the center for mathematics education project was developed...

The Center for Mathematics Education Project was developed at Education Development Center, Inc. (EDC) within the Center for Mathematics Education (CME), with partial support from the National Science Foundation.

Education Development Center, Inc.Center for Mathematics EducationNewton, Massachusetts

This material is based upon work supported by the National Science Foundation under Grant No.

ESI-0242476, Grant No. MDR-9252952, and Grant No. ESI-9617369. Any opinions, findings, and

conclusions or recommendations expressed in this material are those of the author(s) and do not

necessarily reflect the views of the National Science Foundation.

Cover Art: 9 Surf Studios; Jim Cummins/Corbis; Stockbyte/Getty Images, Inc.

Taken from:

CME Project: GeometryBy the CME Project Development Team

Copyright ©2009 by Educational Development Center, Inc.

Published by Pearson Education, Inc.

Upper Saddle River, New Jersey 07458

CME Common Core Additional Lessons: GeometryBy the CME Project Development Team

Copyright ©2012 by Educational Development Center, Inc.

Published by Pearson Education, Inc.

Upper Saddle River, New Jersey 07458

CME Project Development TeamLead Developer: Al Cuoco

Core Development Team: Anna Baccaglini-Frank, Jean Benson, Nancy Antonellis D’Amato, Daniel Erman,

Brian Harvey, Wayne Harvey, Bowen Kerins, Doreen Kilday, Ryota Matsuura, Stephen Maurer, Sarah Sword,

Audrey Ting, and Kevin Waterman.

Others who contributed include Steve Benson, Paul D’Amato, Robert Devaney, Andrew Golay,

Paul Goldenberg, Jane Gorman, C. Jud Hill, Eric Karnowski, Helen Lebowitz, Joseph Leverich, Melanie Palma,

Mark Saul, Nina Shteingold, and Brett Thomas.

All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission

in writing from the publisher.

This special edition published in cooperation with Pearson Learning Solutions.

All trademarks, service marks, registered trademarks, and registered service marks are the property of their

respective owners and are used herein for identifi cation purposes only.

Pearson Learning Solutions, 501 Boylston Street, Suite 900, Boston, MA 02116

A Pearson Education Company

www.pearsoned.com

Printed in the United States of America

1 2 3 4 5 6 7 8 9 10 XXXX 17 16 15 14 13 12

000200010271661109

MD

ISBN 10: 1-256-74148-5

ISBN 13: 978-1-256-74148-0

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iii

Contents in BriefIntroduction to the CME Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

CME Project Student Handbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

Go Online . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

Chapter 1 An Informal Introduction to Geometry . . . . . . . . . . . . . . . . . 2

Chapter 2 Congruence and Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Chapter 3 Dissections and Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Chapter 4 Similarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

Chapter 5 Circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

Chapter 6 Using Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

Chapter 7 Coordinates and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

Chapter 8 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652

TI-Nspire™ Technology Handbook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740

Tables

Math Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753

Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754

Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756

Properties of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757

Postulates and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762

Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817

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Introduction to the CME Project

CME PROJECT

iv CME Project • Geometry

The CME Project, developed by EDC’s Center for Mathematics Education, is a new NSF-funded high school program, organized around the familiar courses of algebra 1, geometry, algebra 2, and precalculus. The CME Project provides teachers and schools with a third alter native to the choice between traditional texts driven by basic skill development and more pro gressive texts that have unfamiliar organizations. This program gives teachers the option of a problem-based, student-centered program, organized around the mathematical themes with which teachers and parents are familiar. Furthermore, the tremendous success of NSF-funded middle school programs has left a need for a high school program with similar rigor and pedagogy. The CME Project fills this need.

The goal of the CME Project is to help students acquire a deep understanding of mathematics. Therefore, the mathematics here is rigorous. We took great care to create lesson plans that, while challenging, will capture and engage students of all abilities and improve their mathematical achievement.

The Program’s Approach The organization of the CME Project provides students the time and focus they need to develop fundamental mathematical ways of thinking. Its primary goal is to develop in students robust mathematical proficiency.

• The program employs innovative instructional methods, developed over decades of classroom experience and informed by research, that help students master mathematical topics.

• One of the core tenets of the CME Project is to focus on developing students’ Habits of Mind, or ways in which students approach and solve mathematical challenges.

