Third Edition

Michael Serra

Project EditorLadie Malek

Project AdministratorShannon Miller

EditorsChristian Aviles-Scott, Dan Bennett, Mary Jo Cittadino,Curt Gebhard

Editorial AssistantsHalo Golden, Erin Gray, Susan Minarcin,Laura Schattschneider, Jason Taylor

Editorial ConsultantsCavan Fang, David Hoppe, Stacey Miceli, Davia Schmidt

Mathematics ReviewersMichael de Villiers, Ph.D., University of Durban, Westville,Pinetown, South Africa

David Rasmussen, Neil’s Harbour, Nova Scotia

Multicultural and Equity ReviewersDavid Keiser, Montclair State University, Upper Montclair,New Jersey

Swapna Mukhopadhyay, Ph.D., San Diego State University,San Diego, California

Accuracy CheckersDudley Brooks, Marcia Ellen Olmstead

Editorial Production ManagerDeborah Cogan

Production EditorKristin Ferraioli

CopyeditorMargaret Moore

Production DirectorDiana Jean Parks

Production CoordinatorAnn Rothenbuhler

Cover DesignerJill Kongabel

Text DesignerMarilyn Perry

Art EditorJason Luz

Photo EditorMargee Robinson

Art and Design CoordinatorCaroline Ayres

IllustratorsJuan Alvarez, Andy Levine, Claudia Newell, Bill Pasini,William Rieser, Sue Todd, Rose Zgodzinski

Technical Art Precision Graphics

Compositor and PrepressTSI Graphics

PrinterVon Hoffman Press

Executive EditorCasey FitzSimons

PublisherSteven Rasmussen

© 2003 by Key Curriculum Press. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by anymeans, electronic, photocopying, recording, or otherwise,without the prior written permission of the publisher.

®The Geometer’s Sketchpad, Dynamic Geometry, and Key Curriculum Press are registered trademarks ofKey Curriculum Press. ™The Discovering Mathematics logoand Sketchpad are trademarks of Key Curriculum Press.

™Fathom Dynamic Statistics is a trademark ofKCP Technologies.

All other trademarks are held by their respective owners.

Key Curriculum Press1150 65th StreetEmeryville, CA 94608editorial@keypress.comhttp://www.keypress.com

Printed in the United States of America

10 9 8 7 6 5 4 06 05

ISBN 1-55953-459-1

iii

Acknowledgments

First, to all the teachers who have used Discovering Geometry, a sincere thank youfor your wonderful support and encouragement. I wish to thank my always

delightful and ever-so-patient students for their insight, humor, and hard work.I also wish to thank the many students across the country who have written to mewith their kind words, comments, and suggestions. And thanks to Kelvin Taylorand the rest of the marketing and sales staff at Key Curriculum Press for theirsuccessful efforts in bringing the first two editions into so many classrooms.

There are two people who have added their touch to earlier editions of DiscoveringGeometry: Steve Rasmussen was editor on the first edition and Dan Bennett waseditor on the second edition. Thank you, Steve and Dan.

This third edition, as you can see from the credits, involved a much larger team.While working on this edition of Discovering Geometry, I was fortunate to have theassistance of Ladie Malek as project editor. Thank you, Ladie, for your creativity,dedication, and especially your patience. To the editorial and production staff andmanagers at Key, the field testers, the advisors, the consultants, and the reviewers,I am grateful for your quality work.

Michael Serra

iv

A Note from the Publisher

When Key Curriculum Press first published Discovering Geometry in 1989,it was unique among high school geometry books because of its discovery

approach. Discovering Geometry still presents concepts visually, and studentsexplore ideas analytically, then inductively, and finally deductively—developinginsight, confidence, and increasingly sophisticated mathematical understanding.As J. Michael Shaughnessy, mathematics professor at Portland State University, said,“This is a book for ‘doers.’ Students constantly do things in this book, both aloneand in groups. If you want your students to become actively involved in the processof learning and creating geometry, then this is the book for you.”

