible answer: ∠BAC, ∠CAD, ∠BAD
Example 2 Measures greater Measures 90° Measures greater Formed by two than 0° and less than 90° and less opposite rays and Find the measure of each angle. than 90° than 180° meaures 180° Then classify each as acute, right, or obtuse. xä EXAMPLE 2 Measuring and Classifying Angles £Îä £xä Îä Find the measure of each angle. Then 7 classify each as acute, right, or obtuse. <
A ∠AOD 9 8 6 m∠AOD = 165°
∠AOD is obtuse. B ∠COD ∠ " A. WXV 30°; acute m∠COD = 165 - 75 = 90° ∠ ∠COD is a right angle. B. ZXW 100°; obtuse
Also available on transparency Use the diagram to find the measure of each angle. Then classify each as acute, right, or obtuse. 2a. ∠BOA 2b. ∠DOB 2c. ∠EOC 40°; acute 125°; obtuse 105°; obtuse INTERVENTION Questioning Strategies 1- 3 Measuring and Constructing Angles 21
EXAMPLE 1 2 • Does the order of the letters matter Teach when you name an angle? Explain. ge07se_c01_0020_0027.indd 21 12/3/05 11:45:48 AM Guided Instruction EXAMPLE 2 Discuss how to name angles. The vertex is Through Visual Cues • Does the order in which the real always the middle letter, or is used as the Use pictures from magazines of angles numbers are subtracted matter? single letter name. Show students how to of different sizes. Ask students to identify Explain. use a protractor to measure angles and the type of angle and estimate the mea- • Do the lengths of the sides of an how to write their measure. Explain that sure. Then have students measure the ∠ ∠ angle affect its measure? the measure of A is written as m A. angles with a protractor. If protractors are Describe each step of the constructions. not available, they can use index cards or Emphasize that congruent angles have origami paper. The edge is already a 90° equal measures and that an angle bisector angle, and anything greater would be an divides an angle into two congruent angles. obtuse angle. A half-fold forms a 45° angle, a tri-fold approximately 30°, and so on. The pictures can be posted by classification and used for reference. Lesson 1-3 21
gge07te_c01_0020_0027.indde07te_c01_0020_0027.indd 2211 112/20/052/20/05 111:10:561:10:56 AAMM Visual Remind students to Congruent angles are angles that have the same show all arcs in their con- measure. In the diagram, m∠ABC = m∠DEF, ÀV ∠ ∠ struction. In Step 4, they so you can write ABC DEF. This is read as >ÀÃ should use the same compass span “angle ABC is congruent to angle DEF.” Arc marks are used to show that the two angles are congruent. BC when drawing the arcs. Have students practice constructing both acute and obtuse angles. Construction Congruent Angle
Construct an angle congruent to ∠A. Additional Examples Example 3 m∠DEG = 115°, and m∠DEF = 48°. Find m∠FEG. 67°
Use a Place the compass Using the same Place the compass Use a straightedge to point at A and compass setting, point at B and open straightedge to draw a ray with draw an arc that place the compass it to the distance BC. draw DF . endpoint D. intersects both sides point at D and draw Place the point of ∠ ∠D ∠A Also available on transparency of A. Label the an arc that intersects the compass at E and intersection points the ray. Label the draw an arc. Label B and C. intersection E. its intersection with the first arc F. INTERVENTION Questioning Strategies The Angle Addition Postulate is very similar to the Segment Addition Postulate that you learned in the previous lesson. EXAMPLE 3 • What is actually being added in Postulate 1-3-2 Angle Addition Postulate the example? Are you adding the If S is in the interior of ∠PQR, then - angles or the measures of the m∠PQS + m∠SQR = m∠PQR. angles? , (∠ Add. Post.) * + Reading Math Review the different meanings of m∠ A = m∠B and ∠ A ∠B. Have students EXAMPLE 3 Using the Angle Addition Postulate ENGLISH practice reading the sym- LANGUAGE m∠ABD = 37° and m∠ABC = 84°. Find m∠DBC. LEARNERS bols aloud. m∠ABC = m∠ABD + m∠DBC ∠ Add. Post.
84° = 37° + m∠DBC Substitute the given values. - - Subtract 37 from both sides. −−−−37 −−−−−−−−−−−37 Simplify. 47° = m∠DBC
9 3. m∠XWZ= 121° and m∠XWY = 59°. 8 Find m∠YWZ. 62°
7 <
22 Chapter 1 Foundations for Geometry
ge07se_c01_0020_0027.indd 22 12/3/05 11:45:51 AM ge07se_c01_0020_0027.indd 23 12/3/05 11:45:54 AM
22 Chapter 1
gge07te_c01_0020_0027.indde07te_c01_0020_0027.indd 2222 66/14/06/14/06 22:07:52:07:52 PPMM An angle bisector is a ray that divides an angle into two " " Ê ,,", congruent angles. JK bisects ∠LJM; thus ∠LJK ∠KJM. ,/ In the constructions, if students’ compass settings are not tightly Construction Angle Bisector fixed, the compass setting may change without students being Construct the bisector of ∠A. aware. Stress to students that they must check compass tightness and keep the same fixed compass setting for accuracy. Reading Math Explain the bi is a prefix meaning
“two” and sect ENGLISH means “to cut,” as LANGUAGE LEARNERS into sections.
___› Place the point of the compass at A Without changing the compass Use a straightedge to draw AD. ___› and draw an arc. Label its points of setting, draw intersecting AD bisects ∠A. intersection with ∠A as B and C. arcs from B and C. Label the intersection of the arcs as D. Additional Examples
Example 4
KM bisects ∠ JKL, m∠ JKM = (4x EXAMPLE 4 Finding the Measure of an Angle + 6)°, and m∠MKL = (7x - 12)°. ∠ ∠ = ( + ) BD bisects ABC, m ABD 6x 3 °, ∠ ∠ = ( - )° ∠ Find m JKM. 30° and m DBC 8x 7 . Find m ABD. Step 1 Find x. ∠ = ∠ mABD m DBC Def. of ∠ bisector ( + ) = ( - ) 6x 3 ° 8x 7 ° Substitute the given values. + + −−−−−−−7 −−−−−−7 Add 7 to both sides. 6x + 10 = 8x Simplify. Also available on transparency - - −−−−−−−6x −−−−−−6x Subtract 6x from both sides. 10 = 2x Simplify. 10 2x _ = _ Divide both sides by 2. 2 2 INTERVENTION = 5 x Simplify. Questioning Strategies Step 2 Find m∠ABD. m∠ABD = 6x + 3 EXAMPLE 4 =( ) + 6 5 3 Substitute 5 for x. • When do you set the angle = 33° Simplify. measures equal to solve for the variable? Find the measure of each angle. • How can you solve for the variable ___› 4a. QS bisects ∠PQR, m∠PQS = ( 5y - 1)°, and if you are given the measures of m∠PQR = ( 8y + 12)°. Find m∠PQS. 34° the bisected angle and one of the __› 4b. JK bisects ∠LJM, m∠LJK = (-10x + 3)°, and congruent angles? m∠KJM = (-x + 21)°. Find m∠LJM. 46°
1- 3 Measuring and Constructing Angles 23
3 Close ge07se_c01_0020_0027.indd 22 12/3/05 11:45:51 AM ge07se_c01_0020_0027.indd 23 12/3/05 11:45:54 AM Summarize and INTERVENTION Review with students the parts of an angle, Diagnose Before the Lesson the names of the types of angles, and 1-3 Warm Up, TE p. 20 how to measure angles. Review the Angle Addition Postulate and how it relates to Monitor During the Lesson the bisector of an angle. Check It Out! Exercises, SE pp. 20–23 Questioning Strategies, TE pp. 21–23
Assess After the Lesson 1-3 Lesson Quiz, TE p. 27 Alternative Assessment, TE p. 27
Lesson 1-3 23
gge07te_c01_0020_0027.indde07te_c01_0020_0027.indd 2233 66/14/06/14/06 22:08:43:08:43 PPMM Answers to Think and Discuss G.RP.5, G.CM.11, G.R.1 1. Two with the same measure THINK AND DISCUSS are . All rt. measure 90°, so 1. Explain why any two right angles are congruent. any 2 rt. are . ___› 2. BD bisects ∠ABC. How are m∠ABC, m∠ABD, and m∠DBC related? 2. m∠ ABD = m∠DBC = __1 m∠ ABC 2 3. GET ORGANIZED Copy and >}À> i>ÃÕÀi >i 3. See p. A2. complete the graphic organizer. In the cells sketch, measure, VÕÌiÊ}i and name an example of each ,} ÌÊ}i angle type. "LÌÕÃiÊ}i -ÌÀ>} ÌÊ}i
1-3 1-3 ExercisesExercises Exercises KEYWORD: MG7 1-3
KEYWORD: MG7 Parent Assignment Guide GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. Assign Guided Practice exercises 3. ∠AOB, ∠BOA, 1. ∠A is an acute angle. ∠O is an obtuse angle. ∠R is a as necessary. or ∠1; ∠BOC, right angle. Put ∠A, ∠O, and ∠R in order from least ∠ ∠ ∠ ∠COB, or ∠2; to greatest by measure. A, R, O If you finished Examples 1–2 ∠ ∠ AOC or COA 2. ∠ Basic 11–14, 19–26, 33–35 Which point is the vertex of BCD? Which rays form the sides of ∠BCD? C ; CB , CD Average 11–14, 19–27, 32–36 £Ó Advanced 11–14, 19–27, 32–36 SEE EXAMPLE 1 3. Music Musicians use a metronome to keep time as p. 20 they play. The metronome’s needle swings back and If you finished Examples 1–4 forth in a fixed amount of time. Name all of the " Basic 11–26, 29–31, 33–35, angles in the diagram. 38, 41–45, 51–58 SEE EXAMPLE 2 Use the protractor to find the measure of each / Average 11–37, 39–46, 48–58 1 p. 21 angle. Then classify each as acute, right, or obtuse. Advanced 11, 12–22 even, 23, 4. ∠VXW 15°; acute 26–58 5. ∠TXW 105°; obtuse - 6 Homework Quick Check 6. ∠RXU 110°; obtuse Quickly check key concepts. ,7 8 Exercises: 11, 12, 16, 18, 20, 26, 30 SEE EXAMPLE 3 L is in the interior of ∠JKM. Find each of p. 22 the following. 7. m∠JKM if m∠JKL = 42° and m∠LKM = 28° 70° 8. m∠LKM if m∠JKL = 56.4° and m∠JKM = 82.5° 26.1°
SEE EXAMPLE 4 Multi-Step BD bisects ∠ABC. Find each of the following. p. 23 9. m∠ABD if m∠ABD = (6x + 4)° and m∠DBC = (8x - 4)° 28° 10. m∠ABC if m∠ABD = (5y - 3)° and m∠DBC = (3y + 15)° 84°
24 Chapter 1 Foundations for Geometry
,%33/. *À>VÌViÊ ,%33/. *À>VÌViÊ £Î -EASURING AND #ONSTRUCTING !NGLES1-3 PRACTICE A £Î -EASURING AND #ONSTRUCTING !NGLES1-3 PRACTICE B 5SE THE FIGURE FOR %XERCISES n $RAW YOUR ANSWER ON THE FIGURE 1 5SE A COMPASS AND STRAIGHTEDGE TO !N ANGLE IS A FIGURE FORMED BY TWO RAYS WITH A COMMON ???k % CONSTRUCT ANGLE BISECTOR $' VERTEX ' ENDPOINT CALLED THE 3 .AME THE TWO RAYS THAT FORM 0 ??? ??? $ & k k 0 2 ge07se_c01_0020_0027.indd 5 01 AND 02 # 11/23/06 1:26:18 PM 4 .AME EIGHT DIFFERENT ANGLES IN THE FIGURE 5SE THE ANGLE SYMBOL AND THREE LETTERS TO NAME 0 IN TWO WAYS ! # !"# !"$ !$" " 102 AND 201 !$# #"$ AND #$" .AME A POINT THAT IS IN THE INTERIOR OF 0 ! $ POINT 3 9 # &IND THE MEASURE OF EACH ANGLE " AND CLASSIFY EACH AS ACUTE #OMPLETE THE STATEMENT 8 RIGHT OBTUSE OR STRAIGHT : ! TOOL USED TO MEASURE AND DRAW ANGLES
IS CALLED A PROTRACTOR 7 &IND THE MEASURE OF EACH ANGLE 4HEN TELL WHETHER $ % ! EACH IS ACUTE RIGHT OBTUSE OR STRAIGHT 97: 87: 978
#%! !%" $%! RIGHT OBTUSE ACUTE RIGHT ACUTE STRAIGHT 4 IS IN THE INTERIOR OF 012 &IND EACH OF THE FOLLOWING
#OMPLETE THE ANGLE M014 IF M012 AND M214 5SE A COMPASS AND STRAIGHTEDGE TO FINISH CONSTRUCTING )(* CONGRUENT TO -,. M012 IF M012 X M214 X AND M014 X ?? k M012 IF 14 BISECTS 012 M214 X AND M014 X . * ,ONGITUDE IS A MEASUREMENT OF POSITION AROUND THE EQUATOR OF THE %ARTH ,ONGITUDE IS MEASURED IN DEGREES MINUTES AND SECONDS %ACH DEGREE CONTAINS MINUTES AND EACH MINUTE CONTAINS , - ( ) ( ) SECONDS -INUTES ARE INDICATED BY THE SYMBOL AND SECONDS ARE INDICATED BY THE SYMBOL 7ILLIAMSBURG 6! IS LOCATED AT KEYWORD: MG7 Resources -ARC DOESNT THINK THAT THE ANGLE OF THE FRONT SEAT 2OANOKE 6! IS LOCATED AT &IND THE 9 IN HIS MOMS CAR IS VERY COOL SO HE TILTS THE SEAT DIFFERENCE OF THEIR LONGITUDES IN DEGREES MINUTES AND SECONDS BACK M:79 AND M978 8 4O CONVERT MINUTES AND SECONDS INTO DECIMAL PARTS OF A DEGREE &IND M:78 DIVIDE THE NUMBER OF MINUTES BY AND THE NUMBER OF SECONDS BY 4HEN ADD THE NUMBERS TOGETHER 7RITE THE LOCATION OF 7 2OANOKE 6! AS A DECIMAL TO THE NEAREST THOUSANDTHS OF A DEGREE 24 Chapter 1 :
gge08te_ny_c01_0020_0027.indde08te_ny_c01_0020_0027.indd 2424 112/21/062/21/06 11:32:09:32:09 PPMM PRACTICE AND PROBLEM SOLVING " " Ê ,,", Independent Practice 11. Physics Pendulum clocks have been used since ,/ For See 1656 to keep time. The pendulum swings back Exercises Example and forth once or twice per second. Name all of 11 1 In Exercise 32, students may multi- the angles in the diagram. ply the wrong angle by 4. They can 12–14 2 ∠1 or ∠JMK; ∠2 or ∠LMK; ∠M or ∠JML £Ó 15–16 3 avoid making this error by making a ∠ 17–18 4 drawing of PRS and Q.
Extra Practice Visual For Exercises 4–6 Skills Practice p. S4 Use the protractor to find the measure of each angle. and 12–14, show students Application Practice p. S28 Then classify each as acute, right, or obtuse. how to extend the rays 12. ∠CGE 13. ∠BGD 14. ∠AGB using index cards or sticky notes. 90°; rt. 93°; obtuse 20°; acute
T is in the interior of ∠RSU. Find each of Inclusion For Exercises the following. 12–14, use acetate or trac- 15. m∠RSU if m∠RST = 38° and ing paper to draw individ- m∠TSU = 28.6° 66.6° ual angles, and then measure with 16. m∠RST if m∠TSU = 46.7° and m∠RSU = 83.5° a protractor on the acetate sheet. 36.8° Color overlays also help students Multi-Step SP bisects ∠RST. Find each of the following. see the angles more clearly. 17. m∠RST if m∠RSP= (3x - 2)° and m∠PST = (9x - 26)° 20° Algebra For Exercise 18, 18. m∠RSP if m∠RST = __5y° and m∠PST = (y + 5)° 25° 2 have students multiply both sides of the equation Estimation Use the following information for Exercises 19–22. by 2 to turn the fraction into a whole Assume the corner of a sheet of paper is a right angle. number. Use the corner to estimate the measure and classify each angle in the diagram. 19. ∠BOA acute20. ∠COA rt. Answers
21. ∠EOD acute22. ∠EOB obtuse 28. First construct the bisector of the " given ∠. Then choose one of the Use a protractor to draw an angle with each of the following measures. smaller that was constructed 23–26. Check 23. 33° 24. 142° 25. 90° 26. 168° and construct its bisector. The 1 students’ drawings. resulting will have __ the mea- 27. Surveying A surveyor at point S discovers 4 sure of the original ∠. that the angle between peaks A and B is 3 times as large as the angle between peaks B and C. The surveyor knows that ∠ASC is a right angle. Find m∠ASB and m∠BSC. 67.5°; 22.5° -
28. Math History As far back as the 5th century B.C., mathematicians have been fascinated by the problem of trisecting an angle. It is possible to construct an angle __1 with 4 the measure of a given angle. Explain how to do this.
Find the value of x. 1 29. m∠AOC = 7x - 2, m∠DOC = 2x + 8, m∠EOD = 27 16_ 3 30. m∠AOB = 4x - 2, m∠BOC = 5x + 10, m∠COD = 3x - 8 10 31. m∠AOB = 6x + 5, m∠BOC = 4x - 2, m∠AOC = 8x + 21 9 32. Multi-Step Q is in the interior of right ∠PRS. If m∠PRQ " is 4 times as large as m∠QRS, what is m∠PRQ? 72°
1- 3 Measuring and Constructing Angles 25
,%33/. *À>VÌViÊ £Î -EASURING AND #ONSTRUCTING !NGLES1-3 PRACTICE C
5SE THE FIGURE FOR %XERCISES n $
"!% .AME THE OBTUSE ANGLE # % "!# AND $!% .AME TWO ACUTE ANGLES "
"!$ AND #!% .AME TWO RIGHT ANGLES ge07se_c01_0020_0027.indd 25 ! 12/19/05 5:26:58 PM +EISHA HAS A STRAIGHTEDGE AND A COMPASS BUT NO PROTRACTOR 7HAT KIND OF ANGLE CAN +EISHA DRAW EXACTLY WITH ONLY A STRAIGHTEDGE
STRAIGHT ANGLE $ESCRIBE HOW +EISHA CAN DRAW AN EXACT ANGLE USING ONLY A STRAIGHTEDGE AND A COMPASS
&IRST +EISHA CAN DRAW A STRAIGHT ANGLE 3HE CAN THEN BISECT
THE STRAIGHT ANGLE TO MAKE TWO RIGHT ANGLES +EISHA CAN THEN
BISECT THE RIGHT ANGLES TO MAKE A ANGLE
$RAW YOUR ANSWER IN THE SPACE PROVIDED #ONSTRUCT A ANGLE USING ONLY A STRAIGHTEDGE AND A COMPASS !N ACUTE ANGLE MEASURES X 7RITE AN INEQUALITY TO DESCRIBE THE RANGE OF ALL POSSIBLE VALUES OF X X
5SE THE CHART TO COMPLETE THE EXERCISE $IVE $IFFICULTY )N DIVING A SOMERSAULT IS A FULL FORWARD "ACK ? SOMERSAULT ROTATION AND A TWIST IS A FULL TURN 4HE CHART &ORWARD ? SOMERSAULT TWIST NAMES SOME OF THE MOST DIFFICULT PLATFORM ? ? DIVES ACCORDING TO THE )NTERNATIONAL 2EVERSE SOMERSAULT TWISTS 3WIMMING &EDERATION .AME THE DIVE IN "ACK ? SOMERSAULT ? TWISTS WHICH THE DIVER MOVES THROUGH A TOTAL OF EXACTLY BACK ?? SOMERSAULT ?? TWISTS ??k ??k (* BISECTS )(, (+ BISECTS *(, AND M)(+ &IND M)(,
???k M879 IS TWICE M87: 4ELL IF 7: MUST BE THE ANGLE BISECTOR OF 879 ???k .O 7: DOES NOT HAVE TO BE THE ANGLE BISECTOR OF 879 Lesson 1-3 25
gge07te_c01_0020_0027.indde07te_c01_0020_0027.indd 2255 112/20/052/20/05 110:44:000:44:00 AAMM Exercise 33 involves 33. This problem will prepare you for the Multi-Step using algebra to Test Prep on page 34. An archaeologist standing at L find the measures O looks for clues on where to dig for artifacts. of congruent angles formed by an 3xº J a. What value of x will make the angle between O angle bisector. This exercise prepares the pottery and the arrowhead measure 57°? 9 (2x + 12)º students for the Multi-Step Test Prep b. What value of x makes ∠LOJ ∠JOK? 12 on page 34. K c. What values of x make ∠LOK an acute angle? 0 < x < 15.6 Kinesthetic For Exercise 40, have students investi- gate how to find the bisec- Data Analysis Use the circle graph for Exercises 34–36. ∠ ∠ ∠ ∠ tor of the angle using patty paper or 34. Find m AOB, m BOC, m COD, and m DOA. /Þ«iÃÊvÊ ÃÊÊ-ÌÀi a Mira. Draw an angle on a piece of Classify each angle as acute, right, or obtuse. patty paper, and then fold the paper 35. m∠COD = 72°; 35. What if...? Next year, the music store will use so that the two sides of the angle m∠BOC = 90° some of the shelves currently holding jazz music overlap. The crease created defines a to double the space for rap. What will m∠COD ,V ,ÊEÊ ∠ Óx¯ Îä¯ bisector of the angle. To use a Mira, and m BOC be next year? place the Mira on the vertex of the " 36. Suppose a fifth type of music, salsa, is added. angle so that one side is reflected >ââ 37. No; an obtuse If the space is divided equally among the five types, Îx¯ onto the other side. Then draw the ,>« ∠ measures greater what will be the angle measure for each type of £ä¯ Mira line. than 90°, so it music in the circle graph? 72° cannot be 37. Critical Thinking Can an obtuse angle be congruent to an acute angle? In Exercise 44, if stu- to an acute ∠ Why or why not? (less than 90°). dents chose C, they 38. The measure of an obtuse angle is (5x + 45)°. What is the largest value for x? 27 divided 90° by 2 and ___ › ∠ then added 30 instead of dividing 39. Write About It FH bisects EFG. Use the Angle Addition Postulate to explain why m∠EFH = __1 m ∠EFG. the original angle by 2. 39. m∠EFG = 2 m ∠EFH + m∠HFG 40. Multi-Step Use a protractor to draw a 70° angle. Then use a compass and = 2m∠EFH, so straightedge to bisect the angle. What do you think will be the measure of Answers m∠EFH = _1 m∠EFG each angle formed? Use a protractor to support your answer. 34. m∠ AOB = 90°, rt.; 2 Check students’ constructions. Each ∠ should be 35°. m∠BOC = 126°, obtuse; ∠ = m COD 36°, acute; ___› m∠DOA = 108°, obtuse 41. m∠UOW = 50°, and OV bisects ∠UOW. What is m∠VOY? 7 8 6 25° 130° 65° 155° {ä 19" 42. What is m∠UOX? 50° 115° 140° 165°
___› 43. BD bisects ∠ABC, m∠ ABC = (4x + 5)°, and m∠ ABD = (3x - 1)°. What is the value of x? 2.2 3 3.5 7
44. If an angle is bisected and then 30° is added to the measure of the bisected angle, the result is the measure of a right angle. What is the measure of the original angle? 30° 60° 75° 120°
45. Short Response If an obtuse angle is bisected, are the resulting angles acute or obtuse? Explain. The are acute. An obtuse ∠ measures between 90° and 180°. Since __1 of 180 is 90, the resulting must measure less than 90°. 2 26 Chapter 1 Foundations for Geometry
,%33/. ,i>`}Ê-ÌÀ>Ìi}ià ,%33/. ,iÌi>V £Î "UILDING 6OCABULARY1-3 READING STRATEGIES £Î -EASURING AND #ONSTRUCTING !NGLES1-3 RETEACH 4HE FOUR TYPES OF ANGLES ARE DESCRIBED IN THE TABLE BELOW !N ANGLE IS A FIGURE MADE UP OF TWO RAYS OR SIDES THAT HAVE A COMMON ENDPOINT CALLED THE VERTEX OF THE ANGLE !NGLE $EFINITION %XAMPLE 8 !CUTE MEASURES GREATER THAN 4HE SIDES ARE?? ?? BUT LESS THAN k k 98 AND 9: 4HE VERTEX IS 9 9 : ge07se_c01_0020_0027.indd2IGHT 7 ANGLE 11/23/06 1:26:28 PM ge07se_c01_0020_0027.indd 27 12/3/05 11:46:07 AM 4HERE ARE FOUR WAYS TO NAME THIS ANGLE 9 5SE THE VERTEX 89: OR :98 5SE THE VERTEX AND A POINT ON EACH SIDE /BTUSE MEASURES GREATER THAN 5SE THE NUMBER BUT LESS THAN
.AME EACH ANGLE IN THREE WAYS 3TRAIGHT FORMED BY TWO OPPOSITE RAYS + AND MEASURES 0 2 )DENTIFY EACH ANGLE AS ACUTE RIGHT OBTUSE OR STRAIGHT 5SE THE CHART ABOVE TO HELP YOU 1 ( * 1 012 * (*+ +*(
.AME THREE DIFFERENT ANGLES IN THE FIGURE #
$ OBTUSE ACUTE RIGHT !"$ !"# $"# ! "
!NGLE ACUTE RIGHT OBTUSE STRAIGHT -ODEL A A RIGHT STRAIGHT OBTUSE A A
4ELL WHAT TYPE OF ANGLE HAS THE GIVEN MEASURE 0OSSIBLE A A A A -EASURES ACUTE OBTUSE #LASSIFY EACH ANGLE AS ACUTE RIGHT OBTUSE OR STRAIGHT OBTUSE STRAIGHT .-0 RIGHT ACUTE OBTUSE 1 0 $RAW AN ACUTE ANGLE $RAW AN OBTUSE ANGLE 1-.