• The program builds on lessons learned from high-performing countries: develop an idea thoroughly and then revisit it only to deepen it; organize ideas in a way that is faithful to how they are organized in mathematics; and reduce clutter and extraneous topics.

• It also employs the best American models that call for grappling with ideas and problems as preparation for instruction, moving from concrete problems to abstractions and general theories, and situating mathematics in engaging contexts.

• The CME Project is a comprehensive curriculum that meets the dual goals of mathematical rigor and accessibility for a broad range of students.

About CMEEDC’s Center for Mathematics Education, led by mathematician and teacher Al Cuoco, brings together an eclectic staff of mathematicians, teachers, cognitive scientists, education researchers, curriculum developers, specialists in educational technology, and teacher educators, internationally known for leadership across the entire range of K–16 mathematics education. We aim to help students and teachers in this country experience the thrill of solving problems and building theories, understand the history of ideas behind the evolution of mathematical disciplines, and appreciate the standards of rigor that are central to mathematical culture.

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CME Project • Geometry v

National Advisory Board The National Advisory Board met early in the project, providing critical feedback on the instructional design and the overall organization. Members include

Richard Askey, University of Wisconsin Edward Barbeau, University of Toronto Hyman Bass, University of MichiganCarol Findell, Boston University Arthur Heinricher, Worcester Polytechnic InstituteRoger Howe, Yale UniversityBarbara Janson, Janson AssociatesKenneth Levasseur, University of Massachusetts, LowellJames Madden, Louisiana State University, Baton RougeJacqueline Miller, Education Development CenterJames Newton, University of MarylandRobert Segall, Greater Hartford Academy of Mathematics and ScienceGlenn Stevens, Boston UniversityHerbert Wilf, University of PennsylvaniaHung-Hsi Wu, University of California, Berkeley

Core Mathematical Consultants Dick Askey, Ed Barbeau, and Roger Howe have been involved in an even more substantial way, reviewing chapters and providing detailed and critical advice on every aspect of the program. Dick and Roger spent many hours reading and criticizing drafts, brainstorming with the writing team, and offering advice on everything from the logical organization to the actual numbers used in problems. We can’t thank them enough.

Teacher Advisory Board The Teacher Advisory Board for the CME Project was essential in help ing us create an effective format for our lessons that embodies the philosophy and goals of the program. Their debates about pedagogi cal issues and how to develop mathematical top ics helped to shape the distinguishing features of the curriculum so that our lessons work effective ly in the classroom. The advisory board includes

Jayne Abbas, Richard Coffey, Charles Garabedian, Dennis Geller, Eileen Herlihy, Doreen Kilday, Gayle Masse, Hugh McLaughlin, Nancy McLaughlin, Allen Olsen, Kimberly Osborne, Brian Shoemaker, and Benjamin Sinwell

Field-Test Teachers Our field-test teachers gave us the benefit of their classroom experi ence by teaching from our draft lessons and giv ing us extensive, critical feedback that shaped the drafts into realistic, teachable lessons. They shared their concerns, questions, challenges, and successes and kept us focused on the real world. Some of them even welcomed us into their classrooms as co-teachers to give us the direct experience with students that we needed to hone our lessons. Working with these expert professionals has been one of the most gratifying parts of the development—they are “highly qualified” in the most profound sense.

California Barney Martinez, Jefferson High School, Daly City; Calvin Baylon and Jaime Lao, Bell Junior High School, San Diego; Colorado Rocky Cundiff, Ignacio High School, Ignacio; Illinois Jeremy Kahan, Tammy Nguyen, and Stephanie Pederson, Ida Crown Jewish Academy, Chicago; Massachusetts Carol Martignette, Chris Martino, and Kent Werst, Arlington High School, Arlington; Larry Davidson, Boston University Academy, Boston; Joe Bishop and Carol Rosen, Lawrence High School, Lawrence; Maureen Mulryan, Lowell High School, Lowell; Felisa Honeyman, Newton South High School, Newton Centre; Jim Barnes and Carol Haney, Revere High School, Revere; New Hampshire Jayne Abbas and Terin Voisine, Cawley Middle School, Hooksett; New Mexico Mary Andrews, Las Cruces High School, Las Cruces; Ohio James Stallworth, Hughes Center, Cincinnati; Texas Arnell Crayton, Bellaire High School, Bellaire; Utah Troy Jones, Waterford School, Sandy; Washington Dale Erz, Kathy Greer, Karena Hanscom, and John Henry, Port Angeles High School, Port Angeles; Wisconsin Annette Roskam, Rice Lake High School, Rice Lake.