The mathematics we learn and teach in school changes over time, driven by newscientific discoveries, new research in education, changing societal needs, and by the use of new technology in work and in education. The effectiveness ofDiscovering Geometry’s investigative approach has been substantiated in manythousands of classrooms and is reflected in the Principles and Standards for SchoolMathematics, the guiding document of the National Council of Teachers ofMathematics (NCTM). In this, the third edition, you will find many of the text’soriginal and hallmark features—plus a host of improvements. The layout is easierto follow, and the additional examples from art and science will be a motivatingcomplement to your curriculum. We have carefully analyzed the exercises foroptimal practice and real-world interest. There are more opportunities to reviewalgebra and more ways to use technology, especially The Geometer’s Sketchpad®software, in the projects and homework assignments. These changes will give you—the student, parent, or teacher—greater flexibility in attaining your educationalgoals. They will enable more students to succeed in high school geometry andachieve continued success in future mathematics courses, other areas of education,and eventual careers.

Experience as well as sound educational research on how geometric thinkingdevelops during adolescence tells us that, regardless of the subject or level, studentslearn mathematics best when they understand the concepts. The positive feedbackwe have received over the years for Discovering Geometry has inspired us to createan entire series. Key Curriculum Press now offers the Discovering Mathematicsseries, a complete program of algebra, geometry, and advanced algebra. Throughthe investigations that are the heart of the series, students discover many importantmathematical principles. In the process, they come to believe in their ability tosucceed at mathematics, they understand the course content more deeply, and theyrealize that they can re-create their discoveries if they need to.

If you are a student, we hope that as you work through this course you gainknowledge for a lifetime. If you are a parent, we hope you enjoy watching yourstudent develop mathematical power. If you are a teacher, we hope you find thatDiscovering Geometry makes a significant positive impact in your classroom.Whether you are learning, guiding, or teaching, please share your trials andsuccesses with the professional team at Key Curriculum Press.

Steven Rasmussen, PresidentKey Curriculum Press

v

A Note to Students from the Author xiv

0.1 Geometry in Nature and in Art 20.2 Line Designs 70.3 Circle Designs 100.4 Op Art 130.5 Knot Designs 16

Project: Symbolic Art 190.6 Islamic Tile Designs 20

Project: Photo or Video Safari 23Chapter 0 Review 24

Assessing What You’ve Learned 26

1.1 Building Blocks of Geometry 28Investigation: Mathematical Models 29Project: Spiral Designs 35

Using Your Algebra Skills 1: Midpoint 361.2 Poolroom Math 38

Investigation: Virtual Pool 411.3 What’s a Widget? 47

Investigation: Defining Angles 491.4 Polygons 541.5 Triangles and Special Quadrilaterals 59

Investigation: Triangles and Special Quadrilaterals 60Project: Drawing the Impossible 66

1.6 Circles 67Investigation: Defining Circle Terms 69

1.7 A Picture Is Worth a Thousand Words 731.8 Space Geometry 80

Investigation: Space Geometry 82Exploration: Geometric Probability I 86

Activity: Chances Are 86Chapter 1 Review 88

Assessing What You’ve Learned 92

Contents

CHAPTER

1Introducing Geometry 27

CHAPTER

0Geometric Art 1

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2.1 Inductive Reasoning 94Investigation: Shape Shifters 96

2.2 Deductive Reasoning 100Investigation: Overlapping Segments 102

2.3 Finding the nth Term 106Investigation: Finding the Rule 106Project: Best-Fit Lines 111

2.4 Mathematical Modeling 112Investigation: Party Handshakes 112

Exploration: The Seven Bridges of Königsberg 118Activity: Traveling Networks 118

2.5 Angle Relationships 120Investigation 1: The Linear Pair Conjecture 120Investigation 2: Vertical Angles Conjecture 121

2.6 Special Angles on Parallel Lines 126Investigation 1: Which Angles Are Congruent? 126Investigation 2: Is the Converse True? 128Project: Line Designs 132

