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.

Taken from:CME Project: Geometry, Algebra 2, Algebra 1, PrecalculusBy the CME Project Development TeamCopyright © 2009 by Education Development Center, Inc.Published by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

CME Common Core Additional Lessons: Geometry, Precalculus, Algebra 2, Algebra 1By the CME Project Development TeamCopyright © 2012 by Education Development Center, Inc.Published by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

CME Project Development Team

Lead Developers: Al Cuoco and Bowen Kerins

Core Development Team: Anna Baccaglini-Frank, Jean Benson, Nancy Antonellis D’Amato, Daniel Erman, Paul Goldenberg, Brian Harvey, Wayne Harvey, Doreen Kilday, Ryota Matsuura, Stephen Maurer, Nina Shteingold, Sarah Sword, Audrey Ting, Kevin Waterman.

All trademarks, service marks, registered trademarks, and registered service marks are the property of their respective owners and are used herein for identification purposes only.

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Copyright © 201 by 3 Pearson Learning SolutionsAll rights reserved.

Permission in writing must be obtained from the publisher before any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system.

000200010271656491_CH00_FM_pi-xxv.pdf Page 2

iii

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

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

CME Project Student Handbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Chapter 1 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Chapter 2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Chapter 3 Quadratics and Complex Numbers. . . . . . . . . . . . . . . . . . . . . . 150

Chapter 4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

Chapter 5 Applications of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

Chapter 6 Congruence and Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

Chapter 7 Similarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

Chapter 8 Circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

Chapter 9 Using Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704

Chapter 10 Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800

Honors Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830

TI-Nspire™ Technology Handbook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990

Tables

Math Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003

Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004

Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006

Properties of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007

Postulates and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014

Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094


Introduction to the CME Project

CME PROJECT

iv Mathematics II

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.


Table of Contents v Mathematics II 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

Radicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.02 Defi ning Square Roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.03 Arithmetic with Square Roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.04 Conventions for Roots—Simplifi ed Forms . . . . . . . . . . . . . . . . . . . . . 16 1.05 Rational and Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.06 Roots, Radicals, and the nth Root . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Working with Exponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.07 Getting Started. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.08 Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.09 Zero and Negative Exponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.10 Sequences and Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.11 Defi ning Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

1A

1B

vi Mathematics II

Real Numbers


Polynomials

Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

The Need for Identities—Equivalent Expressions . . . . . . . . . . 64v 2.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.02 Form and Function—Showing Expressions are Equivalent . . . . . . . 68 2.03 The Zero-Product Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.04 Transform ing Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Polynomials and Their Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.05 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.06 Anatomy of a Polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.07 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.08 Arithmetic with Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Factoring to Solve: Quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2.09 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2.10 Factoring a Difference of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . 117 2.11 Factoring Sums and Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 2.12 Factoring by Completing the Square. . . . . . . . . . . . . . . . . . . . . . . . 133 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Project: Using Mathematical Habits Differences of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

2A

2B

2C

Contents vii

2


Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

The Quadratic Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.02 Making it Formal—Deriving the Quadratic Formula . . . . . . . . . . . 155 3.03 Building the Quadratic Formula from its Roots . . . . . . . . . . . . . . . 162 3.04 Factoring Nonmonic Quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Quadratic Graphs and Applications. . . . . . . . . . . . . . . . . . . . . . . 174 3.05 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 3.06 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3.07 Graphing Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 3.08 Jiffy Quadratics: Parabolas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Introduction to the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . 202 3.09 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 3.10 Extending the Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 3.11 Making the Extension: The Square Root of –1 . . . . . . . . . . . . . . . . 210 3.12 Extension to Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 3.13 Reciprocals and Division. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

3 Quadratics and Complex Numbers

3C

3B

3A

viii Mathematics II


Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 4.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 4.02 Two Types of Defi nitions—Closed Form and Recursive . . . . . . . . . 238 4.03 Constant Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 4.04 Tables and Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 4.05 Difference Tables for Polynomial Functions . . . . . . . . . . . . . . . . . . 259 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

About Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 4.06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 4.07 Getting Precise About Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 4.08 Algebra with Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 4.09 Inverses: Doing and Undoing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 4.10 Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 4.11 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 4.12 Graphs of Exponential Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 316 4.13 Tables of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

Transforming Basic Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 4.14 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 4.15 More Basic Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 4.16 Translating Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 4.17 Scaling and Refl ecting Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

4 Functions

4A

4B

4C

4D

Contents ix


Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 5.0 Binomials and Pascal’s Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

Probability and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 378 5.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 5.02 Probability and Pascal’s Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 5.03 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .388 5.04 Probabilities of Compound Events . . . . . . . . . . . . . . . . . . . . . . . . . 395 5.05 Polynomial Powers and Counting . . . . . . . . . . . . . . . . . . . . . . . . . . 401 5.06 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 5.07 Lotteries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

5

x Mathematics II

Applications of Probability

5A


Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 6.0 Triangle Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