Special thanks go to our colleagues at Pearson, most notably Elizabeth Lehnertz, Joe Will, and Stewart Wood. The program benefits from their expertise in every way, from the actual mathematics to the design of the printed page.

Contributors to the CME Project


1Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.0 Habits of Mind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Picturing and Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.02 Drawing 3-D Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.03 Drawing and Describing Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.04 Drawing From a Recipe— Reading and Writing Directions for Drawings. . . . . . . . . . . . . . . . . 19 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Constructing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.05 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.06 Compasses, Angles, and Circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Geometry Software. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.07 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.08 Drawings vs. Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.09 Drawing UnMessUpable Figures—Building Constructions . . . . . . . 42 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Invariants—Properties and Values That Don’t Change . . . . . 48 1.10 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.11 Numerical Invariants in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.12 Spatial Invariants—Shape, Concurrence, and Collinearity . . . . . . . 58 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Project: Using Mathematical Habits Folding Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

1A

1B

1C

vi Geometry

1D

An Informal Introduction to Geometry


Congruence and Proof

Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

The Congruence Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.02 Length, Measure, and Congruence. . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.03 Corresponding Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.04 Triangle Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Proof and Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.05 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.06 Deduction and Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.07 Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.08 The Parallel Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Writing Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 2.09 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.10 What Does a Proof Look Like? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 2.11 Analyzing the Statement to Prove . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.12 Analysis of a Proof— Understanding or Writing: Which Comes First? . . . . . . . . . . . . . . 125 2.13 The Reverse List— Working Back From What You Want to Prove. . . . . . . . . . . . . . . . 131 2.14 Practicing Your Proof-Writing Skills . . . . . . . . . . . . . . . . . . . . . . . . 135 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Quadrilaterals and Their Properties . . . . . . . . . . . . . . . . . . . . . . 142 2.15 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 2.16 General Quadrilaterals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 2.17 Properties of Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 2.18 Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 2.19 Classifying Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Project: Using Mathematical Habits Dividing Into Congruent Pieces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Cumulative Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

2A

2B

2D

2C

Contents vii

2


Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Cut and Rearrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 3.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 3.02 Do the Cuts Really Work? Using Properties of the Shapes Being Cut . . . . . . . . . . . . . . . . . . . . 174 3.03 Cutting Algorithms— Ways to Rearrange Into Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . 181 3.04 Checking an Algorithm— Justifying the Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 3.05 The Midline Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Area Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 3.06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 3.07 What Is Area, Anyway? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 3.08 Area by Dissection— Parallelograms, Triangles, and Trapezoids . . . . . . . . . . . . . . . . . . . . 205 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Proof by Dissection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 3.09 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 3.10 The Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 3.11 Pick a Proof— Other Proofs of the Pythagorean Theorem. . . . . . . . . . . . . . . . . . . 224 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Measuring Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 3.12 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 3.13 Surface Area: Prisms and Pyramids. . . . . . . . . . . . . . . . . . . . . . . . . 235 3.14 Surface Areas: Cylinders and Cones . . . . . . . . . . . . . . . . . . . . . . . . 240 3.15 Volumes of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Project: Using Mathematical Habits Surface Area and Volume of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 254

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

3

viii Geometry

Dissections and Area

3D

3C

3B

3A


Chapter Opener. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

Scaled Copies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 4.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4.02 Scale Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 4.03 What Is a Well-Scaled Drawing? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 4.04 Testing for Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 4.05 Checking for Scaled Copies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Curved or Straight? Just Dilate! . . . . . . . . . . . . . . . . . . . . . . 288 4.06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 4.07 Making Scaled Copies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 4.08 Ratio and Parallel Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

The Side-Splitter Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . 304 4.09 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 4.10 Nested Triangles—One Triangle Inside Another . . . . . . . . . . . . . . . 307 4.11 Proving the Side-Splitter Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 313 4.12 The Side-Splitter Theorems (continued). . . . . . . . . . . . . . . . . . . . . . 317 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Defi ning Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 4.13 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 4.14 Similar Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 4.15 Tests for Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 4.16 Areas of Similar Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