Using Your Algebra Skills 2: Slope 133Exploration: Patterns in Fractals 135

Activity: The Sierpinski Triangle 136Chapter 2 Review 138

Assessing What You’ve Learned 140

3.1 Duplicating Segments and Angles 142Investigation 1: Copying a Segment 143Investigation 2: Copying an Angle 144

3.2 Constructing Perpendicular Bisectors 147Investigation 1: Finding the Right Bisector 147Investigation 2: Right Down the Middle 148

3.3 Constructing Perpendiculars to a Line 152Investigation 1: Finding the Right Line 152Investigation 2: Patty-Paper Perpendiculars 153

3.4 Constructing Angle Bisectors 157Investigation 1: Angle Bisecting by Folding 157Investigation 2: Angle Bisecting with Compass 158

3.5 Constructing Parallel Lines 161Investigation: Constructing Parallel Lines by Folding 161

Using Your Algebra Skills 3: Slopes of Parallel and Perpendicular Lines 165

3.6 Construction Problems 168

CHAPTER

3Using Tools of Geometry 141

CHAPTER

2Reasoning in Geometry 93

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Exploration: Perspective Drawing 172Activity: Boxes in Space 173

3.7 Constructing Points of Concurrency 176Investigation 1: Concurrence 176Investigation 2: Incenter and Circumcenter 177

3.8 The Centroid 183Investigation 1: Are Medians Concurrent? 183Investigation 2: Balancing Act 184

Exploration: The Euler Line 189Activity: Three Out of Four 189Project: Is There More to the Orthocenter? 190

Chapter 3 Review 191Mixed Review 194Assessing What You’ve Learned 196

4.1 Triangle Sum Conjecture 198Investigation: The Triangle Sum 199

4.2 Properties of Special Triangles 204Investigation 1: Base Angles in an Isosceles Triangle 205Investigation 2: Is the Converse True? 206

Using Your Algebra Skills 4: Writing Linear Equations 2104.3 Triangle Inequalities 213

Investigation 1: What Is the Shortest Path from A to B? 214Investigation 2: Where Are the Largest and Smallest Angles? 215Investigation 3: Exterior Angles of a Triangle 215Project: Random Triangles 218

4.4 Are There Congruence Shortcuts? 219Investigation 1: Is SSS a Congruence Shortcut? 220Investigation 2: Is SAS a Congruence Shortcut? 221

4.5 Are There Other Congruence Shortcuts? 225Investigation: Is ASA a Congruence Shortcut? 225

4.6 Corresponding Parts of Congruent Triangles 230Project: Polya’s Problem 234

4.7 Flowchart Thinking 2354.8 Proving Isosceles Triangle Conjectures 241

Investigation: The Symmetry Line in an Isosceles Triangle 242Exploration: Napoleon’s Theorem 247

Activity: Napoleon Triangles 247Project: Lines and Isosceles Triangles 248

Chapter 4 Review 249Take Another Look 253Assessing What You’ve Learned 254

CHAPTER

4Discovering and Proving Triangle Properties 197

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5.1 Polygon Sum Conjecture 256Investigation: Is There a Polygon Sum Formula? 256

5.2 Exterior Angles of a Polygon 260Investigation: Is There an Exterior Angle Sum? 260

Exploration: Star Polygons 264Activity: Exploring Star Polygons 264

5.3 Kite and Trapezoid Properties 266Investigation 1: What Are Some Properties of Kites? 266Investigation 2: What Are Some Properties of Trapezoids? 268Project: Drawing Regular Polygons 272

5.4 Properties of Midsegments 273Investigation 1: Triangle Midsegment Properties 273Investigation 2: Trapezoid Midsegment Properties 274Project: Building an Arch 278

5.5 Properties of Parallelograms 279Investigation: Four Parallelogram Properties 279

Using Your Algebra Skills 5: Solving Systems of Linear Equations 2855.6 Properties of Special Parallelograms 287

