DOCUMENT RESUME ED 135 622 SE 021 992 AUTHOE Allen, Frank E.; And Others TITL1 Geometry, Student's Text, Part II, Unit 14. INSTITUTION Stanford Univ., Calif. School Mathematics Study Group. SPONS AGENCY National Science Foundation, Washington, D.C. PUB DATE 61 NOTE 395p.; For related documents, see SE 021 987-022 002 and EL 130 870-877 EDRS PRICE MF-$0.83 BC-$20.75 Plus Postage. DESCRIPTORS *Curriculum; Elementary Secondary Education; *Geometry; Instruction; *Instructional Materials; Mathematics Education; *Secondary School Mathematics; *Textbooks IDENTIFIEES *School Mathematics Study Group ABSTRACT Unit 14 in the SMSG secondary school mathematics series is a student text covering the following topics in geometry: areas of polygonal regions, similarity, circles and spheres, characterization of sets, constructions, areas of circles and sectors, volumes of solids, and plane coordinate geometry. Appendices cover Eratosthenes° measurement of the earth, rigid motions, proof of the two-circle theorem, trigonometry, and regular polyhedra. (DT) *********************************************************************** Documents acquired by ERIC include many informal unpublished * materials not available from other sources. ERIC makes every effort * * to obtain the best copy available. Nevertheless, items of margirtal * * reproducibility are often encountered and this affects the quality .* * of the microfiche and hardcopy reproductions ERIC makes available * * via the ERIC Document,Reproduction Service (EDRS). EDRS is not * responsible for the quality of the original document. Reproductions * * supplied by EDRS are the best that can be made from the original. *************4********************************************************* STUDENT'S TEXT 1111111111111111111111110116 U S DEPARTMENT OF HEALTH, EDUCATION WELFARE NATIONAL INSTITUTE OF EDUCATION THIS DOCUMENT HAS BEEN REPRO- DUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIGIN- ATING IT POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILY REPRE. SENT OFFICIAL NATIONAL INSTITUTE OF EDUCATION POSITION OR POLICY GEOMETRY PART II SMSG SCHOOL MATHEMATICS STUDY GROUP C) YALE UNIVERSITY PRESS School Mathematics Study Group Geometry Unit 14 3 Geometry Student's Text, Part II Prepared under the supervision of the Panel on Sample Textbooks of the School Mathematics Study Group: Frank B. Allen Lyons Township High School Edwin C. Douglas Taft School Donald E. Richmond Williams College Charles E. Rickart Yale University Henry Swain ,New Trier Township High School Robert J. Walker Cornell University New Haven and London, Yale University Press Copyright © 1960, 1961 by Yale University. Printed in the United States of America. All rights reserved. This book may not be reproduced, in whole or in part, in any form, without written permission from the publishers. Financial support for the School Mathematics Study Group has been provided by the National Science Foundation. Below are listed the names of all those who participated in any of the writing sessions at which the following SMSG texts were prepared: First Course in Algebra, Geometry, Intermediate Mathematics, Elementary Functions, and Introduction to Matrix Algebra. H.W. Alexander, Earlham College B.C. Jurgensen, Culver Military Academy, F.B. Allen, Lyons Township High School, La Culver, Indiana Grange, Illinois Joseph Lehner, Michigan State University Alexander Beck, Olney High School, Phila- Marguerite Lehr, Bryn Mawr College delphia, Pennsylvania Kenneth Leisenring, University of Michigan E.F. Beckenbach, University of California Howard Levi, Columbia University at Los Angeles Eunice Lewis, Laboratory High School, E.G. Begle, School Mathematics Study Group, University of Oklahoma Yale University M.A. Linton, William Penn Charter School, Paul Berg, Stanford University Philadelphia, Pennsylvania Emil Berger, Monroe High School, St. Paul, A.E. Livingston, University of Washington Minnesota L.H. Loomis, Harvard University Arthur Bernhart, University of Oklahoma R.V. Lynch, Phillips Exeter Academy, R.H. Bing, University of Wisconsin Exeter, New Hampshire A.L. Blakers, University of Western W.K. McNabb, Hockaday School, Dallas, Australia Texas A.A. Blank, New York University K.G. Michaels, North Haven High School, Shirley Boselly, Franklin High School, North Haven, Connecticut Seattle, Washington E.E. Moise, University of Michigan K.E. Brown, Department of Health, Educa- E.P. Northrop, University of Chicago tion, and Welfare, Washington, D.C. O.J. Peterson, Kansas State Teachers J.M. Calloway, Carleton College College, Emporia, Kansas Hope Chipman, University High School, Ann B.J. Pettis, University of North Carolina Arbor, Michigan B.S. Pieters, Phillips Academy, Andover, R.R. Christian, University of British Massachusetts