RIGHT . , - 0-1 $RAW A RIGHT ANGLE $RAW A STRAIGHT ANGLE ACUTE 26 Chapter 1 #HECK STUDENTS DRAWINGS
gge08te_ny_c01_0020_0027.indde08te_ny_c01_0020_0027.indd 2626 112/21/062/21/06 11:32:57:32:57 PPMM CHALLENGE AND EXTEND " " Ê ,,", 46. Find the measure of the angle formed by the hands of a clock when it is 7:00. 150° ,/ ___› 47. QS bisects ∠PQR, m∠PQR = (x2)°, and m∠PQS = (2x + 6)°. Find all the possible measures for ∠PQR. 36° or 4° For Exercise 47, students may not try the negative value for x. Remind 48. For more precise measurements, a degree can be divided into 60 minutes, and each students to check both values of x minute can be divided into 60 seconds. An angle measure of 42 degrees, 30 minutes, by substituting them back into the and 10 seconds is written as 42°30 10 . Subtract this angle measure from the expression. measure 81°2415. 38°545 49. If 1 degree equals 60 minutes and 1 minute equals 60 seconds, how many seconds Visual For Exercise 46, are in 2.25 degrees? 8100 use a representation of 50. ∠ABC ∠DBC. m∠ABC = (__3x + 4)° and m∠DBC = (2x - 27__ 1)°. Is ∠ABD a straight a clock, such as a paper 2 4 plate with arrows attached with a angle? Explain. = No; x 62.5, and substituting this value into the brad to represent the hands. Given ∠ expressions for the measures gives a sum of 195.5. a measurement, students can show SPIRAL REVIEW the possible time on the clock. Given 51. What number is 64% of 35? 22.4 a time, students can form and then 52. What percent of 280 is 33.6? (Previous course) 12% classify the angle.
Sketch a figure that shows_ each of the following. (Lesson 1-1) ___› Journal 53. a line that contains AB and CB ACB _ Have students name and give 54. two different lines that intersect MN MN the definition of at least 5 other 55. 55. a plane and a ray that intersect only at Q words that use bi to mean “two.”
Q Find the length of each segment. (Lesson 1-2) ÓÝÊ Ê{ _ _ _ Ý ÎÝ
56. JK 2 57. KL 6858. JL Have students create a diagram containing each type of angle intro- Using Technology Segment and Angle Bisectors duced in this lesson. They should identify the angles in their diagram, 1. Construct_ the bisector 2. Construct the bisector of ∠BAC. measure each with their protrac- of MN . tor, choose one to copy, and bisect another.
1-3
Classify each angle as acute, right, or obtuse. 7 _ 8 a. Draw MN and construct the midpoint B. a. Draw ∠BAC. ___› b. Construct a point A not on the segment. b. Construct the angle bisector AD and measure -1/ _ ‹___› ∠DAC and ∠DAB. ∠ c. Construct_ bisector AB and measure MB 1. XTS acute and NB. c. Drag the angle and observe m∠DAB and 2. ∠WTU right m∠DAC. d. Drag M and N and observe MB and NB. 3. K is in the interior of ∠LMN, m∠LMK = 52°, and m∠KMN 1- 3 Measuring and Constructing Angles 27 = 12°. Find m∠LMN. 64°
*ÀLiÊ-Û} >i}i ,%33/. 1-3 PROBLEM SOLVING ,%33/. 1 -3 CHALLENGE £Î &INDING THE -EASURE OF AN !NGLE £Î ,OCI AND #ENTRAL !NGLES 4. BD bisects ∠ ABC, m∠ ABD = 0ROJECTION DRAWINGS ARE OFTEN USED TO REPRESENT THREE DIMENSIONAL ! LOCUS IS THE SET OF ALL POINTS THAT SATISFY ONE OR MORE GIVEN CONDITIONS MOLECULES 4HE PROJECTION DRAWING OF A METHANE MOLECULE IS SHOWN 4HE PLURAL OF LOCUS IS LOCI 1 ° ? _ BELOW ALONG WITH THE ANGLES THAT ARE FORMED IN THE DRAWING + ∠ = 4HE VERTICAL LINE THAT BISECTS 89 AT THE RIGHT IS AN EXAMPLE ( y 10) , and m DBC -ETHANE -OLECULE OF A LOCUS 4HIS LINE CAN BE DESCRIBED AS THE LOCUS OF ALL 2 POINTS IN THE PLANE THAT ARE EQUALLY DISTANT OR EQUIDISTANT 89 0ROJECTION $RAWING !NGLES &ORMED FROM THE ENDPOINTS OF THE SEGMENT (y + 4)°. Find m∠ ABC. 32° ge07se_c01_0020_0027.indd 26 12/3/05 11:46:02 AM ge07se_c01_0020_0027.indd 27 $RAW A FIGURE TO ILLUSTRATE A LOCUS OF ALL POINTS IN THE INTERIOR 12/3/05 11:46:07 AM H , OF %&' THAT ARE EQUIDISTANT FROM THE SIDES OF THE ANGLE % C + 5. Use a protractor to draw an 120° H ' H * H & ( ' angle with a measure of 165°.
.AME FIVE DIFFERENT ANGLES THAT ARE FORMED IN THE DRAWING .AME THE FIGURE THAT YOU DREW IN %XERCISE ANGLE BISECTOR Check students’ work.
0OSSIBLE ANSWER ,+' '+( (+* *+, ,+( ! CENTRAL ANGLE OF A CIRCLE IS AN ANGLE THAT ∠ = ( - ) )F M,+( M*+, AND M(+' 6. m WYZ 2x 5 ° and INTERSECTS A CIRCLE IN TWO POINTS AND HAS ITS WHAT IS M'+, VERTEX AT THE CENTER OF THE CIRCLE ∠ = ( + ) &IND M*+( m XYW 3x 10 °. Find 4HE CIRCLE HAS TWO CUTS THROUGH THE CENTER MAKING FOUR CONGRUENT 4HE DIAGRAM SHOWS THE PROPER WAY TO SIT AT A COMPUTER TO AVOID ANGLES AT THE CENTER CENTRAL ANGLES 5SE THE TABLE FOR %XERCISES n the value of x. 35 STRAINING YOUR BACK OR EYES 5SE THE FIGURE FOR %XERCISES AND .UMBER .UMBER OF 3MALL -EASURE OF %ACH 3MALL OF #UTS #ONGRUENT #ENTRAL !NGLES #ENTRAL !NGLE 4HE TOTAL VIEWING ANGLE IS $!" )F M$!# ?? M!#" WHAT IS THE MEASURE OF THE TOTAL VIEWING ANGLE !"HORIZONTAL 7 # 4HE ??OPTIMUM VIEWING ANGLE IS BELOW THE HORIZONTAL ?? k k $ )F !6 IS DRAWN TO FORM THIS ANGLE WITH !" WHAT IS THE MEASURE OF $!6 (OW IS THE NUMBER OF CONGRUENT SMALL CENTRAL ANGLES RELATED TO THE NUMBER OF CUTS 8 9< )T IS DOUBLE THE NUMBER OF CUTS
#HOOSE THE BEST ANSWER &OR N CUTS THROUGH THE CENTER THE CONGRUENT SMALL CENTRAL ANGLES OF A CIRCLE ??? k HAVE A SUM OF 5SE THE PATTERN IN THE TABLE TO WRITE AN EXPRESSION TO FIND Also available on transparency 12 IS IN THE INTERIOR OF OBTUSE 013 AND 012 IS A RIGHT ANGLE THE MEASURE OF EACH SMALL CENTRAL ANGLE IF THERE ARE N CUTS THROUGH THE CENTER #LASSIFY 312 N OR N ! ACUTE " RIGHT # OBTUSE $ STRAIGHT ?? k (OW MANY CENTRAL ANGLES WOULD THERE BE IN A CIRCLE WITH CUTS 7HAT WOULD 68 BISECT 769 M768 X AND M769 X BE THE MEASURE OF EACH CENTRAL ANGLE IN A CIRCLE THAT HAS CUTS 7HAT IS THE VALUE OF X & ??? ' ??? ( * Lesson 1-3 27
gge07te_c01_0020_0027.indde07te_c01_0020_0027.indd 2277 112/6/052/6/05 112:59:042:59:04 PPMM 1-4 Organizer 1-4 Pairs of Angles Pacing: Traditional 1 day __1 Block 2 day Objectives: Identify adjacent, Objectives Who uses this? vertical, complementary, and Identify adjacent, vertical, Scientists use properties of angle pairs to supplementary angles. complementary, and design fiber-optic cables. (See Example 4.) supplementary angles. Find measures of pairs of angles. Find measures of pairs A fiber-optic cable is a strand of glass as thin of angles. as a human hair. Data can be transmitted @
Pairs of Angles
Adjacent angles are two angles in the same plane with a common vertex and a common side, but no common £ Warm Up interior points. ∠1 and ∠2 are adjacent angles. Ó Simplify each expression. A linear pair of angles is a pair of adjacent angles 1. 90 - ( x + 20) 70 - x whose noncommon sides are opposite rays. ∠3 and ∠4 form a linear pair. {Î 2. 180 - ( 3x - 10) 190 - 3x Write an algebraic expression for each of the following. EXAMPLE 1 Identifying Angle Pairs 3. 4 more than twice a number Tell whether the angles are only adjacent, adjacent 2n + 4 and form a linear pair, or not adjacent. 4. 6 less than half a number NY Performance A ∠1 and ∠2 Î Indicators 1 ∠ ∠ _ n - 6 1 and 2 have a common vertex, B, a common Ó£ 2 G.CM.11 Understand and use appropriate language, side, BC, and no common interior points. { ∠ ∠ Also available on transparency representations, and terminology Therefore 1 and 2 are only adjacent angles. when describing objects, relationships, mathematical B ∠2 and ∠4 −− solutions, and geometric ∠ ∠ diagrams. Also, G.RP.3. 2 and 4 share BC but do not have a common vertex, so ∠2 and ∠4 are not adjacent angles.
C ∠1 and ∠3 ∠1 and ∠3 are adjacent angles. Their noncommon sides, BC and BA , Teacher: Today we are going to learn are opposite rays, so ∠1 and ∠3 also form a linear pair. about complementary angles.
Student: Does that mean the angles Tell whether the angles are only - are nice to each other? adjacent, adjacent and form a , linear pair, or not adjacent. +/nÇ* 1a. ∠5 and ∠6 adj.; lin. pair È x 1 1b. ∠7 and ∠SPU not adj. 1c. ∠7 and ∠8 not adj. New York Performance Indicators 28 Chapter 1 Foundations for Geometry
Process G.CM.11 Understand and use appropriate language, representations, 1 Introduce and terminology when describing
objects, relationships, mathematical ge07se_c01_0028_0033.inddEXPLORATION 1 11/23/06 1:27:26 PM solutions, and geometric diagrams Motivate £{ *>ÀÃÊvÊ}ià G.RP.3 Investigate and evaluate Angles are used in physics, engineering, art and 5SE THE FOLLOWING lGURES TO HELP YOU COMPARE AND conjectures in mathematical terms, CONTRAST EXAMPLES AND NONEXAMPLES OF ADJACENT design. Show students a picture of a quilt pattern using mathematical strategies to reach ANGLES !DJACENT !NGLES .OT !DJACENT !NGLES with angle pairs in it. Discuss the relationships a conclusion between different pairs of angles. Ask students what they think the angle measures must be so the pieces will fit together.
,IST ALL OF THE PAIRS OF ADJACENT ANGLES Explorations and answers are provided in the IN THE FIGURE Explorations binder. #OMPLETE THIS DEFINITION !DJACENT ANGLES ARE TWO ANGLES IN THE SAME PLANE WITH A COMMON ? AND A COMMON ? BUT WITH NO COMMON INTERIOR POINTS
28 Chapter 1 / - 1--
gge08te_ny_c01_0028_0033.indde08te_ny_c01_0028_0033.indd 2828 112/21/062/21/06 112:39:502:39:50 PPMM Complementary and Supplementary Angles " " Ê ,,",
Complementary angles are two angles ,/ whose measures have a sum of 90°. ∠A and ∠B are complementary. In Example 2, students may not xÎÂ apply the Distributive Property cor- ÎÇÂ Supplementary angles are two angles rectly. Stress that in the expression whose measures have a sum of 180°. 180 - (2y + 20) , both 2y and 20 ∠A and ∠C are supplementary. £ÓÇÂ must be multiplied by -1 to get 160 - 2y.
You can find the complement of an angle that measures x° by subtracting its measure from 90°, or ( 90 - x)°. You can find the supplement of an angle that measures x° by subtracting its measure from 180°, or (180 - x)°. Additional Examples
Example 1 EXAMPLE 2 Finding the Measures of Complements and Supplements Tell whether the angles are only Find the measure of each of the following. adjacent, adjacent and form a ∠ A complement of M linear pair, or not adjacent. ( ) 90 - x ° ÓÈ°n 90° - 26.8° = 63.2°
B supplement of ∠N (180 - x)° ÓÞÊ ÊÓä® - ( + )° = - -  180° 2y 20 180° 2y 20 A. ∠ AEB and ∠BED lin. pair =- (160 2y)° and adjacent B. ∠ AEB and ∠BEC only adj. Find the measure of each of the following. 2a. complement of ∠E (102 - 7x)° C. ∠DEC and ∠ AEB not adj. ££È°x ∠ 1 ÇÝÊÊ£Ó® 2b. supplement of F 63_ °  2 Example 2 Find the measure of each of the EXAMPLE 3 Using Complements and Supplements to Solve Problems following. An angle measures 3 degrees less than twice the measure of its A. complement of ∠F 31° complement. Find the measure of its complement. 2 Step 1 Let m∠A = x°. Then ∠B, its complement, measures (90 - x)°. x Step 2 Write and solve an equation.  m∠A = 2m∠B - 3 x = 2(90 - x) - 3 Substitute x for m∠A and 90 - x for m∠B. B. supplement of ∠G x= 180 - 2x -3 Distrib. Prop. (170 - 7x)° x= 177 - 2x Combine like terms. + + −2x −2x Add 2x to both sides. ÇÝÊ Ê£ä® 3x = 177 Simplify.  _3x = _177 Divide both sides by 3. 3 3 Example 3 = Simplify. x 59 An angle is 10° more than 3 The measure of the complement, ∠B, is (90 - 59)° = 31°. times the measure of its comple- ment. Find the measure of the __1 3. An angle’s measure is 12° more than 2 the measure of its complement. 20° supplement. Find the measure of the angle. 68° Also available on transparency 1- 4 Pairs of Angles 29
2 INTERVENTION Teach Questioning Strategies ge07se_c01_0028_0033.indd 29 12/3/05 11:50:04 AM Guided Instruction EXAMPLE 1 Discuss vertical, complementary, supple- Through Cooperative Learning • What type of angle is formed by a mentary, and adjacent angles and linear Each student should create a simple puzzle pair of adjacent angles that form a pairs. Have students use mental math to by dividing a square into six regions using linear pair? find complements and supplements of segments. Students should measure all of angles before using variables or expres- the angles formed and write several hints EXAMPLES 2–3 sions. for the puzzle solver. For example, a hint • Do two angles have to be adjacent might be that one corner of the square is Cognitive Strategies One way to be complementary or supple- made up of a 15° angle and a 75° angle. mentary? to help students remember that Students should then exchange puzzles complementary angles add to 90° with a partner and try to reassemble the and that supplementary angles add to 180° original square. is that 90 comes before 180 and C comes before S in the alphabet. Lesson 1-4 29
gge07te_c01_0028_0033.indde07te_c01_0028_0033.indd 2299 112/5/052/5/05 110:41:060:41:06 PPMM Visual The following is EXAMPLE 4 Problem-Solving Application another way for students Light passing through a fiber optic cable reflects to remember which sum off the walls in such a way that ∠1 ∠2. ∠1 is 90 and which is 180: Draw a line and ∠3 are complementary, and ∠2 and ∠4 along the C in complementary to are complementary. 4 make a 9 for 90, and add a line If m∠1 = 38°, find m∠2, m∠3, and m∠4. 2 1 through the S in supplementary to 3 make an 8 for 180. 1 Understand the Problem C O° MPLEMENTARY The answers are the measures of ∠2, ∠3, and ∠4. List the important information: Light 1 S 0° UPPLEMENTARY • ∠1 ∠2 • ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. • m∠1 = 38°
Additional Examples 2 Make a Plan
Example 4 If ∠1 ∠2, then m∠1 = m∠2. If ∠3 and ∠1 are complementary, then m∠3 = (90 - 38)°. Light passing through a fiber- If ∠4 and ∠2 are complementary, then m∠4 = (90 - 38)°. optic cable reflects off the walls of the cable in such a way that 3 Solve ∠1 ∠2, ∠1 and ∠3 are com- By the Transitive Property of Equality, if m∠1 = 38° and m∠1 = m∠2, then plementary, and ∠2 and ∠4 are m∠2 = 38°. Since ∠3 and ∠1 are complementary, m∠3 = 52°. Similarly, ∠ = complementary. If m 1 47°, since ∠2 and ∠4 are complementary, m∠4 = 52°. find m∠2, m∠3, and m∠4. 47°; 43°; 43° 4 Look Back + = ∠ ∠ Example 5 The answer makes sense because 38° 52° 90°, so 1 and 3 are complementary, and ∠2 and ∠4 are complementary. Thus m∠2 = 38°, m∠3 = 52°, and m∠4 = 52°. 4. What if...? Suppose m∠3 = 27.6°. Find m∠1, m∠2, and m∠4. m∠1 = m∠2 = 62.4°; m∠4° = 27.6° Name the pairs of vertical angles. Another angle pair relationship exists between two angles whose sides form two pairs of ∠HML and ∠ JMK; ∠HMJ and opposite rays. Vertical angles are two £ ∠ Ó LMK nonadjacent angles formed by two intersecting Î { Also available on transparency lines. ∠1 and ∠3 are vertical angles, as are ∠2 and ∠4.
EXAMPLE 5 Identifying Vertical Angles INTERVENTION Name one pair of vertical angles. Questioning Strategies Do they appear to have the same measure? Check by measuring with a protractor. EXAMPLE 4 ∠EDF and ∠GDH are vertical angles • How are ∠3 and ∠4 related? and appear to have the same measure. Explain. Possible answer: Check m∠EDF ≈ m∠GDH ≈ 135°. ∠EDG and ∠FDH; EXAMPLE 5 m∠EDG ≈ m∠FDH 5. Name another pair of vertical angles. Do they appear to have • Can vertical angles have a common ≈ 45° the same measure? Check by measuring with a protractor. side or common interior points? Explain. 30 Chapter 1 Foundations for Geometry
Inclusion Remind stu- dents that the shape of the 3 letter x suggests the idea Close of vertical angles. ge07se_c01_0028_0033.indd 30 12/3/05 11:50:06 AM Summarize and INTERVENTION Review the different angle pairs and illus- Diagnose Before the Lesson trations in the lesson. Remind students 1-4 Warm Up, TE p. 28 that a linear pair is a pair of adjacent and supplementary angles, but not all comple- Monitor During the Lesson mentary and supplementary angles are Check It Out! Exercises, SE pp. 28–30 adjacent. Then have students identify Questioning Strategies, TE pp. 29–30 what kind of angle is a complement of an acute angle, a supplement of an obtuse angle, and the supplement of a right angle. Assess After the Lesson acute, acute, right 1-4 Lesson Quiz, TE p. 33 Alternative Assessment, TE p. 33
30 Chapter 1
gge07te_c01_0028_0033.indde07te_c01_0028_0033.indd 3300 112/5/052/5/05 110:41:110:41:11 PPMM Answers to Think and Discuss G.CM.5, G.R.1 THINK AND DISCUSS 1. All rt. measure 90°, so the sum of the measures of any 2 rt. 1. Explain why any two right angles are supplementary. is 180°. Therefore any 2 rt. 2. Is it possible for a pair of vertical angles to also be adjacent? Explain. are supp. 3. GET ORGANIZED Copy and complete the graphic organizer below. In 2. Vert. cannot be adj. because each box, draw a diagram and write a definition of the given angle pair. the def. of vert. states that «iiÌ>ÀÞ `>ViÌÊ>}ià *>ÀÃÊv they are nonadj. formed by }ià >}ià intersecting lines. -Õ««iiÌ>ÀÞ i>ÀÊ«>À 3. >}ià See p. A2. 6iÀÌV>Ê>}iÃ
1-4 1-4 Exercises Exercises KEYWORD: MG7 1-4 Exercises
KEYWORD: MG7 Parent GUIDED PRACTICE Assignment Guide Vocabulary Apply the vocabulary from this lesson to answer each question. (90 - x)°; 1. An angle measures x°. What is the measure of its complement? What is the measure Assign Guided Practice exercises (180 - x)° of its supplement? as necessary. 2. ∠ABC and ∠CBD are adjacent angles. Which side do the angles have in common? BC If you finished Examples 1–3 Basic 14–22, 25, 27–30, 34 SEE EXAMPLE 1 Tell whether the angles are only adjacent, adjacent p. 28 and form a linear pair, or not adjacent. Average 14–22, 25, 27–35 3. ∠1 and ∠2 adj.; 4. ∠1 and ∠3 not adj. Ó Advanced 14–22, 25, 27–35, 37 Î £ lin. pair { 5. ∠2 and ∠4 6. ∠2 and ∠3 only adj. If you finished Examples 1–5 not adj. Basic 14–22, 24–30, 33, 34, SEE EXAMPLE 2 Find the measure of each of the following. 37–43, 47–55 p. 29 7. supplement of ∠A 98.8° 8. complement of ∠A 8.8° Average 14–44, 47–55 ∠ ∠ ÈÝÊÊx®Â 9. supplement of B 10. complement of B Advanced 14–55 (185 - 6x)° (95 - 6x)° n£°Ó SEE EXAMPLE 3 11. Multi-Step An angle’s measure is 6 degrees more p. 29 than 3 times the measure of its complement. Homework Quick Check Find the measure of the angle. 69° Quickly check key concepts. Exercises: 14, 18, 22, 24, 28, SEE EXAMPLE 4 12. Landscaping A sprinkler swings back and 30, 34 p. 30 forth between A and B in such a way that ∠1 ∠2. ∠1 and ∠3 are complementary, Î{ and ∠2 and ∠4 are complementary. £Ó If m∠1 = 47.5°, find m∠2, m∠3, and m∠4. m∠2 = 47.5°; m∠3 = m∠4 = 42.5°
SEE EXAMPLE 5 13. Name each pair of vertical angles. p. 30 ∠ABE, ∠CBD; ∠ABC, ∠EBD
1- 4 Pairs of Angles 31
ge07se_c01_0028_0033.indd 4 11/23/06 1:27:35 PM
KEYWORD: MG7 Resources
Lesson 1-4 31
gge08te_ny_c01_0028_0033.indde08te_ny_c01_0028_0033.indd 3131 112/21/062/21/06 11:35:20:35:20 PPMM Exercise 33 involves PRACTICE AND PROBLEM SOLVING using algebra to Independent Practice Tell whether the angles are only adjacent, adjacent find measures of For See and form a linear pair, or not adjacent. Ó Exercises Example £ complementary angles, congruent ∠ ∠ adj.; ∠ ∠ Î 14–17 1 14. 1 and 4 15. 2 and 3 adj.; lin. pair { angles, and linear pairs. This exercise lin. pair 18–21 2 ∠ ∠ ∠ ∠ prepares students for the Multi-Step 16. 3 and 4 17. 3 and 1 not adj. 22 3 only adj. Test Prep on page 34. 23 4 Given m∠A = 56.4° and m∠B = (2x - 4)°, find the measure of each of the following. 24 5 18. supplement of ∠A 123.6° 19. complement of ∠A 33.6° Answers Extra Practice 20. supplement of ∠B (184 - 2x)° 21. complement of ∠B (94 - 2x)° 33a. Skills Practice p. S4 J K Application Practice p. S28 22. Multi-Step An angle’s measure is 3 times the measure of its complement. Find the measure of the angle and the measure of its complement. 67.5°; 22.5° A H 23. Art In the stained glass pattern, ∠1 ∠2. ∠1 and ∠3 are complementary, and ∠2 and b. K ∠4 are complementary. If m∠1 = 22.3°, find m∠2, m∠3, and m∠4. 3 4 m∠2 = 22.3°; m∠3 = m∠4 = 67.7° 1 2 J A H c. J K 24. Name the pairs * 1 + of vertical angles. A ∠ ∠ ∠ ∠ / H PTU, VTR; UTQ, STV ; - , ∠QTR, ∠PTS; ∠PTQ, ∠STR; 6 ∠UTR, ∠PTV ; ∠QTV, ∠UTS 25. Probability The angle measures 30°, 60°, 120°, and 150° are written on slips of paper. You choose two slips of paper at random. What is the probability that the angle measures are supplementary? _1 3 Multi-Step ∠ABD and ∠BDE are supplementary. Find the measures of both angles. 26. m∠ABD = 5x°, m∠BDE = (17x - 18)° 45°; 135° 27. m∠ABD = (3x + 12)°, m∠BDE = (7x - 32)° 72°; 108° 28. m∠ABD = (12x - 12)°, m∠BDE = (3x + 48)° 103.2°; 76.8° 32. The measure of an acute ∠ is less Multi-Step ∠ABD and ∠BDC are complementary. Find the measures of both angles. than 90°. Therefore ∠ = + ∠ = - the measure of 29. m ABD (5y 1)°, m BDC (3y 7)° 61°; 29° its supp. must be 30. m∠ABD = (4y + 5)°, m∠BDC = (4y + 8)° 43.5°; 46.5° between 90° and 31. ∠ = ( - )° ∠ = 10°; 80° 180°, which means m ABD y 30 , m BDC 2y° the supp. is an 32. Critical Thinking Explain why an angle that is supplementary to an acute obtuse ∠. angle must be an obtuse angle.