Investigation 1: What Can You Draw with the Double-Edged Straightedge? 287

Investigation 2: Do Rhombus Diagonals Have Special Properties? 288

Investigation 3: Do Rectangle Diagonals Have Special Properties? 289

5.7 Proving Quadrilateral Properties 294Project: Japanese Puzzle Quilts 299

Chapter 5 Review 300Take Another Look 303Assessing What You’ve Learned 304

6.1 Chord Properties 306Investigation 1: How Do We Define Angles in a Circle? 307Investigation 2: Chords and Their Central Angles 307Investigation 3: Chords and the Center of the Circle 309Investigation 4: Perpendicular Bisector of a Chord 309

6.2 Tangent Properties 313Investigation 1: Going Off on a Tangent 313Investigation 2: Tangent Segments 314

6.3 Arcs and Angles 319Investigation 1: Inscribed Angle Properties 319Investigation 2: Inscribed Angles Intercepting the Same Arc 320

CHAPTER

6Discovering and Proving Circle Properties 305

CHAPTER

5Discovering and Proving Polygon Properties 255

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Investigation 3: Angles Inscribed in a Semicircle 320Investigation 4: Cyclic Quadrilaterals 321Investigation 5: Arcs by Parallel Lines 321

6.4 Proving Circle Conjectures 325Using Your Algebra Skills 6: Finding the Circumcenter 3296.5 The Circumference/Diameter Ratio 331

Investigation: A Taste of Pi 332Project: Needle Toss 336

6.6 Around the World 3376.7 Arc Length 341

Investigation: Finding the Arcs 342Project: Racetrack Geometry 345

Exploration: Cycloids 346Activity: Turning Wheels 346

Chapter 6 Review 349Mixed Review 352Take Another Look 355Assessing What You’ve Learned 356

7.1 Transformations and Symmetry 358Investigation: The Basic Property of a Reflection 360

7.2 Properties of Isometries 366Investigation 1: Transformations on a Coordinate Plane 367Investigation 2: Finding a Minimal Path 367

7.3 Compositions of Transformations 373Investigation 1: Reflections over Two Parallel Lines 374Investigation 2: Reflections over Two Intersecting Lines 375Project: Kaleidoscopes 378

7.4 Tessellations with Regular Polygons 379Investigation: The Semiregular Tessellations 380

7.5 Tessellations with Nonregular Polygons 384Investigation 1: Do All Triangles Tessellate? 384Investigation 2: Do All Quadrilaterals Tessellate? 385Project: Penrose Tilings 388

7.6 Tessellations Using Only Translations 3897.7 Tessellations That Use Rotations 3937.8 Tessellations That Use Glide Reflections 398Using Your Algebra Skills 7: Finding the Orthocenter

and Centroid 401Chapter 7 Review 404

Assessing What You’ve Learned 408

CHAPTER

7Transformations and Tessellations 357

x

8.1 Areas of Rectangles and Parallelograms 410Investigation: Area Formula for Parallelograms 412Project: Random Rectangles 416

8.2 Areas of Triangles, Trapezoids, and Kites 417Investigation 1: Area Formula for Triangles 417Investigation 2: Area Formula for Trapezoids 417Investigation 3: Area Formula for Kites 418Project: Maximizing Area 421

8.3 Area Problems 422Investigation: Solving Problems with Area Formulas 422

8.4 Areas of Regular Polygons 426Investigation: Area Formula for Regular Polygons 426

Exploration: Pick’s Formula for Area 430Activity: Dinosaur Footprints and Other Shapes 431

8.5 Areas of Circles 433Investigation: Area Formula for Circles 433

8.6 Any Way You Slice It 437Exploration: Geometric Probability II 442

Activity: Where the Chips Fall 442Project: Different Dice 444

8.7 Surface Area 445Investigation 1: Surface Area of a Regular Pyramid 448Investigation 2: Surface Area of a Cone 449

Exploration: Alternative Area Formulas 453Activity: Calculating Area in Ancient Egypt 453