Invariants—Properties and Values that Don’t Change . . . . 434 6.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 6.02 Numerical Invariants in Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . 437 6.03 Spatial Invariants—Shape, Concurrence, and Collinearity. . . . . . .444 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

Proof and Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 6.04 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 6.05 Deduction and Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 6.06 Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 6.07 The Parallel Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

Writing Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 6.08 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 6.09 What Does a Proof Look Like?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 6.10 Analyzing the Statement to Prove . . . . . . . . . . . . . . . . . . . . . . . . . 487 6.11 Analysis of a Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 6.12 The Reverse List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 6.13 Practicing Your Proof-Writing Skills. . . . . . . . . . . . . . . . . . . . . . . . . 499 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

Quadrilaterals and Their Properties . . . . . . . . . . . . . . . . . . . . . . 506 6.14 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 6.15 General Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 6.16 Properties of Quadrilaterals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 6.17 Parallelograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 6.18 Classifying Quadrilaterals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

Congruence and Proof6

6A

6B

6C

Contents xi

6D


Chapter Opener. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

Scaled Copies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 7.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 7.02 Scale Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 7.03 What is a Well-Scaled Drawing? . . . . . . . . . . . . . . . . . . . . . . . . . . .544 7.04 Testing for Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .548 7.05 Checking for Scaled Copies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

Curved or Straight? Just Dilate! . . . . . . . . . . . . . . . . . . . . . . 558 7.06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 7.07 Making Scaled Copies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 7.08 Ratio and Parallel Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572

The Side-Splitter Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . 574 7.09 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 7.10 Nested Triangles—One Triangle Inside Another . . . . . . . . . . . . . . 577 7.11 Proving the Side-Splitter Theorems. . . . . . . . . . . . . . . . . . . . . . . . . 583 7.12 The Side-Splitter Theorems (continued) . . . . . . . . . . . . . . . . . . . . . 587 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

Defi ning Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 7.13 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 7.14 Similar Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 7.15 Tests for Similar Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .600 7.16 Areas of Similar Triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

Project: Using Mathematical Habits Midpoint Quadrilaterals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614

Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618

7 Similarity

7A

7B

7C

7D

xii Mathematics II


Chapter Opener. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

Area and Circumference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 8.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 8.02 Area and Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 8.03 Connecting Area, Circumference. . . . . . . . . . . . . . . . . . . . . . . . . . . 635 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641

Circles and 3.14159265358 . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 8.04 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 8.05 An Area Formula for Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 8.06 Circumference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 8.07 Arc Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

Classical Results About Circles . . . . . . . . . . . . . . . . . . . . . . . 662 8.08 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 8.09 Arcs and Central Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 8.10 Chords and Inscribed Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 8.11 Circumscribed and Inscribed Circles. . . . . . . . . . . . . . . . . . . . . . . . .680 8.12 Secants and Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686 8.13 The Power of a Point, Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699

Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

8

8A

8C

Contents xiii

Circles

8B


Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704

Some Uses of Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 9.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 9.02 An Inequality of Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 9.03 Similarity in Ancient Greece. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 9.04 Concurrence of Medians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 9.05 Midpoint and Distance Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . 728 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735

Exploring Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 736 9.06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737 9.07 Some Special Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 9.08 Some Special Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 9.09 Finding Triangle Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760

Volume Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762 9.10 Volumes of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 9.11 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769 9.12 Cavalieri’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 9.13 Proving Volume Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780 9.14 Volume of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798

9

xiv Mathematics II

Using Similarity

9A

9B

9C


Chapter Opener. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800

10.0 Equations of Circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802

Coordinate Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804 10.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 10.02 Equations as Point-Testers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .808 10.03 Coordinates and Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 10.04 The Power of a Point, Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827

Chapter Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829

10

10A

Contents xv

Analytic Geometry


Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830

The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832 H.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 H.02 Graphing Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 H.03 Arithmetic in the Complex Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . 842 H.04 Magnitude and Direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861

Complex Numbers, Geometry, and Algebra . . . . . . . . . . . . 862 H.05 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 H.06 Multiplying Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865 H.07 Conjugates and Roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879

Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880 H.08 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881 H.09 Solving Systems Systematically . . . . . . . . . . . . . . . . . . . . . . . . . . . .884 H.10 Solving Again, in Matrix Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889 H.11 Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905

Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906 H.12 Right Angle Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907 H.13 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913 H.14 Extending the Domain, Part 1—0° to 360°. . . . . . . . . . . . . . . . . . . 916 H.15 Extending the Domain, Part 2—All Real Numbers. . . . . . . . . . . . . 922 H.16 The Pythagorean Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928 H.17 Solving Trigonometric Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . 933 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937

Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . 938 H.18 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939 H.19 Graphing Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941 H.20 Graphing the Tangent Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 947 H.21 The Angle-Sum Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 959

Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960 H.22 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961 H.23 Slicing Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .964 H.24 Conics at the Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971 H.25 Conics Anywhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .980 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989

H

xvi Mathematics II

Honors Appendix

HonorsAppendix

A

HonorsAppendix

B

HonorsAppendix

C

HonorsAppendix

D

HonorsAppendix

E

HonorsAppendix

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