33. This problem will prepare you for the Multi-Step Test Prep on page 34. H is in the interior of ∠JAK. m∠JAH = (3x - 8)°, and m∠KAH = (x + 2)°. Draw a picture of each relationship. Then find the measure of each angle. a. ∠JAH and ∠KAH are complementary angles. m∠JAH = 64°; m∠KAH = 26° b. ∠JAH and ∠KAH form a linear pair.m∠JAH = 131.5°; m∠KAH = 48.5° c. ∠JAH and ∠KAH are congruent angles. m∠JAH = m∠KAH = 7°
1-4 PRACTICE A 1-4 PRACTICE C
*À>VÌViÊ ,%33/. 1-4 PRACTICE B £{ 0AIRS OF !NGLES 32 Chapter 1 Foundations for Geometry 012 AND 312 FORM A LINEAR PAIR &IND THE SUM OF THEIR MEASURES ???k .AME THE RAY THAT 012 AND 312 SHARE 12 ,%33/. ,i>`}Ê-ÌÀ>Ìi}ià ,%33/. ,iÌi>V £{ 5SE A 'RAPHIC /RGANIZER1-4 READING STRATEGIES £{ 0AIRS OF !NGLES1-4 RETEACH 5SE THE FIGURES FOR %XERCISES n 9 X 4HE GRAPHIC ORGANIZER BELOW OUTLINES THE DIFFERENT POSSIBILITIES FOR A PAIR OF ANGLES !NGLE 0AIRS SUPPLEMENT OF : !DJACENT !NGLES ,INEAR 0AIRS 6ERTICAL !NGLES : COMPLEMENT OF 9 X ADJACENTTWO ANGLES IN THE SAME LINEARADJACENT HAVE THE SAME VERTEX AND ADJACENT ANGLES WHOSE NONADJACENT ANGLES FORMED PLANE WITH A COMMON VERTEX AND ANGLES WHOSE SHARE A COMMON SIDE NONCOMMON SIDES ARE BY TWO INTERSECTING LINES !N ANGLE MEASURES DEGREES LESS THAN THREE TIMES ITS SUPPLEMENT &IND THE A COMMON SIDE BUT NO COMMON NONCOMMON SIDES OPPOSITE RAYS INTERIOR POINTS ARE OPPOSITE RAYS MEASURE OF THE ANGLE ge07se_c01_0028_0033.indd 32 12/3/05 11:50:11 AM ge07se_c01_0028_0033.indd 33 12/3/05 11:50:13 AM !N ANGLE IS ITS OWN COMPLEMENT &IND THE MEASURE OF A SUPPLEMENT TO THIS ANGLE COMPLEMENTARYTWO VERTICALTWO ANGLES WHOSE MEASURES !NGLE PAIRS NONADJACENT AND ARE ADJACENT AND ARE VERTICAL HAVE A SUM OF CAN BE ANGLES FORMED AND ARE ADJACENT AND FORM A LINEAR PAIR ANGLES $%& AND &%' ARE COMPLEMENTARY M$%& X AND M&%' X BY TWO INTER SECTING LINES M$%& M&%' 4ELL WHETHER AND IN EACH FIGURE ARE ONLY ADJACENT ARE ADJACENT AND FORM A &IND THE MEASURES OF BOTH ANGLES SUPPLEMENTARYTWO LINEAR PAIR OR ARE NOT ADJACENT ANGLES WHOSE MEASURES $%& AND &%' ARE SUPPLEMENTARY M$%& X AND M&%' X HAVE A SUM OF M$%& M&%' &IND THE MEASURES OF BOTH ANGLES
5SE THE FIGURE FOR %XERCISES n )N SEVERAL NICKELS WERE MINTED TO COMMEMORATE THE ,OUISIANA 0URCHASE AND ,EWIS AND #LARKS EXPEDITION INTO THE !MERICAN 7EST /NE ADJACENT AND FORM NICKEL SHOWS A PIPE AND A HATCHET CROSSED TO SYMBOLIZE PEACE BETWEEN A LINEAR PAIR ONLY ADJACENT NOT ADJACENT )DENTIFY EACH PAIR OF ANGLES AS COMPLEMENTARY SUPPLEMENTARY LINEAR THE !MERICAN GOVERNMENT AND .ATIVE !MERICAN TRIBES VERTICAL OR ADJACENT 5SE THE GRAPHIC ORGANIZER ABOVE TO HELP YOU 4ELL WHETHER THE INDICATED ANGLES ARE ONLY ADJACENT ARE ADJACENT AND .AME A PAIR OF VERTICAL ANGLES +EEP IN MIND THAT THERE MAY BE MORE THAN ONE ANSWER FOR EACH EXERCISE FORM A LINEAR PAIR OR ARE NOT ADJACENT 0OSSIBLE ANSWER AND AND ONLY ADJACENT .AME A LINEAR PAIR OF ANGLES 0OSSIBLE ANSWER AND COMPLEMENTARY VERTICAL AND NOT ADJACENT !"# AND #"$ FORM A LINEAR PAIR AND HAVE EQUAL MEASURES 4ELL IF !"# IS ACUTE RIGHT OR OBTUSE RIGHT AND ADJACENT AND FORM A LINEAR PAIR ??k +,- AND -,. ARE COMPLEMENTARY ,- BISECTS +,. &IND THE MEASURES OF +,- .AME EACH OF THE FOLLOWING AND -,. SUPPLEMENTARY LINEAR OR SUPPLEMENTARY 0OSSIBLE ANSWERS A PAIR OF VERTICAL ANGLES AND AND
A LINEAR PAIR 0OSSIBLE ANSWER AND ADJACENT COMPLEMENTARY 32 Chapter 1 AN ANGLE ADJACENT TO
gge07te_c01_0028_0033.indde07te_c01_0028_0033.indd 3322 112/5/052/5/05 110:41:170:41:17 PPMM Determine whether each statement is true or false. If false, explain why. If students chose F 34. F; the supp. 34. If an angle is acute, then its complement must be greater than its supplement. for Exercise 40, they must be greater F; vert. cannot be adj. , did not substitute the 35. A pair of vertical angles may also form a linear pair. than the comp. so they cannot form a lin. pair. value for x in the expression 2x and 36. If two angles are supplementary and congruent, the measure of each angle is 90°. T find the larger angle. If they chose 37. If a ray divides an angle into two complementary angles, then the original angle D for Exercise 41, they wrote the + = is a right angle. T equation 3y 2y 180 instead of multiplying 3y by 2. 38. Write About It Describe a situation in which two angles are both congruent and complementary. Explain. The 2 must both measure 45°. 45° + 45° = 90°, so the are comp. and .
39. What is the value of x in the diagram? 15 45 ÝÂ ÝÂ
30 90
40. The ratio of the measures of two complementary angles is 1 : 2. What is the measure of the larger angle? (Hint: Let x and 2x represent the angle measures.) 30° 45° 60° 120°
41. m∠A = 3y, and m∠B = 2m∠A. Which value of y makes ∠A supplementary to ∠B? 10 18 20 36 Journal 42. The measures of two supplementary angles are in the ratio 7 : 5. Which value is the measure of the smaller angle? (Hint: Let 7x and 5x represent the angle measures.) Have students draw and then define vertical angles in their own words. 37.5 52.5 75 105 Ask them to give examples of verti- cal angles used in real-world CHALLENGE AND EXTEND structures. 43. How many pairs of vertical angles are in the diagram? 12 44. The supplement of an angle is 4 more than twice its complement. Find the measure of the angle. 4° 45. An angle’s measure is twice the measure of its complement. Have students name and draw the The larger angle is how many degrees greater than the smaller angle? 30° various pairs of angles introduced in 46. The supplement of an angle is 36° less than twice the supplement of the this lesson. Ask them to write and complement of the angle. Find the measure of the supplement. 168° solve a word problem using comple- mentary or supplementary angles on index cards, and then rotate for SPIRAL REVIEW other students to solve. Solve each equation. Check your answer. (Previous course) 47. 4x + 10 = 42 8 48. 5m - 9 = m + 4 3.25 49. 2( y + 3) = 12 3 50. - (d + 4) = 18 -22 1-4 Y is between X and Z, XY = 3x + 1, YZ = 2x - 2, and XZ = 84. Find each of the following. (Lesson 1-2) m∠ A = 64.1°, and m∠B = 51. x 17 52. XY 5253. YZ 32 (4x - 30) °. Find the measure of each of the following. XY bisects ∠WYZ. Given m∠WYX = 26°, find each of the following. (Lesson 1-3) 1. supplement of ∠ A 115.9° 54. m∠XYZ 26°55. m∠WYZ 52° 2. complement of ∠B (120 - 4x)° 1- 4 Pairs of Angles 33 3. Determine whether this state-
,%33/. *ÀLiÊ-Û} ,%33/. >i}i £{ 0AIRS OF !NGLES1-4 PROBLEM SOLVING £{ )NVESTIGATING !NGLE -EASUREMENT1-4 CHALLENGE ment is true or false. If false, 5SE THE DRAWING OF PART OF THE %IFFEL 4OWER FOR %XERCISES n &OR GREATER ACCURACY IN ANGLE MEASUREMENT EACH DEGREE CAN BE DIVIDED INTO SIXTY EQUAL PARTS CALLED MINUTES %ACH MINUTE CAN BE FURTHER DIVIDED INTO explain why. If two angles are .AME A PAIR OF ANGLES THAT APPEAR TO BE SIXTY EQUAL PARTS CALLED SECONDS COMPLEMENTARY 8 'IVEN A DECIMAL ANGLE MEASURE YOU 'IVEN AN ANGLE MEASURE IN DEGREES AND MINUTES complementary and congru- 0OSSIBLE ANSWER !," AND ",# CAN CONVERT IT TO DEGREES AND MINUTES YOU CAN CONVERT IT TO A DECIMAL MEASURE 9 ! - .AME A PAIR OF SUPPLEMENTARY ANGLES ge07se_c01_0028_0033.indd 32 12/3/05 11:50:11 AM ge07se_c01_0028_0033.indd 6 , " 11/23/06 1:27:43 PM ent, then the measure of each 0OSSIBLE ANSWER !-, AND 9-, ??? 7 S D OR )F M#37 WHAT IS M*34 (OW DO + # OR is 90°. False; each is 45°. YOU KNOW 3 4HEY ARE VERTICAL ANGLES : * $ &IND THE MEASURE OF THE COMPLEMENT OF EACH ANGLE USING DEGREES 4 MINUTES AND SECONDS ∠ = ∠ = 2 & % m XYZ 2x° and m PQR )F M&+" WHAT IS M"+, (OW DO (8x - 20) °. YOU KNOW 37.76° 4HE ANGLES ARE SUPPLEMENTARY ∠ ∠ 4. If XYZ and PQR are supple- &IND THE MEASURE OF THE SUPPLEMENT OF EACH ANGLE USING DEGREES MINUTES .AME THREE ANGLES WHOSE MEASURES SUM TO AND SECONDS mentary, find the measure of 0OSSIBLE ANSWER !"- -"+ AND +"# 152.375° each angle. 40°; 140° #HOOSE THE BEST ANSWER ! LANDSCAPER USES PAVING STONES FOR A WALKWAY A B 7HICH ARE POSSIBLE ANGLE MEASURES FOR A AND B SO THAT THE STONES DO NOT HAVE SPACE BETWEEN THEM 5. If ∠ XYZ and ∠PQR are com- ! # 5SE THE DIAGRAM FOR %XERCISES n " $ .AME A PAIR OF SUPPLEMENTARY ANGLES + plementary, find the measure 4HE ANGLE FORMED BY A TREE BRANCH AND THE PART OF THE TRUNK ABOVE IT IS 0OSSIBLE ANSWER +,- AND -,. ( * , - 7HAT IS THE MEASURE OF THE ANGLE THAT IS FORMED BY THE BRANCH AND THE PART OF THE TRUNK BELOW IT .AME A PAIR OF VERTICAL ANGLES WHOSE MEASURES HAVE A of each angle. 22°; 68° & ( SUM THAT IS LESS THAN & ' # 0 . ' * 0OSSIBLE ANSWER +,( AND -,. 2 AND 3 ARE COMPLEMENTARY )F M2 X AND M3 X Also available on transparency )F M(*+ X WHAT IS THE MEASURE OF +*- 1 WHICH IS A TRUE STATEMENT 2 ! 2 IS ACUTE # 2 AND 3 ARE RIGHT ANGLES X OR X ? " 2 IS OBTUSE $ M3 M2 3UPPOSE (1 IS DRAWN ON THE FIGURE .AME A PAIR OF VERTICAL ANGLES THAT IS FORMED 0OSSIBLE ANSWER (#+ AND 2#1 Lesson 1-4 33
gge08te_ny_c01_0028_0033.indde08te_ny_c01_0028_0033.indd 3333 112/21/062/21/06 11:36:05:36:05 PPMM SECTION 1A SECTION 1A Euclidean and Construction Tools Can You Dig It? A group of college and high school students participated in an archaeological dig. The team discovered four Organizer fossils. To organize their search, Sierra used a protractor and ruler to make a diagram of Objective: Assess students’ where different members of the group found ability to apply concepts and skills fossils. She drew the locations based on the location of the campsite. The campsite is in Lessons 1-1 through 1-4 in a located at X on XB . The four fossils were real-world format. found at R, T, W, and M. @
4. adj.; adj. and a lin. pair; supp.; comp.; vert.
34 Chapter 1 Foundations for Geometry
INTERVENTION Scaffolding Questions ge07se_c01_0034_0035.indd 34 12/2/05 7:09:20 PM ge07se_c01_0034_0035.indd 35 12/2/05 7:09:24 PM 1. What term describes the placement of 4. How are the angle pairs ∠RXM and and relationship between points? Possible ∠MXB, and ∠RXM and ∠MXT alike? answer: X, R, and T are collinear. How are they different? Both ∠ pairs are 2. How would you compare the distances adj. angles. ∠RXM and ∠MXB are comp. between X and R and between X and T? ∠RXM and ∠MXT are supp. and a lin. pair. The distances are the same. Extension 3. You can assume the measure of which angle? What is the term used to classify What occupations can you name that use the angle? ∠RXT = 180°; it is a straight ∠. angles and directions in a similar manner? Possible answer: Draftspersons use com- passes for greater accuracy in their drawings. KEYWORD: MG7 Resources
34 Chapter 1
gge07te_c01_0034_0035.indde07te_c01_0034_0035.indd 3344 112/5/052/5/05 110:42:170:42:17 PPMM SECTION 1A SECTION Quiz for Lessons 1-1 Through 1-4 1A 1-1 Understanding Points, Lines, and Planes 3. Draw and label each of the following. 1. a segment with endpoints X and Y XY 4. 2. a ray with endpoint M that passes through P MP Organizer 3. three coplanar lines intersecting at a point Objective: Assess students’ 4. two points and a line that lie in a plane mastery of concepts and skills in Use the figure to name each of the following. / 7 Lessons 1-1 through 1-4. 5. three coplanar points 5. Possible 8 6 Ű 6. two lines XZ and WY answer: T, V, W < 7. a plane containing T, V, and X plane 9 Resources 8. a line containing V and Z TVX Assessment Resources 1-2 Measuring and Constructing Segments ,- / 6 Section 1A Quiz Find the length of each segment. { Î Ó £ ä£ÓÎ{x −− −− −− 9. SV 6.510. TR 611. ST 3.5 £°x 12. The diagram represents a {ÝÊ ÊÈ straight highway with three towns, Henri, Joaquin, and Î Kenard. Find the distance INTERVENTION from Henri H to Joaquin J. 30 Check students’ work. −−
13. Sketch, draw, and construct a segment congruent to CD . Resources −− 14. Q is the midpoint of PR , PQ = 2z, and PR = 8z - 12. Find z, PQ, and PR. 3; 6; 12 Ready to Go On? Intervention and
1-3 Measuring and Constructing Angles Enrichment Worksheets 15. Name all the angles in the diagram. * £ Ó Ready to Go On? CD-ROM ∠LMN, ∠NML, or ∠1; ∠NMP, ∠PMN, or ∠2; ∠LMP, ∠PML Classify each angle by its measure. obtuse Ready to Go On? Online 16. m∠PVQ = 21° acute17. m∠RVT = 96° 18. m∠PVS = 143° obtuse 19. RS bisects ∠QRT, m∠QRS = (3x + 8)°, and m∠SRT = (9x - 4)°. Find m∠SRT. 14° 20. Use a protractor and straightedge to draw a 130° angle. Then bisect the angle. Check students’ work. 1-4 Pairs of Angles Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. Î £ 21. ∠1 and ∠2 22. ∠4 and ∠5 23. ∠3 and ∠4 x Ó adj.; lin. pair only adj. not adj. { If m∠T = (5x - 10)°, find the measure of each of the following. 24. supplement of ∠T 25. complement of ∠T (190 - 5x)° (100 - 5x)°
Ready to Go On? 35
NO YES ge07se_c01_0034_0035.indd 34 12/2/05 7:09:20 PM ge07se_c01_0034_0035.indd 35 INTERVENE Diagnose and Prescribe 12/2/05 7:09:24 PMENRICH
READY TO GO ON? Intervention, Section 1A READY TO GO ON? Ready to Go On? Worksheets CD-ROM Online Intervention Enrichment, Section 1A Lesson 1-1 1-1 Intervention Activity 1-1 Worksheets CD-ROM Lesson 1-2 1-2 Intervention Activity 1-2 Diagnose and Online Lesson 1-3 1-3 Intervention Activity 1-3 Prescribe Online Lesson 1-4 1-4 Intervention Activity 1-4
Ready to Go On? 35
gge07te_c01_0034_0035.indde07te_c01_0034_0035.indd 3355 112/21/052/21/05 55:55:32:55:32 PPMM Coordinate and Transformation Tools
One-Minute Section Planner
Lesson Lab Resources Materials
Lesson 1-5 Using Formulas in Geometry Geometry Lab Activities Optional • Apply formulas for perimeter, area, and circumference. 1-5 Geometry Labs quilt patterns □ NY Regents Exam □✔ SAT-10 □✔ NAEP □✔ ACT □✔ SAT
Lesson 1-6 Midpoint and Distance in the Coordinate Plane Optional • Develop and apply the formula for midpoint. coordinate grid with points • Use the Distance Formula and the Pythagorean Theorem to find the representing a house and a distance between two points. school, graph paper □✔ NY Regents Exam □✔ SAT-10 □✔ NAEP □✔ ACT □✔ SAT
Lesson 1-7 Transformations in the Coordinate Plane Required • Identify reflections, rotations, and translations. graph paper, straightedge • Graph transformations in the coordinate plane. Optional ✔ ✔ ✔ □ NY Regents Exam □ SAT-10 □ NAEP □ ACT □ SAT mirror (MK); examples of tessellations, translations, rotations, and reflections; coordinate grid and cut-out triangle, geometry software
1-7 Technology Lab Explore Transformations Technology Lab Activities Required • Use geometry software to perform transformations and explore their 1-7 Lab Recording Sheet geometry software properties. □✔ NY Regents Exam □ SAT-10 □✔ NAEP □✔ ACT □ SAT
MK = Manipulatives Kit
36A Chapter 1
gge08te_ny_c01_0036A_0036B.indde08te_ny_c01_0036A_0036B.indd 336A6A 112/21/062/21/06 112:42:082:42:08 PPMM Section Overview
Formulas in Geometry Lesson 1-5
Finding area and perimeter of figures is an important skill in a variety of occupations.
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Ü >V À Ã
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P = 2 + 2w P = a + b + c C = 2πr P = 4s 1 2 = 2 A = w A = _ bh A = πr A s 2
Midpoint and Distance Lesson 1-6
Some problems are easier to solve when the figure is drawn on a coordinate plane.
Midpoint Formula Distance Formula
The midpoint of A ( x 1 , y 1 ) and B ( x 2 , y 2 ) is The distance between A ( x 1 , y 1 ) and B ( x 2 , y 2 ) is
x + x y + y M ( _1 2 , _1 2 ) . AB = √ ( x - x ) + (y - y ) . 2 2 2 1 2 1
Given A (2, 7) and B (-4, 1) , the midpoint is Given A (2, 7) and B (-4, 1) , the distance is + (- ) 2 4 7 + 1 -2 8 2 2 M ( _ , _ ) = ( _ , _ ) = (-1, 4). AB = √ (-4 - 2) + (1 - 7) = √36 + 36 = √72 ≈ 8.5. 2 2 2 2
Transformations Lesson 1-7
Patterns are formed by translating, reflecting, and rotating figures.
,iviVÌ ,Ì>Ì /À>Ã>Ì Ī Ī Ī Ī Ī Ī Ī Ī Ī Ī Ī *
Each point and its image are the Each point and its image are the All points of a figure move same distance from the line of same distance from the center of the same distance in the same reflection. rotation P. direction.
36B
gge07te_c01_0036A_0036B.indde07te_c01_0036A_0036B.indd 336B6B 112/5/052/5/05 110:44:340:44:34 PPMM 1-5 Organizer 1-5 Using Formulas Pacing: Traditional 1 day in Geometry __1 Block 2 day Objectives: Apply formulas for Objective Why learn this? perimeter, area, and circumference. Apply formulas for Puzzles use geometric-shaped pieces. perimeter, area, and Formulas help determine the amount of circumference. Geometry Lab materials needed. (See Exercise 6.) In Geometry Lab Activities Vocabulary perimeter The perimeter P of a plane figure is the sum of the @
Perimeter and Area RECTANGLE SQUARE TRIANGLE Warm Up Ü V> Evaluate. Round to the nearest à hundredth. Ű L P = 2 + 2w or 2( +w) P= 4s P= a + b + c 2 2 NY Performance 1. 12 144 2. 7.6 57.76 A = w A = s 2 A = __1 bh or ___ bh Indicators 2 2 3. √64 8 4. √54 7.35 G.CN.6 Recognize and apply mathematics to situations in the 5. 3 2 (π) 28.27 6. ( 3π) 2 88.83 outside world. The base b can be any side of a triangle. The height h is a segment from a vertex that forms a right angle with a line containing the base. The height may Also available on transparency be a side of the triangle or in the interior or the exterior of the triangle.
L L L
EXAMPLE 1 Finding Perimeter and Area Q: What is the hidden math term? Find the perimeter and area of each figure. RO A rectangle in which = 17 cm B triangle in which a = 8, and w = 5 cm b = (x + 1) , c = 4x, and h = 6 OT Perimeter is expressed in linear units, such P = 2 + 2wP= a + b + c A: Square root as inches (in.) or = 2(17) + 2(5) = 8 + (x + 1) + 4x meters (m). Area is = 34 + 10 = 44 cm = 5x + 9 expressed in square Inclusion Perimeter con- 1 units, such as square A = wA= _bh tains the word rim, which 2 centimeters (cm 2 ). is a single path, so it is one = (17)(5) = 85 cm 2 = _1 (x + 1)(6) = 3x + 3 dimension. Area is found by mea- 2 suring two quantities, so it is two- = dimensional. 1. Find the perimeter and area of a square with s 3.5 in. P = 14 in.; A = 12.25 in 2
36 Chapter 1 Foundations for Geometry New York Performance Indicators 1 Process Introduce G.CN.6 Recognize and apply ge07se_c01_0036_0041.indd 1 11/23/06 1:31:46 PM mathematics to situations in the EXPLORATION Motivate outside world £x 1Ã}ÊÀÕ>ÃÊÊ iiÌÀÞ Discuss buying tile for a room. Ask students how 2ECALL THAT THE AREA OF A RECTANGLE IS THE PRODUCT OF ITS LENGTH AND WIDTH 9OU CAN USE THIS FACT TO DEVELOP THE FORMULA FOR they would determine how much tile they would THE AREA OF A TRIANGLE need to cover a floor. Solicit from them the idea 5SE A STRAIGHTEDGE TO DRAW 4RACE AROUND THE TRIANGLE TO A TRIANGLE 4HEN CUT OUT THE MAKE A COPY OF IT 4HEN CUT of finding the number of same-sized squares that TRIANGLE OUT THIS SECOND TRIANGLE are needed. Ask them how they could determine the length of baseboard needed to trim the room. !RRANGE THE TWO TRIANGLES #UT OFF ONE END OF THE FIGURE AS SHOWN AND MOVE IT TO THE OTHER SIDE TO MAKE A RECTANGLE
3UPPOSE THE ORIGINAL TRIANGLE HAS BASE B Explorations and answers are provided in the AND HEIGHT H 7HAT ARE THE LENGTH AND WIDTH OF THE RECTANGLE IN TERMS OF B AND H Explorations binder. / Ê Ê - 1-- 3HOW HOW YOU CAN WRITE THE AREA OF THE RECTANGLE IN TERMS OF B AND H 36 Chapter 1 %XPLAIN HOW THE AREA OF THE ORIGINAL TRIANGLE IS RELATED TO THE
gge08te_ny_c01_0036_0041.indde08te_ny_c01_0036_0041.indd 3636 112/21/062/21/06 112:43:272:43:27 PPMM EXAMPLE 2 Crafts Application The Texas Treasures quilt block includes 24 purple triangles. The base and height of each triangle are about 3 in. Find the approximate amount of Additional Examples fabric used to make the 24 triangles. Example 1 The area of one triangle is The total area of the 24 triangles is Find the perimeter and area of A = _1bh = _1 (3)(3) = 4 _1 in 2 . 24(4_1 ) = 108 in 2 . 2 2 2 2 each figure. A. 2. Find the amount of fabric used to make the four rectangles. __1 __1 {Ê° Each rectangle has a length of 6 2 in. and a width of 2 2 in. 65 in 2 ÈÊ° In a circle a diameter is a segment that passes 20 in.; 24 in 2 through the center of the circle and whose ,>`ÕÃ endpoints are on the circle. A radius of a B. >iÌiÀ xÝ circle is a segment whose endpoints are the iÌiÀ ÝÊ Ê{ center of the circle and a point on the circle. The circumference of a circle is the distance ÀVÕviÀiVi È around the circle. 6x + 10; 3x + 12
Circumference and Area of a Circle Example 2
The circumference C of a circle is given by the formula C = πd or C = 2πr. The Queens Quilt block includes The area A of a circle is given by the formula A = πr 2. 12 blue triangles. The base and height of each triangle are about 4 in. Find the approximate The ratio of a circle’s circumference to its diameter is the same for all circles. amount of fabric used to make This ratio is represented by the Greek letter π (pi) . The value of π is irrational. 2 __22 the 12 triangles. 96 in Pi is often approximated as 3.14 or 7 . Example 3 EXAMPLE 3 Finding the Circumference and Area of a Circle Find the circumference and area Find the circumference and area of the circle. of a circle with radius 8 cm. Use C= 2πrA= πr 2 the π key on your calculator. ÎÊV = 2π (3) = 6π = π ( 3) 2 = 9π Then round the answer to the ≈ 18.8 cm ≈ 28.3 cm 2 nearest tenth. 50.3 cm; 201.1 cm 2 3. Find the circumference and area of a circle with radius 14 m. Also available on transparency C ≈ 88.0 m; A ≈ 615.8 m 2
INTERVENTION THINK AND DISCUSS G.CN.1, G.R.1 Questioning Strategies 1. 2 Describe three different figures whose areas are each 16 in . EXAMPLES 1–2 2. GET ORGANIZED Copy and ,iVÌ>}i • How does knowing the formula for complete the graphic ÀÕ>Ã /À>}i organizer. In each shape, the area of a rectangle help you find the area of a triangle? write the formula for its -µÕ>Ài ÀVi area and perimeter. EXAMPLE 3 • Compare the results when you multiply 6 by π and multiply 6 by 1- 5 Using Formulas in Geometry 37 3.14.