Chapter 8 Review 455Take Another Look 459Assessing What You’ve Learned 460

9.1 The Theorem of Pythagoras 462Investigation: The Three Sides of a Right Triangle 462Project: Creating a Geometry Flip Book 467

9.2 The Converse of the Pythagorean Theorem 468Investigation: Is the Converse True? 468

Using Your Algebra Skills 8: Radical Expressions 4739.3 Two Special Right Triangles 475

Investigation 1: Isosceles Right Triangles 475Investigation 2: 30°-60°-90° Triangles 476

Exploration: A Pythagorean Fractal 480Activity: The Right Triangle Fractal 481

9.4 Story Problems 482

CHAPTER

9The Pythagorean Theorem 461

CHAPTER

8Area 409

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9.5 Distance in Coordinate Geometry 486Investigation 1: The Distance Formula 486Investigation 2: The Equation of a Circle 488

Exploration: Ladder Climb 491Activity: Climbing the Wall 491

9.6 Circles and the Pythagorean Theorem 492Chapter 9 Review 496

Mixed Review 499Take Another Look 501Assessing What You’ve Learned 502

10.1 The Geometry of Solids 504Exploration: Euler’s Formula for Polyhedrons 512

Activity: Toothpick Polyhedrons 51210.2 Volume of Prisms and Cylinders 514

Investigation: The Volume Formula for Prisms and Cylinders 515Project: The Soma Cube 521

10.3 Volume of Pyramids and Cones 522Investigation: The Volume Formula for Pyramids and Cones 522Project: The World’s Largest Pyramid 527

Exploration: The Five Platonic Solids 528Activity: Modeling the Platonic Solids 528

10.4 Volume Problems 53110.5 Displacement and Density 535

Project: Maximizing Volume 538Exploration: Orthographic Drawing 539

Activity: Isometric and Orthographic Drawings 54110.6 Volume of a Sphere 542

Investigation: The Formula for the Volume of a Sphere 54210.7 Surface Area of a Sphere 546

Investigation: The Formula for the Surface Area of a Sphere 546Exploration: Sherlock Holmes and Forms of Valid Reasoning 551

Activity: It’s Elementary! 552Chapter 10 Review 554

Take Another Look 557Assessing What You’ve Learned 558

CHAPTER

10Volume 503

xii

Using Your Algebra Skills 9: Proportion and Reasoning 56011.1 Similar Polygons 563

Investigation 1: What Makes Polygons Similar? 564Investigation 2: Dilations on a Coordinate Plane 566Project: Making a Mural 571

11.2 Similar Triangles 572Investigation 1: Is AA a Similarity Shortcut? 572Investigation 2: Is SSS a Similarity Shortcut? 573Investigation 3: Is SAS a Similarity Shortcut? 574

Exploration: Constructing a Dilation Design 578Activity: Dilation Creations 578

11.3 Indirect Measurement with Similar Triangles 581Investigation: Mirror, Mirror 581

11.4 Corresponding Parts of Similar Triangles 586Investigation 1: Corresponding Parts 586Investigation 2: Opposite Side Ratios 588

11.5 Proportions with Area and Volume 592Investigation 1: Area Ratios 592Investigation 2: Volume Ratios 594Project: In Search of the Perfect Rectangle 598

Exploration: Why Elephants Have Big Ears 599Activity: Convenient Sizes 599

11.6 Proportional Segments Between Parallel Lines 603Investigation 1: Parallels and Proportionality 604Investigation 2: Extended Parallel/Proportionality 606

Exploration: Two More Forms of Valid Reasoning 611Activity: Symbolic Proofs 613

Chapter 11 Review 614Take Another Look 617Assessing What You’ve Learned 618

12.1 Trigonometric Ratios 620Investigation: Trigonometric Tables 622

12.2 Problem Solving with Right Triangles 627Project: Light for All Seasons 631

Exploration: Indirect Measurement 632Activity: Using a Clinometer 632

12.3 The Law of Sines 634Investigation 1: Area of a Triangle 634Investigation 2: The Law of Sines 635

12.4 The Law of Cosines 641Investigation: A Pythagorean Identity 641

CHAPTER

12Trigonometry 619

CHAPTER

11Similarity 559

xiii

Project: Japanese Temple Tablets 64612.5 Problem Solving with Trigonometry 647Exploration: Trigonometric Ratios and the Unit Circle 651