Columbia H.O. Pollak, Bell Telephone Laboratories R.J. Clark, St. Paulls School, Concord, Walter Prenowitz, Brooklyn College New Hampshire G.B. Price, University of Kansas P.H. Daus, University of California at Los A.L. Putnam, University of Chicago Angeles Persis O. Redgrave, Norwich Free Academy, R.B. Davis, Syracuse University Norwich, Connecticut Charles DePrima, California Institute of Mina Rees, Hunter College Tec-nology D.E. Richmond, Williams College Mary Dolciani, Hunter College C.E. Rickert, Yale University Edwin C. Douglas, The Taft School, Water- Harry Ruderman, Hunter College High School, town, Connecticut New York City Floyd Downs, East High School, Denver, J.T. Schwartz, New York University Colorado O.E. Stanaitis, St. Olaf College E.A. Dudley, North Haven High School, North Robert Starkey, Cuuberley High Schools, Haven, Connecticut Palo Alto, California Lincoln Durst, The Rice Institute Phillip Stucky, Roosevelt High School, Florence Elder, West Hempstead High School, Seattle, Washington West Hempstead, New York Henry Swain, New Trier Township High W.E. Ferguson, Newton High School, Newton- School, Winnetka, Illinois ville, Massachusetts . Henry Syer, Kent School, Kent, Connecticut N.J. Fine, University of Pennsylvania G.B. Thomas, Massachusetts Institute of Joyce D. Fontaine, North Haven High School, Technology North Haven, Connecticut A.W. Tucker, Princeton University F.L. Friedman, Massachusetts Institute of H.E. Vaughan, University of Illinois Technology John Wagner, University of Texas Esther Gassett, Claremore High School, R.J. Walker, Cornell University Claremore, Oklahoma A.D. Wallace, Tulane University R.K. Getoor, University of Washington E.L. Walters, William Penn Senior High V.H. Haag, Franklin and Marshall College School, York, P-ennsylvania R.R. Hartman, Edina-Morningside Senior High Warren White, North High School, Sheboygan, School, Edina, Minnesota Wisconsin M.H. Heins, University of Illinois D.V. Widder, Harvard University Edwin Hewitt, University of Washington William Wooton, Pierce Junior College, Martha Hildebrandt, Proviso Township High Woodland Hills, California School, Maywood, Illinois J.H. Zant, Oklahoma State University CONTENTS Chapter 11. AREAS OF POLYGONAL REGIONS 317 11- 1. Polygonal Regions 317 11- 2. Areas of Triangles and Quadrilaterals 328 11- 3. The Pythagorean Theorem 339 Review Problems 353 12. SIMILARITY 359 12-1. The Idea of a Similarity 359 12-2. Similarities between Triangles 364 12-3. The Basic Similarity Theorems 367 12-4. Similarities in Right Triangles 391 12-5. Areas.of Similar Triangles 395 Review Problems 401 Review Exercises, Chapters 7 to 12 404 13. CIRCLES AND SPHERES 409 13- 1. Basic Definitions 409 13- 2. Tangent Lines. The Fundamental Theorem for Circles 412 13- 3. Tangent Planes. The Fundamental Theorem for Spheres 423 13- 4. Arcs of Circles 429 13- 5.Lengths of Tangent and Secant Segments 448 Review Problems 457 14. CHARACTERIZATION OF SETS. CONSTRUCTIONS . 461 14- 1. Characterization of Sets 461 14- 2. Basic Characterizations. Concurrence Theorems 464 14- 3. Intersection of Sets 473 14- 4. Constructions with Straight-edge and Compass 475 14- 5. Elementary Constructions 477 14- 6. Inscribed and Circumscribed Circles . 490 14- 7. The Impossible Construction Problems of Antiquity 493 Review Problems 503 15. AREAS OFCIRCLESAND SECTORS 505 15-1. Polygons 15-2. RegularPolygons 510 15-3. The Circumference of a Circle. The Number r 516 15-4. Area ofa Circle 520 15-5. Lengthsof Arcs. Areas of Sectors . 525 Review Problems 530 Chapter 16. VOLUMES OF SOLIDS 533 533 16- 1. Prisms 540 16- 2. Pyramids 16- 3. Volumes of Prisms and Pyramids, Cavalierils Principle 546 16- 4. Cylinders and Cones 553 16- 5. Spheres; Volume and Area 559 Review Problems 564 PLANE COORDINATE GEOMETRY 567 17. 567 17- 1. Introduction 17- 2. Coordinate Systems in a Plane 567 . 572 17- 3. How to Plot Points on GraphPaper . 576 17- 4. The Slope of a Non-Vertical Line . 17- 5. Parallel and Perpendicular Lines . 583 17- 6.The Distance Formula 588 17- 7. The Mid=Point Formula 592 17- 8. Proofs of Geometric Theorems 595 17- 9. The Graph of a Condition 600 604 17-10. How to Describe a Line by anEquation 17-11. Various Forms of the Equation of aLine. 611 17-12. The General Form of the Equation of a Line 613 17-13. Intersection of Lines 617 17-14. Circles 621 Review Problems 628 Review Exercises, Chapters 13 to 17 630 . A-29 Appendix VII. How Eratosthenes Measured theEarth A-31 Appendix VIII. Rigid Motion _ 1. The General Idea of aRigid Motion. A-31 2. Rigid Motion of Segments A-35 3. Rigid Motion of Rays, Angles and Triangles A-37 4. Rigid Motion of Circles and Arcs A-40 5. Reflections A-42 A-51 Appendix IX. Proof of the Two-Circle Theorem . A-57 Appendix X. Trigonometry 1. Trigonometric