Transcription
DG3SM586 5th06fm.qxd 8/2/06 6:45 PM Page iDiscoveringGeometryAn Investigative ApproachSolutions Manual
DG3SM586 6th07fm.qxd1/3/074:34 PMPage iiTeacher’s Materials Project Editor: Joan LewisEditor: Heather DeverProject Manager: Erin GrayWriters: Abby Tanenbaum, Stacey MiceliEditorial Assistants: Lori Dixon, Karen Rodriguez’GAccuracy Checker: Dudley BrooksProduction Editors: Jacqueline Gamble, Angela ChenCopyeditor: Margaret MooreEditorial Production Manager: Deborah CoganProduction Director: Diana Jean ParksProduction Coordinator: Ann RothenbuhlerText Designers: Jenny Somerville, Garry HarmanArt Editor: Jason LuzComposition, Technical Art: Interactive Composition CorporationArt and Design Coordinator: Caroline AyresCover Designer: Jill KongabelPrinter, Prepress: Data ReproductionsExecutive Editor: Casey FitzSimonsPublisher: Steven Rasmussen 2003 by Key Curriculum Press. All rights reserved.Cover Photo Credits: Background image: Doug Wilson/Westlight/Corbis.Construction site image: Sonda Dawes/The Image Works. All other images:Ken Karp Photography.Limited Reproduction PermissionThe publisher grants the teacher who purchases Discovering Geometry: AnInvestigative Approach, Solutions Manual the right to reproduce material for use inhis or her own classroom. Unauthorized copying of Discovering Geometry: AnInvestigative Approach, Solutions Manual constitutes copyright infringement and isa violation of federal law. Key Curriculum Press is a registered trademark of Key Curriculum Press.All registered trademarks and trademarks in this book are the property oftheir respective holders.Key Curriculum Press1150 65th StreetEmeryville, CA keypress.comPrinted in the United States of America10 9 8 7 609 08 07ISBN 978-1-55953-586-1
DG3SM586 5th06fm.qxd 8/2/06 6:45 PM Page iiiIntroductionContentsIntroductionvii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Chapter 0Lesson 0.1: Geometry in Nature and in ArtLesson 0.2: Line Designs . . . . . . . . . . . . . . . .Lesson 0.3: Circle Designs . . . . . . . . . . . . . . .Lesson 0.4: Op Art . . . . . . . . . . . . . . . . . . . . . .Lesson 0.5: Knot Designs . . . . . . . . . . . . . . . .Lesson 0.6: Islamic Tile Designs . . . . . . . . . .Chapter 0 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1124456Chapter 1Lesson 1.1: Building Blocks of Geometry . . . . . . .Using Your Algebra Skills 1: Midpoint . . . . . . . . . .Lesson 1.2: Poolroom Math . . . . . . . . . . . . . . . . . . . .Lesson 1.3: What’s a Widget? . . . . . . . . . . . . . . . . . .Lesson 1.4: Polygons . . . . . . . . . . . . . . . . . . . . . . . . . .Lesson 1.5: Triangles and Special Quadrilaterals .Lesson 1.6: Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . .Lesson 1.7: A Picture Is Worth a Thousand WordsLesson 1.8: Space Geometry . . . . . . . . . . . . . . . . . . .Chapter 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7. 8. 911131517192124. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Chapter 2Lesson 2.1: Inductive Reasoning . . . . . . . . . .Lesson 2.2: Deductive Reasoning . . . . . . . . .Lesson 2.3: Finding the nth Term . . . . . . . . .Lesson 2.4: Mathematical Modeling . . . . . . .Lesson 2.5: Angle Relationships . . . . . . . . . . .Lesson 2.6: Special Angles on Parallel LinesUsing Your Algebra Skills 2: Slope . . . . . . . . .Chapter 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2527293032343536Chapter 3Lesson 3.1: Duplicating Segments and Angles . . .Lesson 3.2: Constructing Perpendicular Bisectors .Lesson 3.3: Constructing Perpendiculars to a LineLesson 3.4: Constructing Angle Bisectors . . . . . . . .Lesson 3.5: Constructing Parallel Lines . . . . . . . . . .Using Your Algebra Skills 3: Slopes of Paralleland Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3739414345. . . . . . . . . . . . . . . . .46. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii
DG3SM586 5th06fm.qxd 8/2/06 6:45 PM Page ivLesson 3.6: Construction Problems . . . . . . . . . . . .Exploration: Perspective Drawing . . . . . . . . . . . . .Lesson 3.7: Constructing Points of ConcurrencyLesson 3.8: The Centroid . . . . . . . . . . . . . . . . . . . . .Exploration: The Euler Line . . . . . . . . . . . . . . . . . . .Chapter 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .475050525454Chapter 4Lesson 4.1: Triangle Sum Conjecture . . . . . . . . . . . . . . . . . .Lesson 4.2: Properties of Special Triangles . . . . . . . . . . . . .Using Your Algebra Skills 4: Writing Linear Equations . . .Lesson 4.3: Triangle Inequalities . . . . . . . . . . . . . . . . . . . . . .Lesson 4.4: Are There Congruence Shortcuts? . . . . . . . . . .Lesson 4.5: Are There Other Congruence Shortcuts? . . . .Lesson 4.6: Corresponding Parts of Congruent TrianglesLesson 4.7: Flowchart Thinking . . . . . . . . . . . . . . . . . . . . . . .Lesson 4.8: Proving Isosceles Triangle Conjectures . . . . . .Exploration: Napoleon’s Theorem . . . . . . . . . . . . . . . . . . . . .Chapter 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Take Another Look . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .565861636466686971727274Chapter 5Lesson 5.1: Polygon Sum Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . .Lesson 5.2: Exterior Angles of a Polygon . . . . . . . . . . . . . . . . . . . . . .Lesson 5.3: Kite and Trapezoid Properties . . . . . . . . . . . . . . . . . . . . .Lesson 5.4: Properties of Midsegments . . . . . . . . . . . . . . . . . . . . . . . .Lesson 5.5: Properties of Parallelograms . . . . . . . . . . . . . . . . . . . . . . .Using Your Algebra Skills 5: Solving Systems of Linear EquationsLesson 5.6: Properties of Special Parallelograms . . . . . . . . . . . . . . .Lesson 5.7: Proving Quadrilateral Properties . . . . . . . . . . . . . . . . . . .Chapter 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Take Another Look . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75787981848688919396Chapter 6Lesson 6.1: Chord Properties . . . . . . . . . . . . . . . . . . . . . . .Lesson 6.2: Tangent Properties . . . . . . . . . . . . . . . . . . . . .Lesson 6.3: Arcs and Angles . . . . . . . . . . . . . . . . . . . . . . . .Lesson 6.4: Proving Circle Conjectures . . . . . . . . . . . . . . .Using Your Algebra Skills 6: Finding the CircumcenterLesson 6.5: The Circumference/Diameter Ratio . . . . . . .Lesson 6.6: Around the World . . . . . . . . . . . . . . . . . . . . . .Lesson 6.7: Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exploration: Cycloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Chapter 6 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Take Another Look . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iv97. 99102105107110112113115116120. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DG3SM586 5th06fm.qxd 8/2/06 6:45 PM Page vChapter 7Lesson 7.1: Transformations and Symmetry . . . . . . . . . . . . . . . . . . . . .Lesson 7.2: Properties of Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . .Lesson 7.3: Compositions of Transformations . . . . . . . . . . . . . . . . . . .Lesson 7.4: Tessellations with Regular Polygons . . . . . . . . . . . . . . . . .Lesson 7.5: Tessellations with Nonregular Polygons . . . . . . . . . . . . .Lesson 7.6: Tessellations Using Only Translations . . . . . . . . . . . . . . . .Lesson 7.7: Tessellations That Use Rotations . . . . . . . . . . . . . . . . . . . .Lesson 7.8: Tessellations That Use Glide Reflections . . . . . . . . . . . . .Using Your Algebra Skills 7: Finding the Orthocenterand Centroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Chapter 7 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120122125127129130130131132134Chapter 8Lesson 8.1: Areas of Rectangles and ParallelogramsLesson 8.2: Areas of Triangles, Trapezoids, and KitesLesson 8.3: Area Problems . . . . . . . . . . . . . . . . . . . . . . . .Lesson 8.4: Areas of Regular Polygons . . . . . . . . . . . . .Lesson 8.5: Areas of Circles . . . . . . . . . . . . . . . . . . . . . . .Lesson 8.6: Any Way You Slice It . . . . . . . . . . . . . . . . . .Exploration: Geometric Probability II . . . . . . . . . . . . . .Lesson 8.7: Surface Area . . . . . . . . . . . . . . . . . . . . . . . . .Exploration: Alternative Area Formulas . . . . . . . . . . . .Chapter 8 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Take Another Look . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136138142144146147149149151152155Chapter 9Lesson 9.1: The Theorem of Pythagoras . . . . . . . . . . . . . .Lesson 9.2: The Converse of the Pythagorean TheoremUsing Your Algebra Skills 8: Radical Expressions . . . . . . .Lesson 9.3: Two Special Right Triangles . . . . . . . . . . . . . . .Lesson 9.4: Story Problems . . . . . . . . . . . . . . . . . . . . . . . . . .Lesson 9.5: Distance in Coordinate Geometry . . . . . . . . .Lesson 9.6: Circles and the Pythagorean Theorem . . . . .Chapter 9 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Take Another Look . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156159161162165168171175181Chapter 10Lesson 10.1:Lesson 10.2:Lesson 10.3:Lesson 10.4:Lesson 10.5:Lesson 10.6:Lesson 10.7:The Geometry of Solids . . . . . . . .Volume of Prisms and CylindersVolume of Pyramids and Cones .Volume Problems . . . . . . . . . . . . . .Displacement and Density . . . . .Volume of a Sphere . . . . . . . . . . . .Surface Area of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .182183186189191194197v
DG3SM586 5th06fm.qxd 8/2/06 6:45 PM Page viChapter 10 ReviewTake Another Look. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199203Chapter 11Using Your Algebra Skills 9: Proportion and Reasoning . . . .Lesson 11.1: Similar Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . .Lesson 11.2: Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . .Lesson 11.3: Indirect Measurement with Similar Triangles . .Lesson 11.4: Corresponding Parts of Similar Triangles . . . . .Lesson 11.5: Proportions with Area and Volume . . . . . . . . . . .Exploration: Why Elephants Have Big Ears . . . . . . . . . . . . . . . .Lesson 11.6: Proportional Segments Between Parallel LinesExploration: Two More Forms of Valid Reasoning . . . . . . . . . .Chapter 11 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Take Another Look . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203205208212215219222222226226228Chapter 12Lesson 12.1: Trigonometric Ratios . . . . . . . . . . . . . . . . . .Lesson 12.2: Problem Solving with Right Triangles . .Exploration: Indirect Measurement . . . . . . . . . . . . . . . . .Lesson 12.3: The Law of Sines
The Solutions Manualfor Discovering Geometry: An Investigative Approach contains solutions to the exercises at the end of each lesson and to the Extensions, Improving Your Reasoning Skills, Improving Your Visual Thinking Skills, Improving Your Algebra Skills, Projects, and Take Another Look activities. You can find solutions for the Investigations and Explorations