2 Teach 3 Close ge07se_c01_0036_0041.indd 2 11/23/06 1:31:58 PM Guided Instruction Summarize and INTERVENTION Review the meanings of perimeter, area, Review the formulas for finding the area Diagnose Before the Lesson and circumference, and the parts of a and perimeter (circumference) of a rect- 1-5 Warm Up, TE p. 36 circle. Show examples of triangles having angle, triangle, and circle. interior and exterior heights. Give examples Monitor During the Lesson of finding the area and circumference of a Check It Out! Exercises, SE pp. 36–37 circle, leaving answers in terms of π, and Answers to Think and Discuss Questioning Strategies, TE p. 37 rounding. Remind students that the diam- 1. Possible answer: A rect. with length eter d is twice the radius r or d = 2r. 8 in. and width 2 in.; a square with sides 4 in. long; a with base 4 in. Assess After the Lesson and height 8 in. 1-5 Lesson Quiz, TE p. 41 2. See p. A2. Alternative Assessment, TE p. 41
Lesson 1-5 37
gge08te_ny_c01_0036_0041.indde08te_ny_c01_0036_0041.indd 3737 112/21/062/21/06 11:38:22:38:22 PPMM 1-5 ExercisesExercises 1-5 Exercises KEYWORD: MG7 1-5
KEYWORD: MG7 Parent Assignment Guide GUIDED PRACTICE 1. Both terms Vocabulary Apply the vocabulary from this lesson to answer each question. Assign Guided Practice exercises refer to the dist. 1. Explain how the concepts of perimeter and circumference are related. as necessary. around a figure. 2. For a rectangle, length and width are sometimes used in place of __?__. If you finished Examples 1–3 (base and height or radius and diameter) base and height P = (x + 21) m; Basic 10–17, 19–24, 26–34 SEE EXAMPLE 1 Find the perimeter and area of each figure. A = (2x + 6) m 2 even, 38, 41, 42, 44, p. 36 3. 4. 5. 46–50, 56–62 {Ê £ÎÊ ÞÊÊÎ xÊ {Ê Average 10–30, 32–52, 56–62 P = 4y - 12; Advanced 10–30 even, 31–62 ££Ê = ( - ) 2 P = 30 mm; A = 44 mm 2 A y 3 ÎÊ ÝÊ = y 2 - 6y + 9 Homework Quick Check SEE EXAMPLE 2 6. Manufacturing A puzzle contains a triangular Quickly check key concepts. p. 37 piece with a base of 3 in. and a height of 4 in. Exercises: 10, 14, 24, 34, 36, A manufacturer wants to make 80 puzzles. 38, 42 Find the amount of wood used if each puzzle contains 20 triangular pieces. 9600 in 2
Math Background SEE EXAMPLE 3 Find the circumference and area of each circle. Use the π key on your calculator. Exercise 13 involves quilt p. 37 Round to the nearest tenth. patterns. Plan ahead and 7. 8. 9. ask students whether their families Ó°£Ê ÇÊ° £ÈÊV have quilts they may bring in as examples of different geometric patterns. C ≈ 13.2 m; A ≈ 13.9 m 2 C ≈ 44.0 in.; A ≈ 153.9 in 2 C ≈ 50.3 cm; A ≈ 201.1 cm 2 PRACTICE AND PROBLEM SOLVING Independent Practice Find the perimeter and area of each figure. For See Exercises Example 10. 11. 12. xÝ 10–12 1 Ç°{Ê Ý {Ý ÎÝ 13 2 ÝÊ ÊÈ n 14–16 3 P = 29.6 m; A = 54.76 m 2 P = 4x + 12; A = x 2 + 6xP= 9x + 8; A = 12x Extra Practice 13. Crafts The quilt pattern includes 32 small triangles. Skills Practice p. S5 Each has a base of 3 in. and a height of 1.5 in. Find the Application Practice p. S28 amount of fabric used to make the 32 triangles. 72 in 2
Find the circumference and area of each circle with the given radius or diameter. Use the π key on your calculator. Round to the nearest tenth. 14. C ≈ 75.4 m; 14. r = 12 m 15. d = 12.5 ft 16. d = _ 1 mi A ≈ 452.4 m 2 2 15. C ≈ 39.3 ft; Find the area of each of the following. 2 A ≈ 122.7 ft 2 17. square whose sides are 9.1 yd in length 82.81 yd ( ) 2 + + 16. C ≈ 1.6 mi; 18. square whose sides are x + 1 in length x 2x 1 2 A ≈ 0.2 mi 19. __1 __1 6.1875 in 2 triangle whose base is 5 2 in. and whose height is 2 4 in.
38 Chapter 1 Foundations for Geometry
,%33/. *À>VÌViÊ £x 5SING &ORMULAS IN 'EOMETRY1-5 PRACTICE A #OMPLETE THE STATEMENTS 4HE SUM OF THE SIDE LENGTHS OF A PLANE FIGURE IS CALLED THE PERIMETER
'IVE THE FORMULA FOR THE PERIMETER OF A RECTANGLE 0 W
4HE AREA OF A PLANE FIGURE IS THE NUMBER OF NONOVERLAPPING ge07se_c01_0036_0041.indd 38 SQUARE UNITS OF A GIVEN SIZE THAT EXACTLY COVER THE FIGURE 12/2/05 3:48:28 PM ! ?? BH 4HE FORMULA FOR THE AREA OF A TRIANGLE IS
5SE THE FIGURE FOR %XERCISES n
&IND THE PERIMETER OF THE RECTANGLE YD YD &IND THE AREA OF THE RECTANGLE YD
YD 5SE THE FIGURE FOR %XERCISES n &IND THE PERIMETER OF THE TRIANGLE CM CM CM &IND THE AREA OF THE TRIANGLE CM CM #OMPLETE THE STATEMENTS )N A CIRCLE A DIAMETER IS A SEGMENT THAT PASSES THROUGH THE CENTER OF THE CIRCLE AND THAT HAS ENDPOINTS ARE ON THE CIRCLE
! RADIUS OF A CIRCLE IS A SEGMENT WHOSE ENDPOINTS ARE THE CENTER OF THE CIRCLE AND A POINT ON THE CIRCLE
4HE DIAMETER OF A CIRCLE IS TWICE THE RADIUS
$RAW YOUR ANSWER IN THE SPACE PROVIDED 3KETCH A CIRCLE AND LABEL THE CENTER A DIAMETER AND A RADIUS CENTER RADIUS DIAMETER KEYWORD: MG7 Resources
'IVE THE FORMULA FOR THE AREA OF A CIRCLE ! QR
4HE CIRCUMFERENCE OF A CIRCLE IS THE DISTANCE AROUND THE CIRCLE 38 Chapter 1 'IVE THE FORMULA FOR THE CIRCUMFERENCE OF A CIRCLE # QR OR # QD
gge07te_c01_0036_0041.indde07te_c01_0036_0041.indd 3388 112/20/052/20/05 110:02:370:02:37 AAMM Given the area of each of the following figures, find each unknown measure. " " Ê ,,", 2 20. The area of a triangle is 6.75 m . If the base of the triangle is 3 m, what is the height ,/ of the triangle? 4.5 m 21. A rectangle has an area of 347.13 cm 2 . If the length is 20.3 cm, what is the width of In Exercises 24 and 25, students 2 the rectangle? 17.1 cm may confuse πr in the area for- mula and 2πr in the circumference 22. The area of a circle is 64π. Find the radius of the circle. r = 8 formula. Remind them that area is 23. /////ERROR ANALYSIS///// Below are two statements about the measured in square units, so they area of the circle. Which is incorrect? Explain the error. nÊV should use the formula in which r is ! " squared. ÊrÊûÀÓ ÊrÊûÀÓ ÊrÊûn®Ó ÊrÊû{®Ó A = π( 8) 2 is incorrect. Reading Math In Ê rÊÈ{ûÊVÓÊ ÊrÊ£ÈûÊVÓÊ The radius is 4, not 8. Exercises 24 and 25, A = πr 2 = π( 4) 2 = 16π cm 2 remind students that the direction to leave answers ENGLISH Find the area of each circle. Leave answers in terms of π. in terms of π means that LANGUAGE 2 LEARNERS 24. circle with a diameter of 28 m 196π m π is part of the answer. 25. circle with a radius of 3y 9y 2 π 26. Geography The radius r of the earth at the Exercise 30 involves equator is approximately 3964 mi. Find the perimeters of trian- gles and the circum- distance around the earth at the equator. r Use the π key on your calculator and round ference of a semicircle. This exercise to the nearest mile. 24,907 mi prepares students for the Multi-Step Equator Test Prep on page 58. 27. For a square, 27. Critical Thinking Explain how the formulas for the length and the perimeter and area of a square may be derived width are both s, from the corresponding formulas for a rectangle. = + = so P 2 2w 28. + = Find the perimeter and area of a rectangle whose 2s 2s 4s and length is (x + 1) and whose width is ( x - 3) . = = ( ) = 2 ÝÊÊÎ A w s s s Express your answer in terms of x. 28. P = 4x - 4; 29. Multi-Step If the height h of a triangle is 3 inches less ÝÊ Ê£ A = (x + 1)(x - 3) than the length of the base b, and the area A of the = x 2 - 2x - 3 triangle is 19 times the length of the base, find b and h. b = 41 in.; h = 38 in.
30. This problem will prepare you for the Multi-Step Test Prep on page 58. A landscaper is to install edging around a garden. The edging costs $1.39 for each 24-inch-long strip. {ÊvÌ {ÊvÌ The landscaper estimates it will take 4 hours to ÎÊvÌ ÎÊvÌ install the edging. a. If the total cost is $120.30, what is the cost of the material purchased? $20.85 b. What is the charge for labor? $99.45 c. What is the area of the semicircle to the nearest tenth? 25.1 ft 2 d. What is the area of each triangle? 6 ft 2 e. What is the total area of the garden to the nearest foot? 37 ft 2
1- 5 Using Formulas in Geometry 39
,%33/. *À>VÌViÊ ,%33/. *À>VÌViÊ £x 5SING &ORMULAS IN 'EOMETRY1-5 PRACTICE B £x 5SING &ORMULAS IN 'EOMETRY1-5 PRACTICE C 5SE THE FIGURES FOR %XERCISES n &IND THE LENGTH OF THE SIDES OF A SQUARE WHOSE AREA ! FT AND PERIMETER ARE THE SAME NONZERO NUMBER &IND THE PERIMETER OF TRIANGLE ! FT FT &IND THE LENGTH OF THE RADIUS OF A CIRCLE WHOSE AREA &IND THE AREA OF TRIANGLE ! FT FT AND CIRCUMFERENCE ARE THE SAME NONZERO NUMBER " %XPLAIN WHY THE AREA AND THE PERIMETER OR THE CIRCUMFERENCE 4RIANGLE ! IS IDENTICAL TO TRIANGLE " FT OR ?? FT H &IND THE HEIGHT H OF TRIANGLE " OF A FIGURE CAN NEVER BE EQUAL THEY CAN ONLY HAVE EQUAL NUMBERS ge07se_c01_0036_0041.indd 39 FT 12/2/05 3:48:31 PM !REA IS MEASURED IN SQUARE UNITS PERIMETER IS MEASURED IN LINEAR UNITS &IND THE PERIMETER AND AREA OF EACH SHAPE SQUARE WITH A SIDE M IN LENGTH RECTANGLE WITH LENGTH X AND WIDTH &IND THE MEASUREMENTS 0 M ! M 0 X ! X &AYE HAS FEET OF FENCING TO ENCLOSE A RECTANGULAR GARDEN 3HE WANTS TO !LTHOUGH A CIRCLE DOES NOT HAVE SIDES IT DOES HAVE A PERIMETER ENCLOSE AS MUCH AREA AS POSSIBLE 5SE TRIAL AND ERROR TO FIND THE MAXIMUM 7HAT IS THE TERM FOR THE PERIMETER OF A CIRCLE CIRCUMFERENCE AREA &AYE CAN ENCLOSE WITH ALL FEET OF FENCE .AME THE LENGTH AND WIDTH THAT GIVE THE MAXIMUM AREA &IND THE CIRCUMFERENCE AND AREA OF EACH CIRCLE ! FT FT W FT %XPLAIN WHAT THE ANSWER TO %XERCISE IMPLIES ABOUT THE RELATIONSHIP OF PERIMETER TO AREA FOR RECTANGLES MI X
CM &OR A GIVEN PERIMETER A RECTANGLE WITH SIDES OF EQUAL LENGTH
,EAVE P AS P A SQUARE ENCLOSES THE MAXIMUM AREA 5SE FOR P 5SE FOR P
# MI # CM # QX &AYE DECIDES TO USE HER FEET OF FENCING TO ENCLOSE A CIRCULAR GARDEN &IND THE AREA OF THE GARDEN 5SE ??? FOR P ABOUT FT ! MI ! CM ! QX X &IND THE DIFFERENCE BETWEEN THE AREA &AYE CAN ENCLOSE BY A 4HE AREA OF A SQUARE IS ?? IN &IND THE PERIMETER IN CIRCLE AND THE MAXIMUM AREA SHE CAN ENCLOSE BY A RECTANGLE ABOUT FT 4HE AREA OF A TRIANGLE IS M AND THE HEIGHT IS M &IND THE BASE M %XPLAIN WHAT THE ANSWER TO %XERCISE IMPLIES ABOUT THE RELATIONSHIP OF PERIMETER TO AREA FOR RECTANGLES AND CIRCLES 4HE CIRCUMFERENCE OF A CIRCLE IS P MM &IND THE RADIUS MM )F A RECTANGLE AND A CIRCLE HAVE THE SAME PERIMETER THEN THE
5SE THE FIGURE FOR %XERCISES n FT CIRCLE HAS THE GREATER AREA
,UCAS HAS A FOOT LONG ROPE (E USES ALL THE ROPE TO FT FT ! RECTANGULAR BOX OF TISSUES IS INCHES LONG INCHES WIDE OUTLINE THIS 4 SHAPE IN HIS BACKYARD !LL THE ANGLES IN AND INCHES HIGH &IND THE AREA OF THE SURFACE OF THE BOX IN THE FIGURE ARE RIGHT ANGLES FT FT ! RIGHT TRIAGNLE HAS TWO LEGS WITH LENGTHS A AND B AND A &IND X FT X HYPOTENUSE WITH LENGTH C )N THIS TRIANGLE THE AREA AND X &IND THE AREA ENCLOSED BY THE ROPE FT PERIMETER ARE THE SAME NONZERO NUMBER &IND THE LENGTH A IF B (INT 5SE THE 0YTHAGOREAN 4HEOREM A B C A Lesson 1-5 39
gge07te_c01_0036_0041.indde07te_c01_0036_0041.indd 3399 112/20/052/20/05 110:03:080:03:08 AAMM Diversity Exercise 32 31. Algebra The large rectangle has length a + b and > L gives dimensions of inter- width c + d. Therefore, its area is (a + b)(c + d). national soccer fields. a. Find the area of each of the four small rectangles V The size of the playing field in in the figure. Then find the sum of these areas. ` Paralympics soccer, for athletes Explain why this sum must be equal to the product ( + )( + ) + + + with cerebral palsy, is 75 m long b. (a + 1)(c + 1) ; a b c d . ac ad bc bd and 55 m wide. ac + a + c + 1 b. Suppose b = d = 1. Write the area of the large rectangle as a product of its length and width. Then find the sum of the areas of the four small rectangles. Explain ( + ) 2 Number Sense c. a 1 ; why this sum must be equal to the product (a + 1)(c + 1) . 2 + + For Exercises 41–43, a 2a 1 c. Suppose b = d = 1 and a = c. Write the area of the large rectangle as a product of explain to students that its length and width. Then find the sum of the areas of the four small rectangles. leaving the answer in terms of π Explain why this sum must be equal to the product (a + 1) 2 . gives an exact answer. Point out that 32. Sports The table shows the minimum and maximum dimensions for rectangular the decimal answer of the diameter soccer fields used in international matches. Find the difference in area of the π found by using the key on a calcu- largest possible field and the smallest possible field. 1850 m 2 lator is an approximation. Minimum Maximum If students chose D Length 100 m 110 m for Exercise 47, they Width 64 m 75 m divided by 2 instead of taking the square root. If they chose F for Exercise 48, they found Find the value of each missing measure of a triangle. the length instead of the width. If 33. b = 2 ft; h = ft; A = 28 ft 2 28 ft 34. b = ft; h = 22.6 yd; A = 282.5 yd 2 they chose G for Exercise 50, they 25 yd did not divide by 2 in the equation Find the area of each rectangle with the given base and height. representing the area. 35. 9.8 ft; 2.7 ft 26.46 ft 2 36. 4 mi 960 ft; 440 ft _ 37. 3 yd 12 ft; 11 ft 2 2 9,715,200 ft , or 0.348 mi 25_2 yd 2 or 231 ft 2 Find the perimeter of each rectangle with the given base and height. 3 Answers 38. 21.4 in.; 7.8 in. 58.4 in.39. 4 ft 6 in.; 6 in. 10 ft40. 2 yd 8 ft; 6 ft 13 yd 1 ft 31a. This must be equal to (a + b) (c + d) because the Find the diameter of the circle with the given measurement. Leave answers in terms of π. sum of the areas of the 4 _14 41. C = 14 π 42. A = 100π 2043. C = 50π 50 small rects. equals the area of the large rect. 45. Measure any 44. A skate park consists of a two adjacent rectangular £ÇÊÞ` side as the base. regions as shown. Find the perimeter and area of b. This must be equal to the prod- the park. P = 52 yd; A = 137 yd 2 Then measure the ÊÞ` {ÊÞ` uct ( a + 1) (c +1) because the height of the at 45. Critical Thinking Explain how you would measure {ÊÞ` sum of the areas of the 4 small ∠ a rt. to the base. a triangular piece of paper if you wanted to find its rects. equals the area of the area. large rect. 46. Write About It c. This must be equal to the prod- A student wrote in her journal, “To find the perimeter of a rectangle, add the length and width together and then double this value.” Does her uct (a + 1) 2 because the sum method work? Explain. The method works because adding the length and width of the areas of the 4 small rects. together and doubling the result is 2 ( + w), which is equivalent to 2 + 2w. equals the area of the large rect.
47. Manda made a circular tabletop that has an area of 452 in2. Which is closest to the radius of the tabletop? 9 in. 12 in. 24 in. 72 in.
48. A piece of wire 48 m long is bent into the shape of a rectangle whose length is twice its width. Find the length of the rectangle. 8 m 16 m 24 m 32 m 40 Chapter 1 Foundations for Geometry
,%33/. ,i>`}Ê-ÌÀ>Ìi}ià ,%33/. ,iÌi>V £x 5SE A 'RAPHIC !ID1-5 READING STRATEGIES £x 5SING &ORMULAS IN 'EOMETRY1-5 RETEACH 4HE PERIMETER OF A FIGURE IS THE SUM OF THE LENGTHS OF THE SIDES OF THE FIGURE 4HE AREA 4HE PERIMETER OF A FIGURE IS THE SUM OF THE LENGTHS OF THE SIDES OF A FIGURE IS THE NUMBER OF NONOVERLAPPING SQUARE UNITS OF A GIVEN SIZE THAT EXACTLY 4HE AREA IS THE NUMBER OF SQUARE UNITS ENCLOSED BY THE FIGURE COVER THE FIGURE &IGURE 2ECTANGLE 3QUARE #ONSIDER THE RECTANGLE ON THE GRID BELOW 4O FIND THE PERIMETER OF THIS RECTANGLE COUNT -ODEL S THE SQUARES AS YOU GO AROUND THE OUTSIDE OF Y THE RECTANGLE 4HIS COULD BE DONE AS FOLLOWS WW S S 4 Y ge07se_c01_0036_0041.indd 5 567 S 11/23/06 1:32:24 PM ge07se_c01_0036_0041.indd 41 12/19/05 5:26:01 PM 4 8 4 3 9 X 2 10 0ERIMETER 0 S 0 4 0 W OR W 1 11 14 13 12 X !REA ! W ! S 0 4
4HE PERIMETER OF THE RECTANGLE IS UNITS &IND THE PERIMETER AND AREA OF EACH FIGURE .OW FIND THE AREA OF THE SAME FIGURE 4HE AREA OF A RECTANGLE IS CALCULATED RECTANGLE WITH FT W FT SQUARE WITH S MM BY MULTIPLYING THE LENGTH BY THE WIDTH 4HE LENGTH OF THE RECTANGLE IS UNITS AND THE WIDTH IS UNITS 4HEREFORE FT FT MM MM THE AREA OF THIS RECTANGLE IS SQUARE UNITS IN !NSWER EACH QUESTION CM X X (OW CAN YOU FIND THE PERIMETER OF THIS SQUARE !NSWERS WILL VARY 3TUDENTS SHOULD INDICATE THAT THE IN Y PERIMETER OF THE SQUARE IS THE SUM OF THE LENGTHS OF CM CM X IN X IN THE SIDES X 4HE PERIMETER OF A TRIANGLE IS THE SUM OF ITS SIDE LENGTHS 4HE BASE AND HEIGHT ARE USED TO FIND THE AREA
0ERIMETER !REA C C A A 0 A B C ! ?? BH OR ??? BH H H (OW IS FINDING THE PERIMETER OF A SQUARE LIKE FINDING THE PERIMETER OF A RECTANGLE
!NSWERS WILL VARY 3TUDENTS SHOULD POINT OUT THAT THE PERIMETER OF ANY B B
OBJECT IS THE SUM OF THE LENGTHS OF ALL THE SIDES &IND THE PERIMETER AND AREA OF EACH TRIANGLE 7HAT IS THE FORMULA FOR FINDING THE AREA OF A SQUARE
FT CM !REA LENGTH WIDTH OR !REA LENGTH OF ONE SIDE Y FT CM CM 7HAT IS THE AREA OF THE SQUARE SHOWN ABOVE FT CM !REA LENGTH WIDTH !REA !REA SQUARE UNITS Y FT Y FT CM CM 40 Chapter 1
gge08te_ny_c01_0036_0041.indde08te_ny_c01_0036_0041.indd 4040 112/21/062/21/06 11:39:33:39:33 PPMM 49. Which equation best represents the area A of the triangle? " " Ê ,,", 2 A = 2x + 4x ,/ A = 4x(x + 2) ÝÊ ÊÓ A = 2x 2 + 2 In Exercise 51, some students may A = 4x 2 + 8 {Ý attempt to subtract the area of the rectangle from the area of the circle. 50. Ryan has a 30 ft piece of string. He wants to use the string to lay out the Explain that they must subtract the boundary of a new flower bed in his garden. Which of these shapes would smaller area from the larger area, or use all the string? the result will be a negative number. A circle with a radius of about 37.2 in. A rectangle with a length of 6 ft and a width of 5 ft Algebra For Exercise 54, A triangle with each side 9 ft long you might need to review A square with each side 90 in. long that you find the percent of change by dividing the difference by the original.
CHALLENGE AND EXTEND £{Ê° 51. A circle with a 6 in. diameter is stamped out of a rectangular piece of metal as shown. Journal Find the area of the remaining piece of metal. nÊ° Math History Use the π key on your calculator and round Have students draw a rectangle, 2 to the nearest tenth. 83.7 in a triangle, and a circle on a grid P - 2 = + w = _ and explain how they would 52. a. Solve P 2 2w for w. 2 b. Use your result from part a to find the width of a rectangle that has a perimeter find the area and perimeter (or of 9 ft and a length of 3 ft. 1.5 ft circumference) of each figure, using their own dimensions. 53. Find all possible areas of a rectangle whose sides are natural numbers and whose perimeter is 12. 5; 8; 9 The Ahmes Papyrus is 54. Estimation The Ahmes Papyrus dates from approximately 1650 B.C.E. Lacking an ancient Egyptian π source of information a precise value for , the author assumed that the area of a circle with a about mathematics. diameter of 9 units had the same area as a square with a side length of 8 units. Give students the area, perimeter, A page of the Ahmes By what percent did the author overestimate or underestimate the actual area and circumference of three figures. Papyrus is about 1 foot of the circle? overestimated by about 0.6% Then ask students to find the dimen- wide and 18 feet long. 55. Multi-Step The width of a painting is __4 the measure of the length of the painting. sions and draw the figures that Source: scholars.nus.edu.sg 5 If the area is 320 in 2 , what are the length and width of the painting? match the corresponding values. width = 16 in.; length = 20 in. SPIRAL REVIEW Determine the domain and range of each function. (Previous course) 1-5 56. (2, 4) , (-5, 8) , (-3, 4) D: {2, -5, -3}; 57. (4, -2) , ( -2, 8) , (16, 0) D: {4, -2, 16}; R: {4, 8} R: {-2, 8, 0} Find the area and perimeter of Name the geometric figure that each item suggests. (Lesson 1-1) line or each figure. 58. the wall of a classroom plane 59. the place where two walls meet segment 1. 60. Marion has a piece of fabric that is 10 yd long. She wants to cut it into 2 pieces so Ý that one piece is 4 times as long as the other. Find the lengths of the two pieces. (Lesson 1-2) 8 yd; 2 yd _ ÝÊ Ê{ 2 61. Suppose that A, B, and C are collinear points. B is the midpoint of AC . The coordinate x + 4x; 4x + 8 of A is -8, and the coordinate of B is -2.5. What is the coordinate of C? (Lesson 1-2) 3 2. {°nÊV 62. An angle’s measure is 9 degrees more than 2 times the measure of its supplement. Find the measure of the angle. (Lesson 1-4) 123°
1- 5 Using Formulas in Geometry 41 23.04 cm 2 ; 19.2 cm
*ÀLiÊ-Û} >i}i 3. ,%33/. 1-5 PROBLEM SOLVING ,%33/. 1-5 CHALLENGE £x 5SING &ORMULAS IN 'EOMETRY £x 5SING &ORMULAS FOR 2EGULAR 0OLYGONS ÎÝÝÊ ÊÈ 5SE THE TABLE FOR %XERCISES n ! REGULAR POLYGON IS A POLYGON IN WHICH ALL OF THE SIDES ARE ÓÝ CONGRUENT AND ALL OF THE ANGLES ARE CONGRUENT 4HE APOTHEM &IGURE 0ERIMETER OR #IRCUMFERENCE !REA IS THE DISTANCE FROM THE CENTER TO A SIDE 4HE AREA ! OF A REGULAR RECTANGLE 0 W OR W ! W POLYGON WITH A PERIMETER 0 AND APOTHEM A IS GIVEN BY THE FORMULA A ?? TRIANGLE 0 A B C! ?? BH ! 0A £ä ge07se_c01_0036_0041.indd 40 12/2/05 3:48:33 PM ge07se_c01_0036_0041.inddCIRCLE 41 # PR! PR 12/19/05 5:26:01 PM 5SE THE REGULAR POLYGON FOR %XERCISES n 10x; 4x + 16 5SE THE DIAGRAM OF A HOCKEY FIELD FOR %XERCISES n &IND THE PERIMETER AND AREA
sideline 100 yards 0 CM ! CM CM CM )F THE SIDE LENGTH AND APOTHEM ARE DOUBLED Find the circumference and HOW WILL THE PERIMETER AND AREA CHANGE area of each circle. Leave 4HE PERIMETER WILL DOUBLE 4HE AREA WILL BE TIMES GREATER goal -AKE A CONJECTURE ABOUT HOW THE AREA ! OF A REGULAR POLYGON WILL CHANGE IF THE π
center line answers in terms of . 25-yard line 25-yard line SIDE LENGTH AND APOTHEM ARE TRIPLED 7RITE AN EXPRESSION THAT GIVES THE NEW AREA back line 60 yards AFTER YOU MULTIPLY THE SIDE LENGTH AND APOTHEM BY A POSITIVE NUMBER K shooting 16 yards 4 yards circle π 2 π 4HE AREA WILL BE TIMES GREATER K ! 4. radius 2 cm 4 cm; 4 cm 7HAT IS THE PERIMETER OF THE FIELD 7HAT IS THE AREA OF THE FIELD #ALCULATE THE SHADED AREA OF EACH FIGURE !SSUME THAT POLYGONS APPEARING TO π 2 π YD YD BE REGULAR ARE REGULAR 2OUND YOUR ANSWERS TO THE NEAREST TENTH 5. diameter 12 ft 36 ft; 12 ft 7HAT IS THE AREA OF EACH OF THE SHOOTING 7HAT IS THE AREA OF THE FIELD BETWEEN THE CIRCLES 5SE FOR P TWO YARD LINES 20.8 m 3.3 cm 6. The area of a rectangle is 4 cm ABOUT YD YD 12 m 2 #HOOSE THE BEST ANSWER 74.82 in , and the length is ! RECTANGULAR COUNTER FEET WIDE AND FEET LONG HAS A CIRCLE CUT OUT OF IT IN ORDER M CM TO HAVE A SINK INSTALLED 4HE CIRCLE HAS A DIAMETER OF INCHES 7HAT IS THE 12.9 in. Find the width. 5.8 in. APPROXIMATE AREA OF THE REMAINING COUNTERTOP SURFACE 5SE FOR P 9.2 mm ! FT # FT " FT $ FT 3 in. 8 mm Also available on transparency 4HE BASE OF A TRIANGULAR GARDEN MEASURES FEET )TS HEIGHT IS FEET 2.5 in. )F POUNDS OF MULCH ARE NEEDED TO COVER A SQUARE FOOT HOW MANY POUNDS OF MULCH WILL BE NEEDED TO COVER THE GARDEN IN MM & LB ' LB ( LB * LB Lesson 1-5 41
gge07te_c01_0036_0041.indde07te_c01_0036_0041.indd 4411 112/20/052/20/05 110:04:030:04:03 AAMM Organizer Graphing in the See Skills Bank page S56 NY Performance Algebra Coordinate Plane Indicators Pacing: Algebra G.R.1, G.CN.6 Traditional __1 day 2 The coordinate plane is used to name and Block __1 day 4 See Skills Bank Þ>ÝÃ locate points. Points in the coordinate plane { page S56 Objective: Review the are named by ordered pairs of the form (x, y). +Õ>`À>ÌÊ +Õ>`À>ÌÊ coordinate plane. The first number is the x-coordinate. Ó The second number is the y-coordinate. Ý>ÝÃ @`À>ÌÊ +Õ>`À>ÌÊ6 Countdown to quadrants, numbered with Roman numerals { Testing Week 2 placed counterclockwise.