Activity: The Unit Circle 651Project: Trigonometric Functions 654

Exploration: Three Types of Proofs 655Activity: Prove It! 657

Chapter 12 Review 659Mixed Review 662Take Another Look 665Assessing What You’ve Learned 666

13.1 The Premises of Geometry 66813.2 Planning a Geometry Proof 67913.3 Triangle Proofs 68613.4 Quadrilateral Proofs 692

Investigation: Proving Parallelogram Conjectures 692Exploration: Proof as Challenge and Discovery 696

Activity: Exploring Properties of Special Constructions 69613.5 Indirect Proof 698

Investigation: Proving the Tangent Conjecture 69913.6 Circle Proofs 70313.7 Similarity Proofs 706

Investigation: Can You Prove the SSS Similarity Conjecture? 708Using Your Algebra Skills 10: Coordinate Proof 712

Project: Special Proofs of Special Conjectures 717Exploration: Non-Euclidean Geometries 718

Activity: Elliptic Geometry 719Chapter 13 Review 721

Assessing What You’ve Learned 723

Hints for Selected Exercises 725

Index 747

Photo Credits 765

CHAPTER

13Geometry as a Mathematical System 667

xiv

A Note to Students from the Author

Michael Serra

What Makes Discovering Geometry Different?

Discovering Geometry was designed so that you can be actively engaged as youlearn geometry. In this book you “learn by doing.” You will learn to use the tools of geometry and to perform geometry investigations with them. Many of thegeometry investigations are carried out in small cooperative groups in which youjointly plan and find solutions with other students. Your investigations will lead youto the discovery of geometry properties. In addition, you will gradually learn aboutproof, a form of reasoning that will help explain why your discoveries are true.

Discovering Geometry was designed so that you and your teacher can have funwhile learning geometry. It has a lot of “extras.” Each lesson begins with a quotethat I hope you will find funny or thought provoking. I think you’ll enjoy the extra challenges in the Improving Your…Skills puzzles at the end of most lessons.To solve each puzzle, you’ll need clever visual thinking skills or sharp reasoningskills or both. I hope you will find some of the illustrated word problemshumorous. I created them in the hope of reducing any anxiety you might haveabout word problems. In the explorations you will learn about geometricprobability, build geometric solids, find the height of your school building, anddiscover why elephants have big ears. In the projects you will draw the impossible,make kaleidoscopes, design a racetrack, and create a mural. There are also severalgraphing calculator projects, Fathom Dynamic Statistics™ projects, TheGeometer’s Sketchpad explorations, and web links that will allow you to practiceand improve your skills with the latest educational technology.

You can do the projects, the puzzles, and the calculator and computer activitiesindependently, whether or not your class tackles them as a group. Read throughthem as you proceed through the book.

Suggestions for Success

It is important to be organized. Keep a notebook with a section for definitions,a section for your geometry investigations, a section for discoveries, and a section for daily notes and exercises. Develop the habit of writing a summary page whenyou have completed each chapter. Study your notebook regularly.

You will need four tools for the investigations: a compass, a protractor, astraightedge, and a ruler. Some investigations use waxed “patty paper” that can be used as a unique geometry tool. Keep a graphing calculator handy, too.

You will find hints for some exercises in the back of the book. Those exercises aremarked with an . Try to solve the problems on your own first. Refer to the hintsto check your method or as a last resort if you can’t solve a problem.

Discovering Geometry will ask you to work cooperatively with your classmates.When you are working cooperatively, always be willing to listen to each other, toactively participate, to ask each other questions, and to help each other when asked.You can accomplish much more cooperatively than you can individually. And, bestof all, you’ll experience less frustration and have much more fun.

Michael Serra