Project: Using Mathematical Habits Midpoint Quadrilaterals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .344

Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

Cumulative Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

4 Similarity

4A

4B

4C

4D

Contents ix


Chapter Opener. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

Area and Circumference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 5.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 5.02 Area and Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 5.03 Connecting Area, Circumference . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Circles and 3.141592653589793238462643383 . . . . . . . . . . . 374 5.04 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 5.05 An Area Formula for Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 5.06 Circumference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 5.07 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

Classical Results About Circles . . . . . . . . . . . . . . . . . . . . . . . 396 5.08 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 5.09 Arcs and Central Angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .400 5.10 Chords and Inscribed Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .408 5.11 Circumscribed and Inscribed Circles . . . . . . . . . . . . . . . . . . . . . . . . . 414 5.12 Secants and Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 5.13 Power of a Point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

Geometric Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 5.14 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 5.15 Probability as a Ratio of Areas—Monte Carlo Method. . . . . . . . . . 438 5.16 Sets of Measure 0—Probability When an Area Is 0. . . . . . . . . . . . .442 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

Project: Using Mathematical Habits Another Interesting Curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .448

Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

5

x Geometry

Circles

5A

5B

5C

5D


Chapter Opener. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

Some Uses of Similarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 6.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

6.02 An Inequality of Means—!ab # a 1 b2 . . . . . . . . . . . . . . . . . . . . . 460

6.03 Similarity in Ancient Greece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 6.04 Concurrence of Medians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

Exploring Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 6.05 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 6.06 Some Special Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 6.07 Some Special Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .488 6.08 Finding Triangle Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 6.09 Extend the Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 502 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

Volume Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 6.10 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 6.11 Cavalieri’s Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 6.12 Proving Volume Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 6.13 Volume of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .540

Project: Using Mathematical Habits Demonstrating a Volume Relationship. . . . . . . . . . . . . . . . . . . . . . . . . . . 541

Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544


Using Similarity6

6A

6B

6C

Contents xi


Chapter Opener. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 7.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 7.02 Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 7.03 Translations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 7.04 Rotations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 7.05 Congruence and Isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

Geometry in the Coordinate Plane . . . . . . . . . . . . . . . . . . . . 590 7.06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 7.07 Midpoint and Distance Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 7.08 Parallel Lines and Collinear Points . . . . . . . . . . . . . . . . . . . . . . . . . 601 7.09 Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 7.10 Coordinates in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

Connections to Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 7.11 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 7.12 Introduction to Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 7.13 The Vector Equation of a Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 7.14 Using the Vector Equation of a Line. . . . . . . . . . . . . . . . . . . . . . . . . 641 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

Project: Using Mathematical Habits Equations of Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .646

Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

7 Coordinates and Vectors

7A

7B

7C

xii Geometry


Chapter Opener. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652

Making the Least of a Situation . . . . . . . . . . . . . . . . . . . . . . 654 8.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 8.02 Finding the Shortest Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 8.03 Refl ecting to Find Shortest Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

Making the Most of a Situation . . . . . . . . . . . . . . . . . . . . . . 670 8.04 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 8.05 Maximizing Areas, Part 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 8.06 Maximizing Areas, Part 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .684

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

Contour Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686 8.07 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 8.08 Drawing Contour Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 8.09 Contour Lines and Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 8.10 Revisiting the Burning Tent— “Finding the Shortest Path” Using Contour Lines. . . . . . . . . . . . . . 699 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705

Advanced Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 8.11 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 8.12 Reasoning by Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709 8.13 Proving Rich’s Function Is Constant— Sum of Distances to Sides of a Triangle . . . . . . . . . . . . . . . . . . . . . . 713 8.14 The Isoperimetric Problem— The Greatest Area for a Constant Perimeter . . . . . . . . . . . . . . . . . . 718 8.15 The Question of Existence— Is There a Solution for the Isoperimetric Problem? . . . . . . . . . . . . . 723 8.16 Solving the Isoperimetric Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 727 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730

Project: Using Mathematical Habits The Airport Problem— Where Best to Locate an Airport Serving Three Cities . . . . . . . . . . . . . . 731

Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736


8

8A

8C

8D

Contents xiii

Optimization

8B


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