Examples
Teach 1 Name the coordinates of P. Þ Starting at the origin ( 0, 0), you count 1 unit to the right. { Remember * Then count 3 units up. So the coordinates of P are (1, 3) . Students review the parts of the Ó coordinate plane and the associated 2 Plot and label H(-2, -4) on a coordinate plane. Ý terms and locate and plot points. Name the quadrant in which it is located. { Ó ä Ó { INTERVENTION For addi- Start at the origin ( 0, 0) and move 2 units left. Then move Ó 4 units down. Draw a dot and label it H. H is in Quadrant III. tional review and practice on plot- { ting points in the coordinate plane, You can also use a coordinate plane to locate places on a map. see Skills Bank page S56.
Inclusion Have students associate the x-coordinate with right/east and left/ Try This west direction and the y-coordinate
with up/north and down/south Name the coordinates of the Þ direction from the origin. Point out point where the following >}> { >ÜÌ À streets intersect. VÀÞ ( ) Ê Ê that both of the ordered pairs x, y À and (horizontal, vertical) are in 1. Chestnut and Plum (0, 0) >«i Î iÃÌÕÌ ÀÞÊ iÀÀÞ i`>À alphabetical order. 2. Magnolia and Chestnut (0, 4) "> Ó 3. Oak and Hawthorn (3, 2) Ê iiV ÀV Close à 4. Plum and Cedar (-1, 0) £ *i Assess Name the streets that intersect Ý Have students draw and label a *Õ at the given points. coordinate plane. Then have them { Î Ó £ ä £ÓÎ{ 5. (-3, -1) 6. (4, -1) make a design. The students can £ -«ÀÕVi exchange designs and label each 7. (1, 3) 8. (-2, 1) others’ points. 5. Spruce and Beech 6. Spruce and Hickory 7. Maple and Elm 8. Pine and Birch New York Performance Indicators 42 Chapter 1 Foundations for Geometry
Process G.CN.6 Recognize and apply mathematics to situations in the outside world G.R.1 Use physical objects, diagrams, ge07se_c01_0042.indd 1 11/23/06 1:33:03 PM charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts
42 Chapter 1
gge08te_ny_c01_0042.indde08te_ny_c01_0042.indd 4422 112/21/062/21/06 112:44:382:44:38 PPMM 1-6 Midpoint and Distance 1-6 Organizer in the Coordinate Plane Pacing: __1 Traditional 2 day __1 Block 4 day Objectives Why learn this? Objectives: Develop and apply Develop and apply the You can use a coordinate plane to help the formula for midpoint. formula for midpoint. you calculate distances. (See Example 5.) Use the Distance Formula and the Use the Distance Formula Pythagorean Theorem to find the and the Pythagorean Major League baseball fields are laid out Theorem to find the distance between two points. according to strict guidelines. Once you distance between two points. know the dimensions of a field, you can @
endpoints A(x1, y1) and B(x2, y2) ÞÓ is found by ÝÓ]ÊÞÓ® B x x + x y + y _1 2 _1 2 Ý Ê ÊÝ Þ Ê ÊÞ M( , ). £ Ó £ Ó ÛiÀ>}iÊvÊ ÊÊpÊÊÊ]ÊÊÊpÊÊÊÊ 2 2 Ó Ó ® 2. Find CD. 8 Þ£Ê>`ÊÞÓ Þ£ Ý£]ÊÞ£® Ý C D ä Ý£ ÝÓ ÛiÀ>}iÊvÊ È Ó Ý£Ê>`ÊÝÓ 3. Find the coordinate−− of the midpoint of CD. -2 2 EXAMPLE 1 Finding the Coordinates of a Midpoint 4. Simplify. √ (3 - (-1) ) 4 −− Find the coordinates of the midpoint of CD Þ { Also available on transparency with endpoints C(-2, -1) and D(4, 2). {]ÊÓ® x + x y + y M (_1 2 , _1 2 ) To make it easier to 2 2 Ý picture the problem, { ä { -2 + 4 -1 + 2 2 1 plot the segment’s _ , _ = (_ , _) Ó]Ê£® endpoints on a 2 2 2 2 coordinate plane. { = (1, _1) 2 Q: What keeps a square from moving? _ 1. Find the coordinates of the midpoint of EF with endpoints A: Square roots. E(-2, 3) and F(5, -3) . 3 (_ , 0) 2 New York Performance 1- 6 Midpoint and Distance in the Coordinate Plane 43 Indicators
Geometry G.G.48 Investigate, justify, and apply 1 Introduce the Pythagorean Theorem and its converse ge07se_c01_0043_0049.inddEXPLORATION 43 Motivate 12/4/06 11:11:31G.G.66 AM Find the midpoint of a line £È `«ÌÊ>`Ê ÃÌ>ViÊÊ segment, given its endpoints Ì iÊ À`>ÌiÊ*>i Give students a coordinate graph on which a G.G.67 Find the length of a line 4HE COORDINATE PLANE SHOWS GRAPHS house and a school have been graphed. Ask them OF FOUR LINE SEGMENTS THAT ARE EACH segment, given its endpoints NAMED !" #OPY EACH SEGMENT to estimate the distance between the school and ONTO A SHEET OF GRAPH PAPER the house and to estimate the point that is half- &OLD EACH SEGMENT SO THAT ! MATCHES UP WITH " 5SE THE way between these two locations. FOLD TO FIND THE MIDPOINT OF THE SEGMENT )N A TABLE RECORD THE COORDINATES OF EACH SEGMENTS ENDPOINTS AND MIDPOINT #OORDINATES #OORDINATES #OORDINATES OF 3EGMENT OF ! OF " -IDPOINT Explorations and answers are provided in the &OR SEGMENTS AND HOW ARE THE X COORDINATES OF EACH SEGMENTS MIDPOINT RELATED TO THE X COORDINATES OF ITS ENDPOINTS Explorations binder. &OR SEGMENTS AND HOW ARE THE Y COORDINATES OF EACH SEGMENTS MIDPOINT RELATED TO THE Y COORDINATES OF ITS ENDPOINTS
/ Ê Ê - 1-- $ESCRIBE HOW YOU WOULD FIND THE COORDINATES OF THE MIDPOINT Lesson 1-6 43
gge08te_ny_c01_0043_0049.indde08te_ny_c01_0043_0049.indd 4343 112/21/062/21/06 112:46:092:46:09 PPMM EXAMPLE 2 Finding the Coordinates of an Endpoint −− M is the midpoint of AB . A has coordinates (2, 2), and M has coordinates Additional Examples (4, -3). Find the coordinates of B. Step 1 Let the coordinates of B equal ( x, y). Example 1 2 + x 2 + y Step 2 Use the Midpoint Formula: (4, -3) = (_ , _) . Find the −−coordinates of the mid- 2 2
point of PQ with endpoints Step 3 Find the x-coordinate. Find the y-coordinate. P( -8, 3) and Q( -2, 7) . ( -5, 5) 2 + x 2 + y 4 = _ Set the coordinates equal. -3 = _ 2 2 + Example 2 ( ) _2 + x ( ) _2 y −− 2 4 = 2 ( ) Multiply both sides by 2. 2 -3 = 2( ) M is the midpoint of XY . X has 2 2 = + - = + coordinates ( 2, 7) , and M has 8 2 x Simplify. 6 2 y - - - - coordinates ( 6, 1) . Find the coor- −−−−2 −−−−2 Subtract 2 from both sides. −−−−2 −−−−2 ( - ) dinates of Y. 10, 5 6 = x Simplify. -8 = y ( - ) Example 3 The coordinates of B are 6, 8 . _ Find FG and−− JK.−− Then determine 2. S is the midpoint of RT . R has coordinates (-6, -1) , and √ whether FG JK . 5; 3 2 ; no S has coordinates ( -1, 1). Find the coordinates of T.(4, 3) Þ { The Ruler Postulate can be used to find the distance between two points on a number line. The Distance Formula is used to calculate the distance between Ý two points in a coordinate plane. { ä { Distance Formula { In a coordinate plane, the distance d between two points (x1, y1) and (x2, y2) is = - 2 + - 2 d √(x2 x1) (y2 y1) . Also available on transparency
EXAMPLE 3 Using the Distance Formula −− −− INTERVENTION Find AB and CD. Then determine if AB CD . Þ Questioning Strategies Step 1 Find the coordinates of each point.
A(0, 3) , B(5, 1) , C(-1, 1) , and D(-3, -4) EXAMPLE 1 Ý { ä { • Does it matter which point is rep-
resented by ( x 1 , y 1 )? Explain. Step 2 Use the Distance Formula. { = √( - )2 + ( - )2 EXAMPLE 2 d x2 x1 y2 y1 2 2 2 2 • Given the midpoint and one end- AB = √( 5 - 0) + (1 - 3) CD = √ -3 - (- 1) + (- 4 - 1)
point of a segment, how is the 2 2 = √ 5 2 + (-2) = √( -2) + (-5) 2 Midpoint Formula used to find the missing endpoint? = √25 + 4 = √4 + 25 = √29 = √29 EXAMPLE 3 _ _ Since AB = CD, AB CD. • How do you substitute the coor- _ _ dinates of two points into the 3. Find EF and GH. Then determine if −− EF GH−−. Distance Formula? EF = 5; GH = 5; EF GH
44 Chapter 1 Foundations for Geometry
2 Teach Guided Instruction ge07se_c01_0043_0049.indd 44 8/15/05 3:40:37 PM ge07se_c01_0043_0049.indd 45 8/15/05 3:40:39 PM Before teaching the Midpoint Formula, Through Cognitive Strategies remind students how to find the aver- Have students use mental math to esti- age of two numbers. Connect average, a mate the midpoint or distance. Then have measure of central tendency, to the idea of them use a calculator to support their midpoint. Review how to subtract positive answer. and negative numbers to avoid subtract- ing errors in the Distance Formula. When finding midpoints and distances on a coor- dinate plane, relate your instruction to find- ing distances and midpoints on a number line.
44 Chapter 1
gge07te_c01_0043_0049.indde07te_c01_0043_0049.indd 4444 112/5/052/5/05 110:45:590:45:59 PPMM You can also use the Pythagorean Theorem to find the distance between " " Ê ,,", two points in a coordinate plane. You will learn more about the Pythagorean Theorem in Chapter 5. ,/
In a right triangle, the two sides that form the right angle are the legs . Students may incorrectly label the The side across from the right angle that stretches from one leg to the other is sides of a triangle when using the hypotenuse . In the diagram, a and b are the lengths of the shorter sides, a 2 + b 2 = c 2 . Remind them that or legs, of the right triangle. The longest side is called the hypotenuse and sides a and b are interchangeable, has length c. and represent the two shorter sides that form the right angle. The side Theorem 1-6-1 Pythagorean Theorem opposite the right angle is called the In a right triangle, the sum of the squares of the Þ hypotenuse and is labeled as c. { lengths of the legs is equal to the square of the length of the hypotenuse. 2 2 2 a + b = c L V Ý { ä { Additional Examples > Example 4 { Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, EXAMPLE 4 Finding Distances in the Coordinate Plane from D(3, 4) to E(-2, -5). 10.3 Use the Distance Formula and Þ { Also available on transparency the Pythagorean Theorem to find Ó]Êή the distance, to the nearest tenth, from A to B. L V Ý { ä { INTERVENTION > Ó]ÊÓ® Questioning Strategies { EXAMPLE 4 Method 1 Method 2 • Do you think it is easier to use the Pythagorean Theorem or the Use the Distance Formula. Use the Pythagorean Theorem. Substitute the values for the Count the units for sides a and b. Distance Formula to find the dis- coordinates of A and B into tance between two points? Explain. the Distance Formula. Algebra Review with stu- = - 2 + - 2 = = √2 2 AB √(x2 x1) (y2 y1) a 4 and b 5. dents that 3 + 4 ≠ 2 = 2 + 2 2 2 c a b 3 + 4. They must first = √ 2 - (-2 ) + (- 2 - 3) =42 + 52 square each number and then take = √ 2 + (- )2 4 5 = 16 + 25 the square root of the sum. The = √16 + 25 = 41 answer is 5, not 7. = √41 c = √41 ≈ 6.4 c ≈ 6.4 Inclusion Have students label the coordinates of any two points they are Use the Distance Formula and the Pythagorean Theorem ( ) ( ) to find the distance, to the nearest tenth, from R to S. given as x 1 , y 1 and x 2 , y 2 before substituting the numbers into the 4a. R(3, 2) and S(-3, -1) 6.7 Midpoint Formula or Distance (- ) ( - ) 4b. R 4, 5 and S 2, 1 8.5 Formula.
1- 6 Midpoint and Distance in the Coordinate Plane 45
ge07se_c01_0043_0049.indd 44 8/15/05 3:40:37 PM ge07se_c01_0043_0049.indd 45 8/15/05 3:40:39 PM
Lesson 1-6 45
gge07te_c01_0043_0049.indde07te_c01_0043_0049.indd 4455 112/5/052/5/05 110:46:030:46:03 PPMM EXAMPLE 5 Sports Application The four bases on a baseball field form Additional Examples a square with 90 ft sides. When a player T(0,90) S(90,90) throws the ball from home plate to Example 5 second base, what is the distance of the throw, to the nearest tenth? A player throws the ball from first base to a point located between Set up the field on a coordinate plane so that home plate H is third base and home plate and at the origin, first base F has 10 feet from third base. What is coordinates ( 90, 0) , second base S the distance of the throw, to the has coordinates ( 90, 90) , and third H(0,0) F(90,0) nearest tenth? 120.4 ft base T has coordinates ( 0, 90) . The distance HS from home plate to second base is the length of the Also available on transparency hypotenuse of a right triangle. = - 2 + - 2 HS √(x2 x1) ( y2 y1) = √( - )2 + ( - )2 INTERVENTION 90 0 90 0 Questioning Strategies = √ 90 2 + 90 2 = √8100 + 8100 EXAMPLE 5 = √16,200 • How do you find the coordinates of ≈ the endpoints used in the Distance 127.3 ft Formula? 5. The center of the pitching mound has coordinates ( 42.8, 42.8) . • Is the pitcher’s mound the mid- When a pitcher throws the ball from the center of the mound point between home plate and to home plate, what is the distance of the throw, to the second base? Explain. nearest tenth? 60.5 ft
THINK AND DISCUSS G.CM.5, G.CN.6, G.R.1
1. Can you exchange the coordinates ( x1 , y 1) and (x2 , y 2) in the Midpoint Formula and still find the correct midpoint? Explain. 2. A right triangle has sides lengths of r, s, and t. Given that s 2 + t2 = r2 , which variables represent the lengths of the legs and which variable represents the length of the hypotenuse? 3. Do you always get the same result using the Distance Formula to find distance as you do when using the Pythagorean Theorem? Explain your answer. 4. Why do you think that most cities are laid out in a rectangular grid instead of a triangular or circular grid? 5. GET ORGANIZED Copy and complete the graphic organizer below. In each box, write a formula. Then make a sketch that will illustrate the formula.
ÀÕ>Ã
`«Ì ÃÌ>Vi *ÞÌ >}Ài> ÀÕ> ÀÕ> / iÀi
46 Chapter 1 Foundations for Geometry
3 Answers to Think and Discuss Close x + x x + x 1. yes; _1 2 = _2 1 and ge07se_c01_0043_0049.indd 4 2 2 11/23/06 1:33:49 PM Summarize and INTERVENTION y + y y + y _1 2 = _2 1 Review with students how to use the Diagnose Before the Lesson 2 2 Midpoint Formula and Distance Formula 1-6 Warm Up, TE p. 43 2. s and t represent the lengths of the to find the midpoint of a segment and the legs. r represents the length of the hyp. distance between two points, given their Monitor During the Lesson 3. Yes; you can use either method to find coordinates. Check It Out! Exercises, SE pp. 43–46 the dist. between 2 pts. Questioning Strategies, TE pp. 44–46 4. Possible answer: to make locating addresses easier Assess After the Lesson 5. See p. A2. 1-6 Lesson Quiz, TE p. 49 Alternative Assessment, TE p. 49
46 Chapter 1
gge08te_ny_c01_0043_0049.indde08te_ny_c01_0043_0049.indd 4646 112/21/062/21/06 11:41:53:41:53 PPMM 1-6 1-6 ExercisesExercises Exercises KEYWORD: MG7 1-6
KEYWORD: MG7 Parent GUIDED PRACTICE Assignment Guide
1. Vocabulary The −−−− ? is the side of a right triangle that is directly across from the right angle. (hypotenuse or leg) hypotenuse Assign Guided Practice exercises as necessary. SEE EXAMPLE 1 Find the coordinates of the midpoint of each segment. _ p. 43 2. AB with endpoints A(4, -6) and B(-4, 2) (0,-2) If you finished Examples 1–3 _ 1 Basic 12–17, 22, 26, 27, 33 3. CD with endpoints C(0, -8) and D(3, 0) (1__,-4) _ 2 Average 12–17, 22, 24–27, 33, SEE EXAMPLE 2 4. M is the midpoint of LN . L has coordinates ( -3, -1) , and M has coordinates ( 0, 1) . 38 Find the coordinates of N.(3, 3) p. 44 _ Advanced 12–17, 22, 24–27, 33, 5. B is the midpoint of AC. A has coordinates (-3, 4) , and B has coordinates 36, 38 (-1__1 , 1). Find the coordinates of C. (0,-2) 2 If you finished Examples 1–5 SEE EXAMPLE 3 Multi-Step Find the length of the given segments and Þ Basic 12–23, 26–28, 30, { p. 44 determine_ if_ they are congruent. _ _ 32–37, 42–50 6. JK and FG √29 ; √29 ; yes7. JK and RS √29 ; 3 √5 ; no Average 12–39, 42–50 Ý Advanced 12–50 SEE EXAMPLE 4 Use the Distance Formula and the Pythagorean Theorem { ä { to find the distance, to the nearest tenth, between each p. 45 , Homework Quick Check pair of points. { Quickly check key concepts. 8. A(1, -2) and B(-4, -4) 5.4 - Exercises: 12, 14, 16, 18, 21, 9. X(-2, 7) and Y(-2, -8) 15.0 26, 30 10. V(2, -1) and W(-4, 8) 10.8
SEE EXAMPLE 5 11. Architecture The plan for a rectangular living p. 46 room shows electrical wiring will be run in a straight line from the entrance E to a light L at the opposite corner of the room. What is the length of the wire to the nearest tenth? 27.2 ft
PRACTICE AND PROBLEM SOLVING Independent Practice Find_ the coordinates of the midpoint of each segment. For See 12. XY with endpoints X(-3, -7) and Y(-1, 1) ( -2,-3) Exercises Example _ 12–13 1 13. MN with endpoints M(12, -7) and N(-5, -2) (3__1 ,-4__1 ) 14–15 2 _ 2 2 14. M is the midpoint of QR . Q has coordinates ( -3, 5) , and M has coordinates ( 7, -9) . 16–17 3 Find the coordinates of R.(17,-23) 18–20 4 _ 21 5 15. (- - ) ( __1 ) D is the midpoint of CE. E has coordinates 3, 2 , and D has coordinates 22 , 1 . Find the coordinates of C. (8, 4) Extra Practice Þ Skills Practice p. S5 { Multi-Step Find the length of the given segments and Application Practice p. S28 determine_ if_ they are congruent. 16. DE and FG 2√5 ; 2 √5 ; yes Ý _ _ { ä Ó 17. √ √ DE and RS 2 5 ; 29 ; no -
, {
1- 6 Midpoint and Distance in the Coordinate Plane 47
ge07se_c01_0043_0049.indd 47 8/15/05 3:40:46 PM
KEYWORD: MG7 Resources
Lesson 1-6 47
gge07te_c01_0043_0049.indde07te_c01_0043_0049.indd 4477 112/5/052/5/05 110:46:100:46:10 PPMM Exercise 33 involves Use the Distance Formula and the Pythagorean Theorem to find the distance, to the finding the distance nearest tenth, between each pair of points. 8.9 15.5 between points on 10.4 18. U(0, 1) and V(-3, -9) 19. M(10, -1) and N(2, -5) 20. P(-10, 1) and Q(5, 5) a coordinate plane. This exercise 18 in. 21. Consumer Application Televisions and computer screens are usually advertised prepares students for the Multi-Step based on the length of their diagonals. If the height of a computer screen is 11 in. Test Prep on page 58. and the width is 14 in., what is the length of the diagonal? Round to the nearest inch. −− −− −− CD, EF , AB 22. Multi-Step_ _ Use the_ Distance Formula to Þ
Answers order AB, CD, and EF from shortest to longest. { 32. When 2 pts. lie on a horiz. or 4.47 23. Use the Pythagorean Theorem to find the distance vert. line, they share a common from A to E. Round to the nearest hundredth. Ý x-coordinate or y-coordinate. To 3 ( -2a, __a) 24. X has coordinates (a, 3a), and Y has coordinates { ä { find the dist. between the pts., 2 ( ) (-5a, 0) . Find the coordinates of the midpoint of XY . find the difference of the other coordinates. Divide each 25. Describe a shortcut for finding the midpoint of a coord. by 2. segment when one of its endpoints has coordinates (a, b) and the other endpoint is the origin.
On the map, each square of the grid represents { 1 square mile. Find each distance to the nearest i`>À tenth of a mile. ÌÞ 26. Find the distance along Highway 201 from Cedar 7 ä City to Milltown. 6.1 mi { ,ÕÌiÊ£ {
History 27. A car breaks down on Route 1, at the midpoint ivviÀà between Jefferson and Milltown. A tow truck is sent } Ü>ÞÊÓä£ out from Jefferson. How far does the truck travel to ÌÜ reach the car? 2.5 mi - 28. History The Forbidden City in Beijing, China, is the world’s largest palace complex. Surrounded by a wall and a moat, the rectangular complex is 960 m long and 750 m wide. Find the distance, to the nearest meter, from one corner of the complex to the The construction of the opposite corner. 1218 m Forbidden City lasted 29. Critical Thinking Give an example of a line segment with midpoint (0, 0) . for 14 years. It began in Possible answer: seg. with endpts. (1, 2) and (-1,-2) 1406 with an estimated workforce of 200,000 men. The coordinates of the vertices of ABC are A(1, 4) , B(-2, -1), and C(-3, -2). Source: www.wikipedia.com 30. Find the perimeter of ABC to the nearest tenth. 14.5 _ _ 31. The height h to side BC is √2 , and b is the length of BC. What is the area of ABC ?1 32. Write About It Explain why the Distance Formula is not needed to find the distance between two points that lie on a horizontal or a vertical line.
33. This problem will prepare you for the Multi-Step Þ Test Prep on page 58. Tania uses a coordinate { plane to map out plans for landscaping a rectangular patio area. On the plan, one square Ý represents 2 feet. She plans to plant a tree at _ { ä { the midpoint of AC . How far from each corner of the patio does she plant the tree? Round −− to the nearest tenth. Let M be the mdpt. of AC; { = = = ≈ 1-6 PRACTICE A AM MC 5.0 ft; MB MD 6.4 ft. 1-6 PRACTICE C
*À>VÌViÊ ,%33/. 1-6 PRACTICE B £È -IDPOINT AND $ISTANCE IN THE #OORDINATE 0LANE 48 Chapter 1 Foundations for Geometry &IND THE COORDINATES OF THE MIDPOINT OF EACH SEGMENT ? 45 WITH ENDPOINTS 4 AND 5 X Y ,%33/. ,i>`}Ê-ÌÀ>Ìi}ià ,%33/. ,iÌi>V ? ?? ????? 1-6 READING STRATEGIES 1-6 RETEACH 67 WITH ENDPOINTS 6 AND 7X Y S D £È #OMPARE AND #ONTRAST £È -IDPOINT AND $ISTANCE IN THE #OORDINATE 0LANE ? 4HERE ARE TWO WAYS TO FIND THE DISTANCE BETWEEN #ONSIDER THE FOLLOWING POINTS 4HE MIDPOINT OF A LINE SEGMENT SEPARATES THE SEGMENT INTO TWO HALVES 9 IS THE MIDPOINT OF 8: 8 HAS COORDINATES AND TWO POINTS IN A COORDINATE PLANE )T CAN BE FOUND Y 9OU CAN USE THE -IDPOINT &ORMULA TO FIND THE MIDPOINT OF THE SEGMENT 9 HAS COORDINATES n &IND THE COORDINATES OF : USING EITHER ! WITH ENDPOINTS ' AND ( Y $ISTANCE &ORMULA C X ( A X?????? X Y?????? Y ????? ????? 0YTHAGOREAN 4HEOREM - - 5SE THE FIGURE FOR %XERCISES n Y ! B S D S D - IS THE MIDPOINT ^ ? " - &IND !" q 4HERE ARE TWO WAYS TO FIND THE DISTANCE - ?? ?? OF (' ge07se_c01_0043_0049.indd 48 S D ' 8/15/05 3:40:49 PM ge07se_c01_0043_0049.indd 49 8/15/05 3:40:52 PM # X BETWEEN POINTS ! AND " X ^ q - &IND "# " $ISTANCE &ORMULA 0YTHAGOREAN 4HEOREM ^ ^ D X X Y Y A B C &IND #! q q &IND THE COORDINATES OF THE MIDPOINT OF EACH SEGMENT ? ? D IS THE DISTANCE BETWEEN TWO POINTS A AND B ARE THE LEGS OF A RIGHT .AME A PAIR OF CONGRUENT SEGMENTS !" AND "# TRIANGLE Y Y 3 X Y AND X Y ARE THE COORDINATES C IS THE HYPOTENUSE ! " X &IND THE DISTANCES OF TWO POINTS X 4 5SE THE $ISTANCE &ORMULA TO FIND THE DISTANCE TO THE !NSWER EACH QUESTION NEAREST TENTH BETWEEN + AND , 7HAT ARE THE COORDINATES OF POINT ! ON THE GRID ABOVE 5SE THE 0YTHAGOREAN 4HEOREM TO FIND THE DISTANCE TO 7HAT ARE THE COORDINATES OF POINT " THE NEAREST TENTH BETWEEN & AND 'n ? 3UBSTITUTE THE VALUES FOR THE COORDINATES OF ! AND " ^^ 12 WITH ENDPOINTS 1 AND 2 INTO THE $ISTANCE &ORMULA 3OLVE FOR D D 5SE THE FIGURE FOR %XERCISES n ^ q ? D X X Y Y ^ *+ WITH ENDPOINTS * n AND + q 3NOOKER IS A KIND OF POOL OR BILLIARDS PLAYED ON A FOOT BY FOOT TABLE 4HE SIDE POCKETS ARE HALFWAY DOWN THE RAILS LONG SIDES FT q ? ^ 3UPPOSE - IS THE MIDPOINT OF #$ AND # HAS COORDINATES 9OU CAN USE &IND THE DISTANCE TO THE NEAREST TENTH OF A FOOT DIAGONALLY q THE -IDPOINT &ORMULA TO FIND THE COORDINATES OF $ ACROSS THE TABLE FROM CORNER POCKET TO CORNER POCKET ^ X X Y Y q - - ?????? ?????? FT S D SIDE FT POCKET A B X COORDINATE OF $ Y COORDINATE OF $ &IND THE DISTANCE TO THE NEAREST TENTH OF AN INCH DIAGONALLY $RAW A RIGHT TRIANGLE IN THE COORDINATE GRID AS SHOWN #ONNECT C ACROSS THE TABLE FROM CORNER POCKET TO SIDE POCKET POINTS ! AND " 4HIS WILL BE THE HYPOTENUSE OF YOUR TRIANGLE %XTEND X X Y Y ?????? 3ET THE COORDINATES EQUAL ?????? A SEGMENT TO THE RIGHT FROM POINT " AT THE SAME TIME YOU EXTEND A IN SEGMENT DOWN FROM POINT ! 4HESE WILL BE THE OTHER TWO SIDES OF X Y ?????? 2EPLACE X Y WITH ?????? YOUR TRIANGLE 4HEY WILL CONTINUE UNTIL THE SEGMENTS MEET 4HE LENGTH ^ CORNER OF THE SEGMENT THAT YOU EXTENDED FROM POINT ! WILL BE A AND THE C q POCKET X -ULTIPLY BOTH SIDES BY Y LENGTH OF THE SEGMENT THAT YOU EXTENDED FROM POINT " WILL BE B X 3UBTRACT TO SOLVE FOR X AND Y Y 5SE THE 0YTHAGOREAN 4HEOREM TO DETERMINE THE DISTANCE BETWEEN POINTS ! AND " 4HE COORDINATES OF $ ARE %XPLAIN THE DIFFERENCE BETWEEN THE 0YTHAGOREAN 4HEOREM AND THE $ISTANCE &ORMULA ? - IS THE MIDPOINT OF 23 AND 2 HAS COORDINATES 3AMPLE ANSWER $ISTANCE &ORMULA USES A COORDINATE PLANE 7HAT ARE THE COORDINATES OF 3 ? - IS THE MIDPOINT OF 78 AND 8 HAS COORDINATES 0YTHAGOREAN 4HEOREM USES KNOWN MEASURES OF TWO SIDES OF A TRIANGLE 48 Chapter 1 7HAT ARE THE COORDINATES OF 7
gge07te_c01_0043_0049.indde07te_c01_0043_0049.indd 4488 112/5/052/5/05 110:46:130:46:13 PPMM " " Ê ,,", 34. Which segment has a length closest to 4 units? Þ _ _ { ,/ EF_ JK_ GH LM In Exercise 39, students may think Ý that 2 is the only value of a. Remind 35. Find the distance,_ to the_ nearest tenth, between the { ä { them that coordinates can be nega- midpoints of LM and JK . tive, and thus -2 is also a value of a. 1.8 4.0 { 3.6 5.3 If students have dif- ficulty with Exercise 36. What are the coordinates of the midpoint of a line 34, have them mark segment that connects the points (7, -3) and (-5, 6)? off a segment on a piece of paper (6, -4__1) (2, __1) 2 2 that is four units long and compare (2, 3) (1, 1 __1) it to the four segments that are 2 shown on the graph. 37. A coordinate plane is placed over the map of a town. A library is located at ( -5, 1), and a museum is located at (3, 5). What is the distance, to the nearest tenth, from Journal the library to the museum? Have students compare the distance 4.5 5.7 6.3 8.9 and midpoint formulas. Ask them to draw an example on a grid. CHALLENGE AND EXTEND 38. Use the diagram to find _the following. Þ
a. P is_ the midpoint of AB , and R is the midpoint { * of BC . Find the coordinates of Q. Q(2.5, 2) , +Ê Have students draw and measure b. Find the area of rectangle PBRQ. 1.5 Ý the sides of a right triangle on a c. Find DB. Round to the nearest tenth. √13 ≈ 3.6 ä { coordinate plane. They should label 39. The coordinates of X are (a - 5, -2a). The coordinates the vertices and find the locations of Y are ( a + 1, 2a). If the distance between X and Y is of the midpoints of each side. Then ± 10, find the value of a. 2 they should find the distances 40. Find two points on the y-axis that are a distance of 5 units from (4, 2) . (0, 5) ; (0,-1) between the vertices. 41. Given ∠ACB is a right angle of ABC, AC = x, and BC = y, find AB in terms of x and y. AB = √x 2 + y2 1-6 SPIRAL REVIEW Determine if the ordered pair (-1, 4) satisfies each function. (Previous course) 42. y = 3x - 1 43. f(x) = 5 - x2 44. g(x) = x2 - x + 2 1. Find the coordinates−− of the no yes yes midpoint of MN with end- (- ) ( ) BD bisects straight angle ABC, and BE bisects ∠CBD. points M 2, 6 and N 8, 0 . Find the measure of each angle and classify it as acute, (3, 3) −− right, or obtuse. (Lesson 1-3) 2. K is the midpoint of HL. H has ∠ ∠ ∠ 45. ABD 46. CBE 47. ABE coordinates ( 1, -7) , and K 90°; rt. 45°; acute 135°; obtuse has coordinates ( 9, 3) . Find Find the area of each of the following. (Lesson 1-5) the coordinates of L. ( 17, 13) 48. square whose perimeter is 20 in. 25 in 2 49. triangle whose height is 2 ft and whose base is twice its height 4 ft 2 3. Find the distance, to the near- est tenth, between S (6, 5) 50. rectangle whose length is x and whose width is ( 4x + 5) 4x2 + 5x and T (-3, -4) . 12.7
4. The coordinates of the 1- 6 Midpoint and Distance in the Coordinate Plane 49 vertices of ABC are A (2, 5) ,
*ÀLiÊ-Û} >i}i ( - ) (- - ) ,%33/. 1-6 PROBLEM SOLVING ,%33/. 1-6 CHALLENGE B 6, 1 , and C 4, 2 . £È -IDPOINT AND $ISTANCE IN THE #OORDINATE 0LANE £È !PPLYING -IDPOINT AND $ISTANCE &OR %XERCISES AND USE THE DIAGRAM OF A TENNIS COURT 5SE %!"# FOR %XERCISES n Find the perimeter of ABC, ! Y ! SINGLES TENNIS COURT IS A RECTANGLE FEET WIDE &IND THE PERIMETER OF $!"# TO THE NEAREST TENTH AND FEET LONG 3UPPOSE A PLAYER AT CORNER ! ! to the nearest tenth. 26.5 HITS THE BALL TO HER OPPONENT IN THE DIAGONALLY UNITS ? ? ? 4 OPPOSITE CORNER " !PPROXIMATELY HOW FAR DOES FT FT 7HAT ARE THE MIDPOINTS OF !" "# AND #! THE BALL TRAVEL TO THE NEAREST TENTH OF A FOOT # −− −− " ge07se_c01_0043_0049.indd 48 8/15/05 3:40:49 PM ge07se_c01_0043_0049.indd 7 X 11/23/06 1:34:03 PM 0 5. Find the lengths of AB and CD FT 4 4 FT &IND THE PERIMETER OF THE TRIANGLE WHOSE 2 ! DOUBLES TENNIS COURT IS A RECTANGLE FEET WIDE AND VERTICES ARE THE MIDPOINTS OF $!"# 2OUND " FEET LONG )F TWO PLAYERS ARE STANDING IN DIAGONALLY TO THE NEAREST TENTH and determine whether they OPPOSITE CORNERS ABOUT HOW FAR APART ARE THEY TO THE NEAREST TENTH OF A FOOT UNITS are congruent. √10 ; √10 ; yes FT #OMPARE THE PERIMETER OF $!"# TO THE PERIMETER OF THE TRIANGLE WHOSE VERTICES ARE THE MIDPOINTS OF $!"#
! MAP OF AN AMUSEMENT PARK IS SHOWN ON A COORDINATE PLANE WHERE EACH SQUARE 4HE PERIMETER OF %!"# IS TWICE THE PERIMETER OF THE SECOND TRIANGLE OF THE GRID REPRESENTS SQUARE METER 4HE WATER RIDE IS AT THE ROLLER COASTER Þ IS AT AND THE &ERRIS WHEEL IS AT &IND EACH DISTANCE TO THE NEAREST { 5SE QUADRILATERAL 1234 FOR %XERCISES AND Y TENTH OF A METER ? ? 1 &IND THE LENGTHS OF DIAGONALS 13 AND 24 7HAT IS THE DISTANCE BETWEEN THE WATER ! CARICATURE ARTIST IS AT THE MIDPOINT 2OUND TO THE NEAREST TENTH 2 RIDE AND THE ROLLER COASTER BETWEEN THE ROLLER COASTER AND THE &ERRIS 4 M WHEEL 7HAT IS THE DISTANCE FROM THE ARTIST UNITS TO THE &ERRIS WHEEL ? ? Ý &IND THE MIDPOINTS OF 13 AND 24 X M $ESCRIBE WHAT YOU FIND 3 0 4
5SE THE MAP OF THE 3ACRAMENTO :OO ON A COORDINATE PLANE "OTH MIDPOINTS ARE AT 4HIS { ä { 4 FOR %XERCISES n #HOOSE THE BEST ANSWER Sacramento Zoo 4 Y Otters IS THE POINT WHERE THE DIAGONALS 3 4O THE NEAREST TENTH OF A UNIT HOW FAR IS IT FROM Hyenas THE TIGERS TO THE HYENAS INTERSECT ! UNITS # UNITS 3 4HE DIAMETER OF A CIRCLE HAS ENDPOINTS * AND + " UNITS $ UNITS { Gibbons X A 7HAT ARE THE COORDINATES OF THE CENTER "ETWEEN WHICH OF THESE EXHIBITS IS THE DISTANCE 3 0 3 OF THE CIRCLE THE LEAST Tigers B &IND THE CIRCUMFERENCE AND AREA OF THE CIRCLE & TIGERS AND PRIMATES 3 5SE THE P KEY ON YOUR CALCULATOR AND ROUND ' HYENAS AND GIBBONS UNITS UNITS Jaguars TO THE NEAREST TENTH Also available on transparency ( OTTERS AND GIBBONS C $ETERMINE WHETHER OR NOT ' IS A POINT ON THE CIRCLE *USTIFY YOUR ANSWER * TIGERS AND OTTERS Primates 4HE DIAMETER OF THE CIRCLE IS APPROXIMATELY UNITS SO THE RADIUS 3UPPOSE YOU WALK STRAIGHT FROM THE JAGUARS TO THE TIGERS AND THEN TO THE OTTERS 7HAT IS THE TOTAL DISTANCE TO THE NEAREST TENTH OF A UNIT IS HALF THAT DISTANCE OR ABOUT UNITS 4HE DISTANCE FROM THE ! UNITS # UNITS CENTER OF THE CIRCLE TO ' IS UNITS 3O ' IS NOT A POINT ON THE CIRCLE " UNITS $ UNITS Lesson 1-6 49
gge08te_ny_c01_0043_0049.indde08te_ny_c01_0043_0049.indd 4949 112/21/062/21/06 11:42:30:42:30 PPMM 1-7 Organizer 1-7 Transformations in the Pacing: Traditional 1 day Coordinate Plane __1 Block 2 day Objectives: Identify reflections, Objectives Who uses this? rotations, and translations. Identify reflections, Artists use transformations to create rotations, and decorative patterns. (See Example 4.) Graph transformations in the translations. coordinate plane. Graph transformations The Alhambra, a 13th-century palace in in the coordinate plane. Granada, Spain, is famous for the geometric @}i The resulting figure is called the image . A transformation maps the preimage to >}i the image. Arrow notation (→) is used to describe a transformation, and primes Ī Ī ( ) are used to label the image. ̱ ÊÊÊÊÊÊ̱Ī Ī ĪÊÊÊ Warm Up 1. Draw a line that divides a right Transformations angle in half. REFLECTION ROTATION TRANSLATION Ī Ī Ī Ī 2. Draw three different squares Ī with ( 3, 2) as one vertex. Ī Ī NY Performance Ī Ī Ī Check students’ drawings. Ī Indicators * 3. Find the values of x and y if G.G.54 Define, investigate, (3, -2) = ( x + 1, y - 3) justify, and apply isometries in A reflection (or flip) is a A rotation (or turn) A translation (or slide) is x = 2; y = 1 the plane (rotations, reflections, transformation across a line, is a transformation a transformation in which translations, glide reflections). called the line of reflection. about a point P, all the points of a figure Also, G.G.56. Also available on transparency Each point and its image called the center of move the same distance are the same distance from rotation. Each point in the same direction. the line of reflection. and its image are the same distance from P.
EXAMPLE 1 Identifying Transformations Spanish teacher: Why didn’t you translate the verb? Identify the transformation. Then use arrow notation to describe the transformation. Student: I did. I moved it 3 inches right and 2 inches down. A Ī The transformation cannot be a translation because each point and its image are not in the Ī same position. Ī
The transformation is a reflection. EFG → E F G New York Performance Indicators 50 Chapter 1 Foundations for Geometry
Geometry G.G.54 Define, investigate, justify, and apply isometries in the plane 1 Introduce (rotations, reflections, translations, glide reflections) ge07se_c01_0050_0055.indd 1 11/23/06 1:34:40 PM EXPLORATION Motivate G.G.56 Identify specific isometries £Ç /À>ÃvÀ>ÌÃÊÊÌ iÊ by observing orientation, numbers of À`>ÌiÊ*>i Show students examples of tessellations, transla- 5SE GEOMETRY SOFTWARE TO REmECT A TRIANGLE ACROSS A LINE invariant points, and/or parallelism SEGMENT tions, rotations, and reflections in the real world, $RAW !" 4HIS WILL BE THE $RAW N234 ON ONE SIDE OF from magazines or advertisements. Ask students LINE OF REFLECTION !" to demonstrate transformations by standing and sliding two steps right and then rotating 90° clock-
? wise. 3ELECT !" AND CHOOSE #ONSTRUCT ? 44| 4HEN ? -ARK -IRROR FROM THE SELECT 4 4 | AND !" AND USE 4RANSFORM MENU 4HEN THE #ONSTRUCT MENU TO CON SELECT N234 AND CHOOSE STRUCT THEIR INTERSECTION ? ? 8 2EFLECT FROM THE 4RANSFORM -EASURE 48 4 | 8 AND MENU ,ABEL THE NEW TRI 48"
ANGLE AS N2|3|4| Explorations and answers are provided in the Explorations binder. $RAG 4 7HAT DO YOU NOTICE ABOUT THE MEASUREMENTS AS YOU DO SO 50 Chapter 1 / - 1--
gge08te_ny_c01_0050_0055.indde08te_ny_c01_0050_0055.indd 5050 112/21/062/21/06 112:47:002:47:00 PPMM Identify the transformation. Then use arrow notation to describe the transformation. B /Ī -Ī The transformation cannot be a Additional Examples reflection because each point and its image are not the same distance from a line of reflection. Example 1 Identify the transformation. 1 / 1Ī ,Ī Then use arrow notation to , - describe the transformation. → A. Þ The transformation is a 90° rotation. RSTU R S T U {
Identify each transformation. Then use arrow notation to Ī describe the transformation. Ý 1a. 1b. <Ī 9Ī * { ä Ó Ī *Ī Ī Ī <
8Ī 90° rotation, " ABC → A'B'C' Ī "Ī 89 → → translation; MNOP M N O P rotation; XYZ X Y Z B. Þ
EXAMPLE 2 Drawing and Identifying Transformations A figure has vertices at A(-1, 4) , B(-1, 1) , Þ Ý ( ) and C 3, 1 . After a transformation, the image Ī Ī ä - - Ó of the figure has vertices at A ( 1, 4) , Ó B ( -1, -1), and C ( 3, -1) . Draw the preimage Ý Ī Ī and image. Then identify the transformation. { Ī { Ī Plot the points. Then use a straightedge to Ó reflection, DEFG → D'E'F'G' connect the vertices. Ī The transformation is a reflection across the Example 2 x-axis because each point and its image are ( - ) the same distance from the x-axis. A figure has vertices at A 1, 1 , B (2, 3) , and C (4, -2) . After a y 2. A figure has vertices at E(2, 0) , F(2, -1) , G(5, -1) , and H(5, 0) . transformation, the image of the HĪ GĪ After a transformation, the image of the figure has vertices at figure has vertices at A' (-1, -1) , E( 0, 2) , F( 1, 2) , G (1, 5) , and H (0, 5) . Draw the preimage and B' (-2, 3) , and C' (-4, -2) . Draw image. Then identify the transformation. rotation; 90° EĪ FĪ the preimage and image. Then E H x identify the transformation. F G To find coordinates for the image of a figure in a translation, add a to the y x-coordinates of the preimage and add b to the y-coordinates of the preimage. Translations can also be described by a rule such as (x, y) → (x + a, y + b). BĪ B
EXAMPLE 3 Translations in the Coordinate Plane Þ x { Find the coordinates for the image of ABC after the translation (x, y) → (x + 3, y - 4) . AĪ A Draw the image. Ý CĪ C { ä { Step 1 Find the coordinates of ABC. The vertices of ABC are A(-1, 1) , B(-3, 3) , and C(-4, 0) . { reflection across the y-axis Also available on transparency
1- 7 Transformations in the Coordinate Plane 51
INTERVENTION 2 Teach Questioning Strategies ge07se_c01_0050_0055.inddGuided 51 Instruction 12/2/05 3:53:05EXAMPLE PM 1 • How would each image in Explain the concepts of reflection, rotation, Through Concrete Manipulatives Example 1A and Example 1B and translation; the terms preimage and Give students a coordinate grid and a cut- differ if the transformation were image; and prime notation before using out triangle. Have them slide the figure to a translation? examples on a coordinate plane. a new location, rotate it about the origin, and reflect it over a specified line to find EXAMPLE 2 Inclusion Have students look in the image. Show students why a reflection • How did you decide what type of a mirror (MK). They are the pre- is not equivalent to a 180° rotation. Use transformation this was? image before they see themselves notebook paper with a large F marked on in the mirror. The image is their reflection one side to show that the image in a rota- in the mirror. tion is upside down.
Lesson 1-7 51
gge07te_c01_0050_0055.indde07te_c01_0050_0055.indd 5511 112/20/052/20/05 111:14:381:14:38 AAMM Step 2 Apply the rule to find the vertices Þ of the image. { Additional Examples A ( -1 + 3, 1 - 4) = A ( 2, -3) B ( -3 + 3, 3 - 4) = B ( 0, -1) Ý Example 3 C (-4 + 3, 0 - 4) = C ( -1, -4) { ä Ī { Find the coordinates for the Step 3 Plot the points. Then finish drawing Ī image of ABC after the trans- the image by using a straightedge to Ī lation ( x, y) → ( x + 2, y - 1) . connect the vertices. Draw the image. A'( -2, 1) , 3. Find the coordinates for the image Þ B'( -1, 3) , C'( 1, 0) { of JKLM after the translation Þ y ( ) → ( - + ) x, y x 2, y 4 . Draw the image. { Ī JĪ KĪ J ( -1, 5) ; K (1, 5) ; L (1, 0) ; M (-1, 0) Ý { ä { Ī Ī Ý x ä MĪ LĪ { { { Ó EXAMPLE 4 Art History Application Example 4 The pattern shown is similar to a pattern y The figure shows part of a tile on a wall of the Alhambra. Write a rule for the translation of square 1 to square 2. floor. Write a rule for the transla- 2 Step 1 Choose 2 points A‘ tion of hexagon 1 to hexagon 2. 1 Choose a point A on the preimage A Þ { and a corresponding point A on the x image. A has coordinates ( 3, 1) , and ( ) A has coordinates 1, 3 . 3 Ó Ý Step 2 Translate ä To translate A to A , 2 units are £ subtracted from the x-coordinate and 2 units are added to the y-coordinate. Therefore, the translation rule is (x, y) → (x - 3, y + 1.5) (x, y) → (x - 2, y + 2) . Also available on transparency 4. Use the diagram to write a rule for the translation of square 1 to square 3. (x, y) → (x - 4, y - 4)
INTERVENTION Questioning Strategies G.CM.11, G.R.1 EXAMPLE 3 THINK AND DISCUSS • How would the image differ if the 1. Explain how to recognize a reflection when given a figure and its image. translation were ( x, y) → (x - 2, 2. GET ORGANIZED Copy and complete the graphic organizer. In each y + 1) ? box, sketch an example of each transformation.
EXAMPLE 4 /À>ÃvÀ>ÌÃ • To write a rule for the translation, does it matter which point on the ,iviVÌ ,Ì>Ì /À>Ã>Ì figure you choose? Why or why not? 52 Chapter 1 Foundations for Geometry
3 Answers to Think and Discuss Close Possible answers:
ge07se_c01_0050_0055.indd 3 11/23/06 1:34:49 PM and INTERVENTION 1. The preimage and image will be mirror Summarize images of each other. Review the three types of transformations Diagnose Before the Lesson 2. See p. A2. and give an example of each. 1-7 Warm Up, TE p. 50
Monitor During the Lesson Check It Out! Exercises, SE pp. 51–52 Questioning Strategies, TE pp. 51–52
Assess After the Lesson 1-7 Lesson Quiz, TE p. 55 Alternative Assessment, TE p. 55
52 Chapter 1
gge08te_ny_c01_0050_0055.indde08te_ny_c01_0050_0055.indd 5252 112/21/062/21/06 11:43:43:43:43 PPMM 1-7 1-7 ExercisesExercises Exercises KEYWORD: MG7 1-7
KEYWORD: MG7 Parent GUIDED PRACTICE Assignment Guide Vocabulary Apply the vocabulary from this lesson to answer each question. 1. Given the transformation XYZ →XYZ, name the preimage and image of Assign Guided Practice exercises the transformation. Preimage is XYZ; image is X Y Z . as necessary. 2. The types of transformations of geometric figures in the coordinate plane can be If you finished Examples 1–2 described as a slide, a flip, or a turn. What are the other names used to identify Basic 8–10, 13–15, 17–22 these transformations? translation; reflection; rotation Average 8–10, 13–15, 17–22 SEE EXAMPLE 1 Identify each transformation. Then use arrow notation to describe the transformation. Advanced 8–10, 13–15, 17–22, p. 50 3. Ī 4. *Ī +Ī 34 If you finished Examples 1–4 * + Basic 8–18, 26–32, 38–47 Ī Ī -Ī ,Ī Average 8–33, 38-47 reflection; ABC → A B C Advanced 8–47 -, translation; PQRS → P Q R S SEE EXAMPLE 2 5. A figure has vertices at A(-3, 2) , B(-1, -1) , and C(-4, -2). After a transformation, Homework Quick Check p. 51 the image of the figure has vertices at A (3, 2) , B( 1, -1) , and C (4, -2) . Draw the Quickly check key concepts. preimage and image. Then identify the transformation. reflection across the y-axis Exercises: 8, 10–12, 14, 18, y 26, 28 SEE EXAMPLE 3 6. Multi-Step The coordinates of the vertices 4 p. 51 of DEF are D(2, 3) , E(1, 1) , and F(4, 0) . 6. D ( -1, 1) ; Find the coordinates for the image of DEF 2 E ( -2, -1) ; after the translation (x, y) → (x - 3, y - 2) . Answers F (1,-2) Draw the preimage and image. x –4 4 5. y SEE EXAMPLE 4 7. Animation In an animated film, a simple 1 A AĪ p. 52 scene can be created by translating a figure against a still background. Write a rule for x the translation that maps the rocket from – 4 position 1 to position 2. B BĪ (x, y) → (x + 4, y + 4) C C? PRACTICE AND PROBLEM SOLVING Independent Practice Identify each transformation. Then use arrow notation to describe the transformation. 6. y D For See 8. Ī Ī 9. 7 8 Exercises Example rotation; reflection; DĪ 8–9 1 → DEFG D E F G WXYZ → W X Y Z E x 10 2 < 9 F 11 3 EĪ Ī Ī <Ī 9Ī 12 4 FĪ Extra Practice 7Ī 8Ī Skills Practice p. S5 Application Practice p. S28
10. A figure has vertices at J(-2, 3) , K(0, 3) , L(0, 1) , and M(-2, 1). After a transformation, the image of the figure has vertices at J ( 2, 1) , K( 4, 1) , L (4, -1) , and M (2, -1) . Draw the preimage and image. Then identify the transformation. translation
1- 7 Transformations in the Coordinate Plane 53
Answers
10. y ge07se_c01_0050_0055.indd 53 JK 12/2/05 3:53:11 PM
JĪ KĪ M L x
MĪ LĪ
KEYWORD: MG7 Resources
Lesson 1-7 53
gge07te_c01_0050_0055.indde07te_c01_0050_0055.indd 5533 112/5/052/5/05 110:47:400:47:40 PPMM Communicating Math In 11. A ( -1, -1) , 11. Multi-Step The coordinates of the vertices of rectangle ABCD are A(-4, 1) , Exercises 16 and 17, stress B (4, -1) , B(1, 1) , C(1, -2) , and D(-4, -2). Find the coordinates for the image of to students that when C (4, -4) , D (-1, -4) rectangle ABCD after the translation ( x, y) → (x + 3, y - 2) . Draw the preimage reflecting across the x-axis, the x- and the image.
coordinate stays the same, and, 12. Travel Write a rule for the translation y when reflecting across the y-axis, that maps the descent of the hot the y-coordinate stays the same. air balloon. (x, y) → (x + 11, y - 4)
Which transformation is suggested by each x Exercise 28 involves of the following? applying a combina- 13. mountain range and its image on a lake tion of transforma- reflection tions to obtain a given image. This 14. straight line path of a band exercise prepares students for the marching down a street translation Multi-Step Test Prep on page 58. 15. wings of a butterfly reflection
( ) (- ) ( ) Answers Given points F 3, 5 , G 1, 4 , and H 5, 0 , draw FGH and its reflection across each of the following lines. 11. y 16. the x-axis 17. the y-axis 17. AB x y 18. Find the vertices of one of the triangles on the graph. Þ Then use arrow notation to write a rule for translating { AĪ BĪ FĪ F the other three triangles. G GĪ {£ DC Ý A transformation maps A onto B and C onto D. HĪ H { ä { DĪ CĪ x 19. Name the image of A.BA20. Name the preimage of B. ÎÓ 16. G F 21. Name the image of C.DC22. Name the preimage of D. { 18. 1 to 2: 23. Find the coordinates for the image of RST with (x, y) → (x, -y) vertices R(1, -4) , S(-1, -1) , and T(-5, 1) after the 2 to 3: translation ( x, y) → (x - 2, y - 8) . R (-1, -12) ; S (-3, -9) ; T ( -7, -7) H → - ( x, y) ( x, y) 24. Critical Thinking ( ) → ( + + ) HĪ Consider the translations x, y x 5, y 3 and 3 to 4: (x, y) → (x + 10, y + 5) . Compare the two translations. (x, y) → (x, -y) Graph each figure and its image after the given translation. _ GĪ FĪ 25. MN with endpoints M(2, 8) and N(-3, 4) after the translation (x, y) → (x + 2, y - 5) _ 24–27. See p. A11. 26. KL with endpoints K(-1, 1) and L(3, -4) after the translation (x, y) → (x - 4, y + 3) 27. Write About It Given a triangle in the coordinate plane, explain how to draw its image after the translation ( x, y) → (x + 1, y + 1) .
28. This problem will prepare you for the Multi-Step Þ Test Prep on page 58. Greg wants to rearrange the Ī Ī triangular pattern of colored stones on his patio. What combination of transformations could he Ý use to transform CAE to the image on the { ä Ī { coordinate plane? Possible answer: 2 reflections (across the y-axis and across EC ) { 1-7 PRACTICE A 1-7 PRACTICE C
*À>VÌViÊ ,%33/. 1-7 PRACTICE B £Ç 4RANSFORMATIONS IN THE #OORDINATE 0LANE 54 Chapter 1 Foundations for Geometry 5SE THE FIGURE FOR %XERCISES n
Y 4HE FIGURE IN THE PLANE AT RIGHT SHOWS THE PREIMAGE IN THE TRANSFORMATION !"#$ → !"#$ -ATCH THE NUMBER OF THE IMAGE BELOW WITH THE ,%33/. ,i>`}Ê-ÌÀ>Ìi}ià ,%33/. ,iÌi>V NAME OF THE CORRECT TRANSFORMATION X £Ç 5SE A 6ISUAL -ODEL1-7 READING STRATEGIES £Ç 4RANSFORMATIONS IN THE #OORDINATE 0LANE1-7 RETEACH Y Y Y ! " &IGURES IN A COORDINATE PLANE CAN BE MOVED WITHIN THE COORDINATE PLANE 4HESE )N A TRANSFORMATION EACH POINT OF A FIGURE IS MOVED TO A NEW POSITION $ # MOVEMENTS ARE CALLED TRANSFORMATIONS 4HERE ARE THREE TYPES OF TRANSFORMATIONS 2EFLECTION 2OTATION 4RANSLATION DESCRIBED IN THE TABLE BELOW !g "g X #g $g X "g !g X ! * * 2 4RANSFORMATION $EFINITION %XAMPLE 2 $g #g "g !g #g $g 2EFLECTION ! TRANSFORMATION ACROSS A LINE # " + %ACH POINT AND ITS IMAGE ARE " # # " # " , ge07se_c01_0050_0055.indd 54 THE SAME DISTANCE FROM THE 34 12/2/05 3:53:13 PM ge07se_c01_0050_0055.indd 55 12/2/05 3:53:16 PM + LINE OF REFLECTION ! , 3 4 ROTATION TRANSLATION REFLECTION ! $ $ ! $!"# → $!"# $*+, → $*+, $234 → $234 Y ! FIGURE HAS VERTICES AT $ % AND & % & 2OTATION ! TRANSFORMATION AROUND A " ! FIGURE IS FLIPPED OVER ! FIGURE IS TURNED AROUND ! FIGURE IS SLID TO A NEW !FTER A TRANSFORMATION THE IMAGE OF THE FIGURE HAS POINT %ACH POINT AND ITS A LINE A FIXED POINT POSITION WITHOUT TURNING VERTICES AT $ % AND & IMAGE ARE THE SAME DISTANCE & $ X $RAW THE PREIMAGE AND THE IMAGE 4HEN IDENTIFY FROM THE CENTRAL POINT ! #
! "
THE TRANSFORMATION $ )DENTIFY EACH TRANSFORMATION 4HEN USE ARROW NOTATION TO DESCRIBE THE % TRANSFORMATION ROTATION
# & 0 0 ! FIGURE HAS VERTICES AT ' ( ) AND * &IND THE COORDINATES FOR THE IMAGE OF '()* AFTER THE TRANSLATION X Y → X Y 4RANSLATION !LL THE POINTS OF A FIGURE MOVE " - - THE SAME DISTANCE IN THE SAME ' ( ' ( ) * DIRECTION " ! # & Y 5SE THE FIGURE FOR %XERCISE . . ! # ! PARKING GARAGE ATTENDANT WILL MAKE THE MOST MONEY WHEN '( THE MAXIMUM NUMBER OF CARS FITS IN THE PARKING GARAGE 4O FIT X 5SE THE INFORMATION IN THE CHART TO ANSWER THE QUESTIONS BELOW ONE MORE CAR IN THE ATTENDANT MOVES A CAR FROM POSITION TO Y TRANSLATION POSSIBLE ANSWER REFLECTION POSSIBLE ANSWER POSITION 7RITE A RULE FOR THIS TRANSLATION 7HAT TYPE OF TRANSFORMATION MOVES FIGURE ! TO FIGURE " ! O&'( → O&'( O-.0 → O-.0 0OSSIBLE ANSWER TRANSLATION X Y → X Y X 7 "# Y ! FIGURE HAS VERTICES AT 8 9 AND : " $RAW THE IMAGE OF 89: AFTER THE TRANSLATION X Y → X Y 8 AND A ROTATION AROUND X 8 X 7HAT TYPE OF TRANSFORMATION MOVES POINT # TO POINT $ 8 !$ ! " 9 0OSSIBLE ANSWER ROTATION # 9 : 9 7 $ $ # REFLECTION POSSIBLE ANSWER ROTATION POSSIBLE ANSWER 7HAT TYPE OF TRANSFORMATION MOVES FIGURE & TO FIGURE ' O789 → O789 O!"#$ → O!"#$ & ' REFLECTION 54 Chapter 1
gge07te_c01_0050_0055.indde07te_c01_0050_0055.indd 5544 112/20/052/20/05 110:10:170:10:17 AAMM If students chose 29. Which type of transformation maps XYZ to XYZ? 8 D for Exercise 31, remind them to find Reflection Translation 9 the x-coordinate instead of the 9Ī Rotation Not here y-coordinate. If they chose G in 30. DEF has vertices at D(-4, 2), E(-3, -3), and F(1, 4). < Exercise 32, they found the x- <Ī coordinate instead. Which of these points is a vertex of the image of DEF 8Ī after the translation (x, y) → (x - 2, y + 1)? (-2, 1) (-5, -2) (3, 3) (-6, -1)
31. Consider the translation ( 1, 4) → (-2, 3). What number was added to the x-coordinate? Journal -3 -1 1 7 Have students explain and give an example on a coordinate plane of a 32. Consider the translation ( -5, -7) → (-2, -1). What number was added to reflection, a translation, and a rota- the y-coordinate? tion. Have them write a rule for their -3 3 6 8 translation.
CHALLENGE AND EXTEND 33. RST with vertices R(-2, -2) , S(-3, 1) , and T(1, 1) is translated by (x, y) → (x - 1, y + 3) . Then the image, RST, is translated by Have students create a figure and its (x, y) → (x + 4, y - 1) , resulting in R"S"T". image under a reflection, a rotation, a. Find the coordinates for the vertices of R"S"T". R (1, 0) ; S (0, 3) ; T (4, 3) and a translation. Then write rules (x, y) → (x + 3, y + 2) b. Write a rule for a single translation that maps RST to R"S"T ". for each and compare the results. 34. Find the angle through which the minute hand of a clock rotates over a period of 12 minutes. 72° 35. 35. A triangle has vertices A(1, 0) , B(5, 0) , and C(2, 3) . The triangle is rotated 90° y counterclockwise about the origin. Draw and label the image of the triangle. 1-7 BĪ Determine the coordinates for the reflection image of any point A(x, y) across the 1. A figure has vertices at CĪ given line. X( -1, 1) , Y (1, 4) , and Z (2, 2) . AĪ 36. x-axis (x, -y)( 37. y-axis -x, y) x After a transformation, the image of the figure has ver- SPIRAL REVIEW tices at X' (-3, 2) , Y' (-1, 5) , Use factoring to find the zeros of each function. (Previous course) and Z' (0, 3) . Draw the preim- 38. y = x 2 + 12x + 35 -5, -7 39. y = x 2 + 3x - 18 -6, 3 age and the image. Identify the transformation. transla- 40. y = x 2 - 18x + 81 9 41. y = x 2 - 3x + 2 1, 2 tion ∠ = Given m A 76.1°, find the measure of each of the following. (Lesson 1-4) YĪ y 42. supplement of ∠A 103.9° 43. complement of ∠A 13.9° Y
Use the Distance Formula and the Pythagorean Theorem to find the distance, to the ZĪ Z nearest tenth, between each pair of points. (Lesson 1-6) XĪ X x 44. (2, 3) and ( 4, 6) 3.6 45. (-1, 4) and ( 0, 8) 4.1 46. (-3, 7) and ( -6, -2) 9.5 47. (5, 1) and ( -1, 3) 6.3 2. What transformation is sug- gested by the wings of an
1- 7 Transformations in the Coordinate Plane 55 airplane? reflection (- - ) ,%33/. *ÀLiÊ-Û} ,%33/. >i}i 3. Given points P 2, 1 and £Ç 4RANSFORMATIONS IN THE #OORDINATE 0LANE1-7 PROBLEM SOLVING £Ç )NVESTIGATING -ULTIPLE 4RANSFORMATIONS1-7 CHALLENGE −− 5SE THE DIAGRAM OF THE STARTING POSITIONS OF FIVE BASKETBALL PLAYERS )N %XERCISES AND EACH IMAGE WAS THE RESULT OF MORE THAN ONE TRANSFORMATION OF Q( -1, 3) , draw PQ and its FOR %XERCISES AND THE PREIMAGE 3HOW THE STEPS THAT YOU CAN USE TO GET FROM THE PREIMAGE TO THE IMAGE Y 'RAPH THE INTERMEDIATE IMAGE AND DESCRIBE EACH STEP !FTER THE FIRST STEP OF A PLAY PLAYER IS AT 3 reflection across the y-axis. AND PLAYER IS AT 7RITE A RULE TO DESCRIBE (6, 1) Y & Y 3 THE TRANSLATIONS OF PLAYERS AND FROM THEIR STARTING X & & ' ' POSITIONS TO THEIR NEW POSITIONS 5 5 0 2 3 4 4 ' (5, 0.5) ge07se_c01_0050_0055.indd 54 12/2/05 3:53:13 PM ge07se_c01_0050_0055.indd 6 11/23/06 1:35:04 PM y PLAYER X Y → X Y & ( & 4 X X 1 ( 0 ( PLAYER X Y → X Y 5 ' 4 ' Q QĪ &OR THE SECOND STEP OF THE PLAY PLAYER IS TO MOVE TO A POSITION DESCRIBED BY 4 THE RULE X Y → X Y AND PLAYER IS TO MOVE TO A POSITION DESCRIBED ( ( BY THE RULE X Y → X Y 7HAT ARE THE POSITIONS OF THESE TWO PLAYERS AFTER THIS STEP OF THE PLAY 0OSSIBLE ANSWER FIRST A REFLECTION ACROSS THE Y AXIS PLAYER PLAYER 4HEN A TRANSLATION UNITS RIGHT AND UNITS DOWN x 5SE THE DIAGRAM FOR %XERCISES n Y Y ! Y &IND THE COORDINATES OF THE IMAGE OF !"#$ AFTER ! " IT IS MOVED UNITS LEFT AND UNITS UP ! ! # " P PĪ ?? ?? 4 # # " S D S D $ # " X # " X 4HE ORIGINAL IMAGE IS MOVED SO THAT ITS NEW X # " 0 4 COORDINATES ARE ! " ?? # 4. Find the coordinates of the ! ! AND $ ?? )DENTIFY THE TRANSFORMATION 0OSSIBLE ANSWER FIRST A REFLECTION ACROSS THE LINE Y ( ) REFLECTION ACROSS THE Y AXIS image of F 2, 7 after the 4HEN A TRANSLATION UNITS LEFT AND UNITS DOWN 4HE ORIGINAL IMAGE IS TRANSLATED SO THAT THE COORDINATES OF " ARE ?? translation (x, y) → (x + 5, 7HAT ARE THE COORDINATES OF THE OTHER THREE VERTICES OF THE IMAGE AFTER THIS TRANSLATION ?? 4RAPEZOID 7 8 9 : RESULTED AFTER TWO TRANSFORMATIONS 7 8 Y ! # $ - ( ) 4 ) S D -AKE A CONJECTURE ABOUT THE COORDINATES OF THE VERTICES y 6 . 7, 1 OF TRAPEZOID 789 : %XPLAIN 4RIANGLE (*+ HAS VERTICES ( * AND + 7HAT ARE THE : 9 : 9 COORDINATES OF THE VERTICES AFTER THE TRANSLATION X Y → X Y 7 8 9 X 4 0 5 ! ( * + # ( * + Also available on transparency : PREIMAGE REFLECTED ACROSS X AXIS 8 " ( * + $ ( * + 3 7
! SEGMENT HAS ENDPOINTS AT 3 AND 4 !FTER A TRANSFORMATION IMAGE TRANSLATED BY X Y → X Y THE IMAGE HAS ENDPOINTS AT 3 AND 4 7HICH BEST DESCRIBES )S THIS THE ONLY POSSIBLE SOLUTION FOR THE COORDINATES OF 789 : %XPLAIN THE TRANSFORMATION & REFLECTION ACROSS THE Y AXIS ( ROTATION ABOUT THE ORIGIN .O COORDINATES COULD BE 7 8 9 : ' TRANSLATION X Y → X Y * ROTATION ABOUT THE POINT Lesson 1-7 55
gge08te_ny_c01_0050_0055.indde08te_ny_c01_0050_0055.indd 5555 112/21/062/21/06 11:44:00:44:00 PPMM 1-7 Organizer Explore Transformations Use with Lesson 1-7 A transformation is a movement of a figure from its original position Pacing: (preimage) to a new position (image). In this lab, you will use geometry Traditional 1 day software to perform transformations and explore their properties. __1 Block 2 day Objective: NY Performance Use geometry Use with Lesson 1-7 Indicators software to perform trans- G.G.55, G.R.1 KEYWORD: MG7 Lab1 formations and explore their properties. Materials: geometry software Activity 1 1 Construct a triangle using the segment tool. @
Resources 3 Select ABC by clicking on all three segments of Technology Lab Activities the triangle.
1-7 Lab Recording Sheet 4 Choose Translate from the Transform menu, using Marked as the translation vector. What do you Teach notice about the relationship between your preimage and its image? Discuss They appear to be . Have students measure the sides, 5 What happens when you drag a vertex or a side angles, and perimeter of both the of ABC? preimage and the image of ABC. The s move together and stay the same Discuss whether any of these mea- size and shape. 1. The image of the moves in the same direction as the endpt. sures change after a translation or Try This a rotation. no Discuss the arrow 2. The s move together and remain a fixed dist. apart. and prime notation to make sure For Problems 1 and 2 choose New Sketch from the File menu. students understand the difference 1. Construct a triangle and a segment outside the in naming the preimage and image. triangle. Mark this segment as a translation vector Ask students whether the area of the as you did in Step 2 of Activity 1. Use Step 4 of image is affected by either a transla- Activity 1 to translate the triangle. What happens tion or a rotation. no when you drag an endpoint of the new segment? 2. Instead of translating by a marked vector, use Rectangular as the translation vector and translate by a horizontal distance of 1 cm and a vertical distance of 2 cm. Compare this method with the marked vector method. What happens when you drag a side or vertex of the triangle? 3. Select the angles and sides of the preimage and image triangles. Use the tools in the Measure menu to measure length, angle measure, perimeter, and area. What do you think is true about these two figures? Each measurement is the same New York Performance for the preimage and image s. Indicators The s appear to be . 56 Chapter 1 Foundations for Geometry Geometry G.G.55 Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections ge07se_c01_0056_0057.indd 1 11/23/06 1:35:58 PM ge07se_c01_0056_0057.indd 2 11/23/06 1:36:01 PM Process G.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts
56 Chapter 1
gge08te_ny_c01_0056_0057.indde08te_ny_c01_0056_0057.indd 5656 112/21/062/21/06 112:53:342:53:34 PPMM Close Activity 2 Key Concept 1 Construct a triangle. Label the vertices G, H, and I. When you translate or rotate a fig- ure, the lengths of sides, angle mea- 2 Select point H and choose Mark Center from the Transform menu. sures, perimeter, and area do not change. The image is congruent to 3 Select ∠GHI by selecting points G, H, and I in that the preimage. order. Choose Mark Angle from the Transform menu. Assessment Journal Have students explain how the preimage and image of a figure 4 Select the entire triangle GHI by dragging a compare under a translation and a selection box around the figure. rotation. Have them discuss which properties are preserved. 5 Choose Rotate from the Transform menu, using Marked Angle as the angle of rotation.
6 What happens when you drag a vertex or a side of GHI? The and its image rotate by the same ∠ measure and remain the same size and shape.
Try This
For Problems 4–6 choose New Sketch from the File menu. 4. Instead of selecting an angle of the triangle as the rotation angle, draw a new angle outside of the triangle. Mark this angle. Mark ∠GHI as Center and rotate the triangle. What happens when you drag one of the points that form the rotation angle? The image rotates by the same ∠ measure as the marked ∠.
The rotates by the same ∠ measure. When P is inside, the image overlaps the . When P coincides with a vertex, the image also coincides with the vertex. 5. Construct QRS, a new rotation angle, and a point P not on the triangle. Mark P as the center and mark the angle. Rotate the triangle. What happens when you drag P outside, inside, or on the preimage triangle? 6. Instead of rotating by a marked angle, use Fixed Angle as the rotation method and rotate by a fixed angle measure of 30°. Compare this method with the marked angle method. 7. Using the fixed angle method of rotation, can you find an angle measure that will result in an image figure that exactly covers the preimage figure? 360°
6. The rotated by an ∠ of 30°, not by the measure of the marked ∠.
1- 7 Technology Lab 57
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1-7 Technology Lab 57
gge07te_c01_0056_0057.indde07te_c01_0056_0057.indd 5577 112/5/052/5/05 110:49:060:49:06 PPMM SECTION 1B SECTION
1B Coordinate and £ÓÊ° Transformation Tools ÈÊ° Pave the Way Julia wants to use L-shaped paving stones Organizer to pave a patio. Two stones will cover a 12 in. by 18 in. £ÓÊ° Objective: Assess students’ rectangle. ability to apply concepts and skills in Lessons 1-5 through 1-7 in a real-world format. @
58 Chapter 1 Foundations for Geometry
INTERVENTION Scaffolding Questions ge07se_c01_0058_0059.indd 58 8/15/05 12:27:21 PM ge07se_c01_0058_0059.indd 59 8/15/05 12:27:27 PM 1. What is the significance of the way the 3. How can you decide which transforma- sides of the stones are marked? How can tion to use? What is meant by creating you find the area of this figure? The marks different patterns? You can use a reflec- on the sides of the figure mean that these tion, a rotation, or a translation. You just sides are the same length. Right angle need to be sure that the sides of the marks tell you that ∠ A and ∠F measure figures will touch when you repeat 90°. You will use the formulas for the area your pattern. Different patterns can be of a rectangle and the area of a square created using the two tiles and one to find the areas of the two parts of the transformation. figure. Then add the two areas together. Extension 2. What formulas will you use to find the Have students choose a pattern and repro- KEYWORD: MG7 Resources location of the fountain? First you will find the midpoint of a segment. Then use duce it on graph paper. They could also the Distance Formula to find the distance create the tile patterns using geometry software. 58 Chapter 1 between the fountain and each point.
gge07te_c01_0058_0059.indde07te_c01_0058_0059.indd 5588 112/5/052/5/05 110:49:480:49:48 PPMM SECTION 1B SECTION Quiz for Lessons 1-5 Through 1-7 1B 1-5 Using Formulas in Geometry Find the perimeter and area of each figure.
1. ÓäÊ° 2. = ÎÝÊÊ££ = + P 56 in.; £Î P 5x 22; Organizer nÊ° A = 160 in 2 A = 13x + 130 ÓÝÊ ÊÓä Objective: Assess students’ 3. ÈÝ 4. mastery of concepts and skills in P = 19x + 22; Lessons 1-5 through 1-7. ÎÝÊ ÊÓ £{ÝÊÊÓ £ä A = 25x + 70
P = 18x + 4; A = 18x 2 + 12x xÝÊ Ê£{ 5. Find the circumference and area of a circle with a radius of 6 m. Use the π key on Resources your calculator and round to the nearest tenth. C ≈ 37.7 m; A ≈ 113.1 m 2 Assessment Resources Section 1B Quiz 1-6 Midpoint and Distance in the Coordinate Plane −− 6. Find the coordinates for the midpoint of XY with endpoints X(-4, 6) Þ and Y(3, 8) . (-0.5, 7) { −− + 7. J is the midpoint of HK, H has coordinates (6, -2) , and J has coordinates (9, 3). Find the coordinates of K.(12, 8) , Ý 8. Using the Distance Formula, find QR and ST to the nearest tenth. { ä { −− −− −− −− / Then determine if QR ST. QR ≈ 7.3; ST ≈ 7.3; QR ST INTERVENTION
9. Using the Distance Formula and the Pythagorean Theorem, { find the distance, to the nearest tenth, from F(4, 3) to G(-3, -2) . 8.6 - Resources Ready to Go On? 1-7 Transformations in the Coordinate Plane Intervention and Identify the transformation. Then use arrow notation to describe Enrichment Worksheets the transformation. Ready to Go On? CD-ROM 10. Ī reflection; 11. +Ī ,Ī Ready to Go On? Online ABC →ABC +, *Ī -Ī translation; → * - PQRS P Q R S Ī Ī 12. A graphic designer used the translation ( x, y) → (x - 3, y + 2) Þ { to transform square HJKL. Find the coordinates and graph the image of square HJKL. 13. A figure has vertices at X(1, 1) , Y(3, 1) , and Z(3, 4) . Ý After a transformation, the image of the figure has vertices Ó ä { at X( -1, -1) , Y (-3, -1) , and Z( -3, -4). Graph the preimage and image. Then identify the transformation. { 12. H (-1, 3) ; J (2, 3) ; K (2, 0) ; L( -1, 0) Answers Ready to Go On? 59 12–13. See p. A11.
NO YES ge07se_c01_0058_0059.indd 58 8/15/05 12:27:21 PM ge07se_c01_0058_0059.indd 59 INTERVENE Diagnose and Prescribe 8/15/05 12:27:27 PMENRICH
READY TO GO ON? Intervention, Section 1B READY TO GO ON? Ready to Go On? Worksheets CD-ROM Online Intervention Enrichment, Section 1B Lesson 1-5 1-5 Intervention Activity 1-5 Worksheets Diagnose and Lesson 1-6 1-6 Intervention Activity 1-6 CD-ROM Prescribe Online Online Lesson 1-7 1-7 Intervention Activity 1-7
Ready to Go On? 59
gge07te_c01_0058_0059.indde07te_c01_0058_0059.indd 5599 112/20/052/20/05 110:11:570:11:57 AAMM CHAPTER For a complete Study Guide: list of the Review postulates and 1 theorems in this chapter, see p. S82. Organizer Vocabulary acute angle ...... 21 diameter ...... 37 plane ...... 6 Objective: Help students adjacent angles ...... 28 distance ...... 13 point ...... 6 organize and review key concepts angle ...... 20 endpoint ...... 7 postulate ...... 7 and skills presented in Chapter 1. angle bisector ...... 23 exterior of an angle ...... 20 preimage ...... 50 area ...... 36 height ...... 36 radius ...... 37 @
1. A(n) −−−−−− ? divides an angle into two congruent angles. 2. −−−−−− ? are two angles whose measures have a sum of 90°. 3. The length of the longest side of a right triangle is called the −−−−−− ? . Answers 1. angle bisector 1-1 Understanding Points, Lines, and Planes (pp. 6–11) G.CM.11 2. complementary angles 3. hypotenuse EXAMPLES EXERCISES 4. A, F, E, G or C, G, D, B ■ Name the common endpoint of SR and ST . Name each of 5. Possible answer: GC the following. 6. Possible answer: plane AEG ,-/ SR andST are opposite rays with common
endpoint S.
4. four coplanar points 5. line containing B and C 6. plane that contains A, G, and E
60 Chapter 1 Foundations for Geometry
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60 Chapter 1
gge08te_ny_c01_0060_0069.indde08te_ny_c01_0060_0069.indd 6060 112/21/062/21/06 11:45:11:45:11 PPMM ■ Draw and label three coplanar lines Draw and label each of the following. Answers intersecting in one point. 7. line containing P and Q 7. 8. pair of opposite rays both containing C PQ 9. CD intersecting plane P at B 8. * C Ű 9. D B C 1-2 Measuring and Constructing Segments (pp. 13–19) G.PS.5 10. 3.5 EXAMPLES EXERCISES 11. 5 − ■ Find the length of XY . Find each length. 12. 7.6 89 XY = -2 - 1 10. JL 11. HK 13. 22 = -3 = 3 Ó äÓ { Ó äÓ 12. Y is between X and Z, £°x 14. 13; 13; 26 XY = 13.8, and XZ = 21.4. ■ 15. 18; 18; 36 S is between R and T. Find RT. Find YZ. RT = RS + ST xÝÊÊÈ ÓÝÊ 16. ∠VYX: rt; ∠VYZ: obtuse; 13. Q is between P and R. ÎÝÊ ÈÝÊ Ê{ 3x + 2 = 5x - 6 + 2x ∠ XYZ: acute; ∠ XYW: rt.; ,-/ Find PR. 3x + 2 = 7x - 6 *+ , ∠ ZYW: acute; ∠VYW: straight ÎÝÊ ÊÓ x= 2 14. U is the midpoint of £{ÝÊÊÈ − 17. 59° RT = 3(2) + 2 = 8 TV, TU = 3x + 4, and UV = 5x - 2. Find TU, 18. 96° UV, and TV. − 15. E is the midpoint of DF, DE = 9x, and EF = 4x + 10. Find DE, EF, and DF.
1-3 Measuring and Constructing Angles (pp. 20–27) G.G.17
EXAMPLES EXERCISES ■ Classify each angle as acute, right, or obtuse. 16. Classify each angle as acute, right, or obtuse. ∠ ABC acute; ∠ 8 CBD acute; < ∠ABD obtuse; Îx Èx ∠  DBE acute; xx Çä {x ∠ CBE obtuse 697 17. m∠HJL = 116°. − ■ KM bisects ∠JKL, m∠JKM = (3x + 4)°, and Find m∠HJK. m∠MKL = (6x - 5)°. Find m∠JKL. £ÎÝÊ ÊÓä®Â 18. NP bisects ∠MNQ, 3x + 4 = 6x - 5 Def. of ∠ bisector m∠MNP = (6x - 12)°, 3 x + 9 = 6x Add 5 to both sides. and m∠PNQ = (4x + 8)°. £äÝÊ ÊÓǮ 9 = 3x Subtract 3x from both sides. ∠ Find m MNQ. x = 3 Divide both sides by 3. m∠JKL = 3x + 4 + 6x - 5 = 9x -1 = 9(3) - 1 = 26°
Study Guide: Review 61
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Study Guide: Review 61
gge08te_ny_c01_0060_0069.indde08te_ny_c01_0060_0069.indd 6161 112/21/062/21/06 11:45:45:45:45 PPMM Answers 1-4 Pairs of Angles (pp. 28–33) G.CM.11 19. only adj. 20. adj. and a lin. pair EXAMPLES EXERCISES 21. not adj. ■ Tell whether the angles are only adjacent, Tell whether the angles are only adjacent, adjacent 22. 15.4°; 105.4° adjacent and form a linear pair, or not and form a linear pair, or not adjacent. ( ) ( ) adjacent. 23. 94 - 2x °; 184 - 2x ° 19. ∠1 and ∠2 ∠1 and ∠2 are only adjacent. 24. 73° 20. ∠3 and ∠4 Ó ∠ ∠ £ Ó 2 £ 2 and 4 are not adjacent. 25. 14x - 2; 12x - 3x ∠ ∠ Î { Î ∠ 2 and ∠3 are adjacent and 21. 2 and 5 x { 2 26. 4x + 16; x + 8x + 16 form a linear pair. 27. x + 15; 4x - 20 ∠ 1 and ∠4 are adjacent and form + + a linear pair. 28. 10x 54; 100x 140 Find the measure of the complement and ≈ 2 ≈ supplement of each angle. 29. A 1385.4 m ; C 131.9 m ■ Find the measure of the complement and 30. A ≈ 153.9 ft 2 ; C ≈ 44.0 ft supplement of each angle. 22. 23. 31. 12 m 90 - 67.3 = 22.7° 180 - 67.3 = 112.7° ÓÝÊÊ{®Â ÊÈǰΠ90 - (3x - 8) = (98 - 3x)° ÊÇ{°È 180 - (3x - 8) = (188 - 3x)° 24. An angle measures 5 degrees more than 4 times ÎÝÊÊn®Â its complement. Find the measure of the angle.
1-5 Using Formulas in Geometry (pp. 36–41) G.CN.6
EXAMPLES EXERCISES ■ Find the perimeter and area of the triangle. Find the perimeter and area of each figure. P = 2x + 3x + 5 + 10 25. {ÝÊÊ£ 26. £ä = 5x + 15 ÝÊ Ê{ ÓÝ ÎÝ A = _1 (3x + 5)(2x) 2 = 2 + ÎÝÊ Êx 3 x 5x 27. 28. xÝÊ ÊÇ £Ó n ■ Find the circumference and area of the circle Óä to the nearest tenth. ÝÊÊx C = 2π r = 2π (11) Find the circumference and area of each circle to the ££ÊV = 22π nearest tenth. ≈ 69.1 cm 29. 30. 2 A = π r Ó£Ê £{ÊvÌ = π (11)2 = 121π ≈ 380.1 cm 2 31. The area of a triangle is 102 m 2. The base of the triangle is 17 m. What is the height of the triangle?
62 Chapter 1 Foundations for Geometry
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62 Chapter 1
gge08te_ny_c01_0060_0069.indde08te_ny_c01_0060_0069.indd 6262 112/21/062/21/06 11:46:16:46:16 PPMM Answers 1-6 Midpoint and Distance in the Coordinate Plane (pp. 43 –49) G.G.48, G.G.66, G.G.67 32. Y (1, 3) (- ) EXAMPLES EXERCISES 33. B 9, 6 − − 34. A (0, 2) ■ X is the midpoint of CD . C has coordinates Y is the midpoint of AB . Find the missing coordinates (-4, 1), and X has coordinates (3, -2). of each point. 35. 8.5 Find the coordinates of D. 32. A(3, 2) ; B(-1, 4) ; Y( , ) 36. 7.3 -4 + x 1 + y (3, -2) = (_ , _) 33. ( ) ( ) (- ) 37. 2 2 A 5, 0 ; B , ; Y 2, 3 8.1 -4 + x 1 + y 3 = _ -2 = _ 34. A( , ) ; B(-4, 4) ; Y(-2, 3) 38. 90° rotation; 2 2 DEFG → DEFG 6 =-4 + x -4 = 1 + y Use the Distance Formula and the Pythagorean 39. → 10 = x -5 = y translation; PQRS P Q R S Theorem to find the distance, to the nearest tenth, (- ) ( ) ( ) The coordinates of D are (10, -5) . between each pair of points. 40. X 1, 1 ; Y 1, 4 ; Z 2, 3 35. X(-2, 4) and Y(6, 1) ■ Use the Distance Formula and the ( ) (- - ) Pythagorean Theorem to find the distance, 36. H 0, 3 and K 2, 4 to the nearest tenth, from (1, 6) to (4, 2). 37. L(-4, 2) and M(3, -2) 2 2 d= √( 4 - 1) + ( 2 - 6) c2 = a2 + b2 2 = √ 3 2 + ( -4) = 3 2 + 4 2 = √9 + 16 = 9 + 16 = 25 = √25 c = √25 = 5.0 = 5.0
1-7 Transformations in the Coordinate Plane (pp. 50–55) G.G.54, G.G.56
EXAMPLES EXERCISES ■ Identify the transformation. Then use arrow Identify each transformation. Then use arrow notation to describe the transformation. notation to describe the transformation.
Ī 38. Ī Ī
Ī Ī The transformation is a reflection. Ī Ī → ABC A B C 39. * + *Ī +Ī
■ The coordinates of the vertices of rectangle HJKL are H(2, -1) , J(5, -1) , K(5, -3), and - , -Ī ,Ī L(2, -3) . Find the coordinates of the image of rectangle HJKL after the translation 40. The coordinates for the vertices of XYZ are → - + (x, y) (x 4, y 1). X(-5, -4) , Y(-3, -1) , and Z(-2, -2) . Find the H=(2 - 4, -1 + 1) = H (-2, 0) coordinates for the image of XYZ after the → + + J=(5 - 4, -1 + 1) = J( 1, 0) translation (x, y) (x 4, y 5) . K=(5 - 4, -3 + 1) = K (1, -2) L=(2 - 4, -3 + 1) = L (-2, -2)
Study Guide: Review 63
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Study Guide: Review 63
gge08te_ny_c01_0060_0069.indde08te_ny_c01_0060_0069.indd 6363 112/21/062/21/06 11:46:38:46:38 PPMM CHAPTER 1
B
1. Draw and label plane N containing two lines that intersect at B. Organizer 2. Possible answer: D, E, C, A Use the figure to name each of the following. Objective: Assess students’ 3. Possible Answer: BE mastery of concepts and skills 2. four noncoplanar points 3. line containing B and E in Chapter 1. 4. The coordinate of A is -3, and the coordinate of B is 0.5. Find AB.3.5
@
Identify each transformation. Then use arrow notation to describe the transformation. 22. +-Ī ,Ī 23. 877Ī 8Ī 180° rotation; reflection; QRS → QRS WXYZ → WXYZ 9< <Ī 9Ī
,- +Ī y 24. A designer used the translation Þ { (x, y) → (x + 3, y - 3) to transform a BĪ triangular-shaped pin ABC. Find the x coordinates and draw the image Ý of ABC. { ä { A( -2, -2) ; B (1, 1) ; C (2, -2) AĪ CĪ
64 Chapter 1 Foundations for Geometry
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KEYWORD: MG7 Resources
64 Chapter 1
gge07te_c01_0060_0069.indde07te_c01_0060_0069.indd 6644 112/5/052/5/05 110:51:060:51:06 PPMM CHAPTER 1
Organizer FOCUS ON SAT The SAT has three sections: Math, Critical Reading, and On SAT multiple-choice questions, you Objective: Provide practice for Writing. Your SAT scores show how you compare with receive one point for each correct answer, college entrance exams such as the but you lose a fraction of a point for other students. It can be used by colleges to determine SAT. admission and to award merit-based financial aid. each incorrect response. Guess only when you can eliminate at least one of the @
1. Points D, E, F, and G are on a line, in that order. 4. What is the area of the square? Resources If DE = 2, FG = 5, and DF = 6, what is the value (A) 16 Þ of EG(DG)? College Entrance Exam (B) 25 Practice (A) 13 { (C) 32 (B) 18 Questions on the SAT represent (D) 36 (C) 19 Ý the following math strands: (E) 41 (D) 42 Î { Number and Operation, 30–32% Ó (E) 99 Algebra and Functions, 28–32% Geometry and Measurement, 27–30% ∠ ∠ = ( + ) 5. If ∠BFD and ∠AFC are right angles and 2. QS bisects PQR, m PQR 4x 2 °, and Data Analysis, Statistics, and m∠SQR = (3x - 6)°. What is the value of x? m∠CFD = 72°, what is the value of x? Probability, 10–12% (A) 1
(B) 4 Items on this page focus on: (C) 7 ÝÂ • Number and Operation (D) 10 • Algebra and Functions (E) 19 Note: Figure not drawn to scale. • Geometry and Measurement (A) 18 Text References: (B) 36 3. A rectangular garden is enclosed by a brick Item 12345 border. The total length of bricks used to (C) 72 enclose the garden is 42 meters. If the length of (D) 90 Lesson 1-2 1-3 1-5 1-6 1-3 the garden is twice the width, what is the area of the garden? (E) 108 (A) 7 meters (B) 14 meters (C) 42 meters (D) 42 square meters (E) 98 square meters
College Entrance Exam Practice 65
4. Students may choose A or D because they incorrectly calculated the length of ge07se_c01_0060_0069.indd 64 12/2/05 4:04:02 PM ge07se_c01_0060_0069.indd 1. 65Students may choose B because they a side of the square. Remind students 12/2/05 4:04:06 PM have found the incorrect value of EF, of the Distance Formula. possibly by subtracting 2 from 5. 5. Students may choose E because they Remind students to sketch a figure of assumed the angles were complemen- the situation. tary. Remind students that they cannot 2. Students may choose A because they assume information that is not given in did not do the opposite operation. Also the test item. remind students of the definition of bisect and the relationship of bisected angles. 3. Students may choose C or D because they calculated the perimeter of the gar- den. Remind students to read each test item carefully to determine what ques- tion is being asked. College Entrance Exam Practice 65
gge07te_c01_0060_0069.indde07te_c01_0060_0069.indd 6655 112/5/052/5/05 110:51:130:51:13 PPMM CHAPTER 1
Organizer Multiple Choice: Work Backward Objective: Provide opportunities When you do not know how to solve a multiple-choice test item, use the answer to learn and practice common test- choices and work the question backward. Plug in the answer choices to see which taking strategies. choice makes the question true. @
− T is the midpoint of RC , RT = 12x - 8, and TC = 28. What is the value of x? - Resources 4 3 £ÓÝÊÊn Ón 2 28 , / New York State Test Prep − Workbook Since T is the midpoint of RC, then RT = RC, or 12x - 8 = 28. New York Test Practice Find what value of x makes the left side of the equation equal 28. Online Try choice A: If x =-4, then 12x - 8 = 12( -4) - 8 =-56. This choice is not correct because length is always a positive number. KEYWORD: MG7 TestPrep Try choice B: If x = 2, then 12x - 8 = 12( 2) - 8 = 16. Since 16 ≠ 28, choice B is not the answer.
Try choice C: If x = 3, then 12x - 8 = 12( 3) - 8 = 28. This Test Tackler Since 28 = 28, the correct answer is C, 3. focuses on how to work backward to obtain an answer to a multiple- choice test item. When faced with a test item they do not know how to solve, students should be encour- Joel used 6400 feet of fencing to make a rectangular horse pen. The width of the aged not to skip the item, but to pen is 4 times as long as the length. What is the length of the horse pen? use the answer choices provided in order to make an educated guess. 25 feet 640 feet Ű By substituting each answer choice 480 feet 1600 feet into the test question, students {Ű can determine whether the choice Use the formula P = 2 + 2w. P = 6400 and w = 4. You can work backward to makes the test question correct and/ determine which answer choice is the most reasonable. or reasonable. Try choice J: Use mental math. If = 1600, then 4 = 6400. This choice is not reasonable because the perimeter of the pen would then be far greater than 6400 feet.
Try choice F: Use mental math. If = 25, then 4 = 100. This choice is incorrect because the perimeter of the pen is 6400 ft, which is far greater than 2( 25) + 2( 100).
Try choice H: If = 640, then 4 = 2560. When you substitute these values into the perimeter formula, it makes a true statement.
The correct answer is H, 640 ft.
66 Chapter 1 Foundations for Geometry
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66 Chapter 1
gge08te_ny_c01_0060_0069.indde08te_ny_c01_0060_0069.indd 6666 112/21/062/21/06 112:54:552:54:55 PPMM Read each test item and answer the questions When you work a test question backward start Answers that follow. with choice C. The choices are usually listed in order from least to greatest. If choice C is Possible answers: incorrect because it is too low, you do not need Item A 1. yes; the def. of comp. and the to plug in the smaller numbers. The measure of an angle is 3 times as great comp. of an ∠ as that of its complement. Which value is the 2. Multiply the ∠ measure given in measure of the smaller angle? Item D choice A by 3 to get the greater 22.5° 63.5° QRS has vertices at Q(3, 5) , R(3, 9), and ∠ measure. Then determine ( ) S 7, 5 . Which of these points is a vertex of whether the sum of the larger 27.5° 67.5° the image of QRS after the translation ∠ measure and the ∠ measure (x, y) → (x - 7, y - 6)? given in choice A equals 90°. 1. Are there any definitions that you can use to Þ , n Repeat this process with choices solve this problem? If so, what are they? B and C, or until the correct 2. Describe how to work backward to find the È answer is found. +- correct answer. { 3. Plug in H first. If H is too high, the answer is F or G. If H is too Ó low, the answer is J. Ý Item B ä Ó {Èn 4. One-half of 26 equals 13, and In a town’s annual relay marathon race, the 13 minus 4 does not equal 4, (- ) ( ) second runner of each team starts at mile 4, 3 4, 1 so choice F is incorrect; and 13 marker 4 and runs to the halfway point of the (0, 0) (4, -3) minus 6.5 does not equal 4, so 26-mile marathon. At that point the second choice G is incorrect. runner passes the relay baton to the third runner of the team. How many total miles 5. Plug in C first. If C is too high, 7. Explain how to use mental math to find an does the second runner of each team run? the answer is A or B. If C is too answer that is NOT reasonable. 4 miles 9 miles low, the answer is D. 8. Describe, by working backward, how you 6. Subtract the value given in each 6.5 miles 13 miles can determine the correct answer. answer choice from 8, the x-coordinate of the translated pt., 3. Which answer choice should you plug in and see if you get -2, the first? Why? Item E x-coordinate of the original pt. ∠ ∠ = ( + ) 4. Describe, by working backward, how you TS bisects PTR. If m PTS 9x 2 ° and 7. If you add -7 to the x-coordi- ∠ = ( + ) know that choices F and G are not correct. m STR x 18 °, what is the value of x? nate and -6 to the y-coordinate * of any of the original pts., you do not get (0, 0). G is not a - reasonable choice. Item C , Consider the translation ( -2, 8) → (8, -4) . / 8. Add 7 to the x-value of the pt. What number was added to the x-coordinate? in choice F, and add 6 to the - 10 2 y-value of the pt. in choice -12 4 0 20 F. Look to see if this new pt. - 6 10 matches any of the original ver- tices of the . If not, repeat this 9. Explain how to use mental math to find an process until the correct answer 5. Which answer choice should you plug in answer that is NOT reasonable. is found. first? Why? 10. Describe how to use the answer choices 6. Explain how to work the test question to work backward to find which answer backward to determine the correct answer. is reasonable.
Test Tackler 67
Answers to Test Items Answers A. A Possible answers: ge07se_c01_0060_0069.indd 66 12/2/05 4:04:07 PM ge07se_c01_0060_0069.indd B. 67 H 9. If you substitute -10 into ( 9x + 2) , 12/2/05 4:04:10 PM ∠ C. D the result is neg. Since an measure cannot be a neg. value, choice A is not D. F reasonable. E. C 10. Substitute the value of x given in choice B into both ∠ measures, and simplify. If the resulting values are equivalent, then that value of x is the correct answer. Repeat this process with choices C and D, or until the correct answer is found.
KEYWORD: MG7 Resources
Test Tackler 67
gge07te_c01_0060_0069.indde07te_c01_0060_0069.indd 6677 112/5/052/5/05 110:51:230:51:23 PPMM KEYWORD: MG7 TestPrep
Organizer CUMULATIVE ASSESSMENT, CHAPTER 1 Objective: Provide review and practice for Chapter 1 and Multiple Choice Use the diagram for Items 8–10. standardized tests. {Ê Use the diagram for Items 1–3. @
+*
Resources 8. Which of these angles is adjacent to ∠MQN? ∠ ∠ 1. Which points are collinear? QMN QNP Assessment Resources ∠ ∠ A, B, and C A, B, and E NPQ PQN Chapter 1 Cumulative Test B, C, and D B, D, and E 9. What is the area of NQP? New York State Test Prep Workbook 2. What is another name for plane R? 3.7 square meters 7.4 square meters Plane C Plane ACE 6.8 square meters 13.6 square meters New York Test Practice Plane AB Plane BDE Online 10. Which of the following pairs of angles are complementary? 3. Use your protractor to find the approximate ∠MNQ and ∠QNP KEYWORD: MG7 TestPrep measure of ∠ABD. ∠ ∠ 123° 77° NQP and QPN ∠ ∠ 117° 63° MNP and QNP ∠QMN and ∠NPQ
4. S is between R and T. The distance between − R and T is 4 times the distance between S and T. 11. K is the midpoint of JL . J has coordinates If RS = 18, what is RT? (2, -1), and K has coordinates (-4, 3). 24 14.4 What are the coordinates of L? ( - ) (- ) 22.5 6 3, 2 1, 1 (1, -1) (-10, 7) 5. A ray bisects a straight angle into two congruent angles. Which term describes each of the 12. A circle with a diameter of 10 inches has a congruent angles that are formed? circumference equal to the perimeter of a square. Acute Right To the nearest tenth, what is the length of each side of the square? Obtuse Straight 2.5 inches 5.6 inches − 3.9 inches 7.9 inches 6. Which− expression states that AB is congruent to CD? − − AB CD AB = CD 13. The map coordinates of a campground are − − (1, 4), and the coordinates of a fishing pier = AB CD AB CD are (4, 7). Each unit on the map represents 1 kilometer. If Alejandro walks in a straight line 7. The measure of an angle is 35°. What is the from the campground to the pier, how many measure of its complement? kilometers, to the nearest tenth, will he walk? 35° 55° 3.5 kilometers 6.0 kilometers 45° 145° 4.2 kilometers 12.1 kilometers
68 Chapter 1 Foundations for Geometry
ge07se_c01_0060_0069.inddFor Item 9 11, encourage students to draw a 11/23/06 2:21:17 PM ge07se_c01_0060_0069.indd 10 11/23/06 1:38:25 PM diagram. Students who do not draw a dia- gram may be more likely to choose answer A, which −−gives the coordinates of the mid- point of JK , not the coordinates of L.
KEYWORD: MG7 Resources
68 Chapter 1
gge08te_ny_c01_0060_0069.indde08te_ny_c01_0060_0069.indd 6868 112/21/062/21/06 112:55:312:55:31 PPMM For many types of geometry problems, it may be Short Response Short-Response Rubric helpful to draw a diagram and label it with the 23. ABC has vertices A(-2, 0), B(0, 0), and information given in the problem. This method Items 23–25 C(0, 3). The image of ABC has vertices is a good way of organizing the information and A(1, -4), B( 3, -4), and C( 3, -1). 2 Points = The student’s response is helping you decide how to solve the problem. correct and complete. The answer a. Draw ABC and its image ABC on a coordinate plane. contains complete explanations 14. m∠R is 57°. What is the measure of its and/or adequate work when supplement? b. Write a rule for the transformation of ABC using arrow notation. required. 33° 123° (x, y) → (x + 3, y - 4) 1 Point = The student’s response 43° 133° 24. You are given the measure of ∠4. You also know demonstrates only partial under- ∠ the following angles are supplementary: 1 and standing. The response may 15. What rule would you use to translate a triangle ∠2, ∠2 and ∠3, and ∠1 and ∠4. 4 units to the right? include the correct process with £ an incorrect answer, or a correct (x, y) → (x + 4, y) {Ó Î answer without the required work. (x, y) → (x - 4, y) (x, y) → (x, y + 4) Explain how you can determine the measures of 0 Points = The student’s response ∠1, ∠2, and ∠3. demonstrates no understanding (x, y) → (x, y - 4) of the mathematical concepts. The − 25. Marian is making a circular tablecloth from a 16. If WZ bisects ∠XWY, which of the following response may include a correct rectangular piece of fabric that measures 6 yards statements is true? response using an obviously incor- by 4 yards. What is the area of the largest circular m∠XWZ > m∠YWZ piece that can be cut from the fabric? Leave your rect procedure. π m∠XWZ < m∠YWZ answer in terms of . Show your work or explain in words how you found your answer. m∠XWZ = m∠YWZ Extended-Response m∠XWZ m∠YWZ Rubric Extended Response 17. The x- and y-axes separate the coordinate plane Item 26 Demara is creating a design using a computer into four regions, called quadrants. If (c, d) is 26. illustration program. She begins by drawing the 3 Points = The student’s response a point that is not on the axes, such that c < 0 rectangle shown on the coordinate grid. is correct and complete. The and d < 0, which quadrant would contain response contains complete expla- point (c, d)? Þ+ , { nations and/or adequate work I III * - when required. II IV Ý 2 Points = The student’s response { ä { demonstrates only partial under- Gridded Response standing. The response addresses 18. The measure of ∠1 is 4 times the measure of most aspects of the task, and/or { its supplement. What is the measure, in degrees, may include complete procedures of ∠1? 144 and reasoning but an incorrect a. Demara translates rectangle PQRS using the rule (x, y) → (x - 4, y - 6). On a copy of the answer. 19. The exits for Market St. and Finch St. are coordinate grid, draw this translation and label 1 Point = The student’s response 3.5 miles apart on a straight highway. The exit each vertex. for King St. is at the midpoint between these two addresses some elements of the exits. How many miles apart are the King St. and b. Describe one way that Demara could have task correctly but contains multiple Finch St. exits? 1.75 moved rectangle PQRS to the same position in part a using a reflection and then a translation. flaws. The response may include a correct answer without the (- ) c. On the same coordinate grid, Demara reflects 20. R has coordinates 4, 9 . S has coordinates required work. (4, -6). What is RS? 17 rectangle PQRS across the x-axis. She draws a figure with vertices at (1, -3), (3, -3), (3, -5), 0 Points = The student’s response and (1, -5). Did Demara reflect rectangle PQRS 21. If ∠A is a supplement of ∠B and is a right angle, demonstrates no understanding correctly? Explain your answer. then what is m∠B in degrees? 90 26a. y of the mathematical concepts. The x response may include a correct 22. ∠C and ∠D are complementary. m∠C is 4 times response using an obviously incor- m∠D. What is m∠C? 72 QĪ RĪ rect procedure.
PĪ SĪ Cumulative Assessment, Chapter 1 69
Answers 25. 4π yd 2; possible answer: The largest Because the 2 figures have different circular piece can have a diam. no shapes, Demara did not perform the 23a. y larger than the width of the fabric. The reflection correctly. ge07se_c01_0060_0069.indd 9 11/23/06 2:21:17 PM ge07se_c01_0060_0069.indd 10 C 11/23/06 1:38:25 PM width of the fabric is 4 yd. If the diam. of the circular piece is 4 yd, then its ( 2) x radius is 2 yd and its area is π 2 = π 2 AB CĪ 4 yd . 26b. Possible answer: She could have reflected rect. PQRS across the y-axis and then translated it 6 units down. AĪ BĪ c. No; possible answer: a figure and its 24. Possible answer: ∠1 is supp. to ∠4, so reflected image should be the same m∠1 = 180° - m∠4. You continue to size and shape. Rect. PQRS has a subtract from 180° to find the measure length of 2 units and a width of 1 of each ∠. unit. The image Demara drew is a square with a side length of 2 units.
Cumulative Assessment 69
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