ebook img

Geometry: A High School Course PDF

405 Pages·1988·10.33 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Geometry: A High School Course

Books by Serge Lang of Interest for High Schools Geometry: A High School Course (with Gene Murrow) This high school text, inspired by the work and educational interests of a prom inent research mathematician, and Gene Murrow's experience as a high school teacher, presents geometry in an exemplary and, to the student, accessible and attractive form. The book emphasizes both the intellectually stimulating parts of geometry and routine arguments or computations in physical and classical cases. MATH! Encounters with High School Students This book is a faithful record of dialogues between Lang and high school stu dents, covering some of the topics in the Geometry book, and others at the same mathematical level. These encounters have been transcribed from tapes, and are thus true, authentic, and alive. Basic Mathematics This book provides the student with the basic mathematical background neces sary for college students. It can be used as a high school text, or for a college precalculus course. The Beauty of Doing Mathematics Here, we have dialogues between Lang and audiences at a science museum in Paris. The audience consisted of many types of persons, including some high school students. The topics covered are treated at a level understandable by a lay public, but were selected to put people in contact with some more advanced research mathematics which could be expressed in broadly understandable terms. First Course in Calculus This is a standard text in calculus. There are many worked out examples and problems. Introduction to Linear Algebra Although this text is used often after a first course in calculus, it could also be used at an earlier level, to give an introduction to vectors and matrices, and their basic properties. Serge Lang Gene Murrow Geotnetry A High School Course Second Edition With 577 Illustrations Springer Science+Business Media, LLC Serge Lang Gene Murrow Department of Mathematics Riverview Farm Road Yale University Ossining, NY 10562 New Haven, CT 06520 U.S.A. U.S.A. Mathematics Subject Classifications (1980): 50-01, 51M05, 00-01 Library of Congress Cataloging-in-Publication Data Lang, Serg Geometry: a high school course. Includes index. 1. Geometry. 1. Murrow, Gene. Il. Title. QA445.L36 1988 516.2 87-32376 © 1983, 1988 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Inc. in 1988 Softcover reprint of the hardcover 2nd edition 1988 Ali rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC , except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adap tation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Typeset by Composition House Ltd., Salisbury, England. 9 8 7 6 5 4 3 2 1 ISBN 978-1-4419-3084-2 ISBN 978-1-4757-2022-8 (eBook) DOI 10.1007/978-1-4757-2022-8 Contents Introduction . . . ix CHAPTER 1 Distance and Angles . . . . . . . . . . . . . . . . . . . . . . . . . . 1 §1. Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 §2. Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 §3. Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 §4. Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 §5. Right Angles and Perpendicularity . . . . . . . . . . . . . . . . . . . . . . . 36 §6. The Angles of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 §7. An Application: Angles of Reflection. . . . . . . . . . . . . . . . . . . . . . 62 CHAPTER 2 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 §1. Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 §2. Distance Between Points on a Line. . . . . . . . . . . . . . . . . . . . . . . 71 §3. Equation of a Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 CHAPTER 3 Area and the Pythagoras Theorem . . . . . . . . . . . . . . . . . . . . . . . . 81 §1. The Area of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 §2. The Pythagoras Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 CHAPTER 4 The Distance Formula . ... 110 §1. Distance Between Arbitrary Points. . . . . . . . . . . . . . . . . . . . . . . 110 §2. Higher Dimensional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 §3. Equation of a Circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 vi CONTENTS CHAPTER 5 Some Applications of Right Triangles . . 123 §1. Perpendicular Bisector. ........ . 124 §2. Isosceles and Equilateral Triangles . . . . . 136 §3. Theorems About Circles . . . . . . . . . . . . . . . . . 148 CHAPTER 6 Polygons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 §1. Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 §2. Convexity and Angles. . . . . . . . . . . . . . . . . . . . . . . . . . . 165 §3. Regular Polygons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 CHAPTER 7 Congruent Triangles. . . . . . . . . . . . . . . . . . . . . . . 177 §1. Euclid's Tests for Congruence . . . . . . . . . . . . . . . 177 §2. Some Applications of Congruent Triangles . . . . . . . . . 192 §3. Special Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 CHAPTER 8 Dilations and Similarities . . 210 §1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 §2. Change of Area Under Dilation. . . . . . . . . . . . 218 §3. Change of Length Under Dilation . . . . . . . . . . . . . . . . . . . . . . . 232 §4. The Circumference of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . 235 §5. Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 CHAPTER 9 Volumes .. ......... . 261 §1. Boxes and Cylinders. . . . . . . . . . . . . . . . . . . . . . . . 261 §2. Change of Volume Under Dilations . . . . . . . . . . . . . . 270 §3. Cones and Pyramids. . . . . . . . . . . . . . . . . . . . . . . . . . 274 §4. The Volume of the Ball. . . . . . . . . . . . . . . . . . . . . . . . 281 §5. The Area of the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 CHAPTER 10 Vectors and Dot Product . . . . . . . . 295 §1. Vector Addition . . . . . . . . . . . 296 §2. The Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 §3. Perpendicularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 §4. Projections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 §5. Ordinary Equation for a Line . . . . . . . . . . . . . . . . . . . . . . . . 312 §6. The 3-Dimensional Case . . . . . . . 316 §7. Equation for a Plane in 3-Space . . . . . . . . . . . . . . . . . . . . . . 319 CHAPTER 11 Transformations 321 §1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 §2. Symmetry and Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 §3. Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 CONTENTS vii §4. Reflection Through a Line . 332 §5. Reflection Through a Point 337 §6. Rotations. . . . . . . . . . . . 340 §7. Translations ......... . 346 §8. Translations and Coordinates 348 CHAPTER 12 lsometries . . 356 §1. Definitions and Basic Properties. 356 §2. Relations with Coordinates. 363 §3. Composition of Isometries . . . . 367 §4. Definition of Congruence . . . . . 377 §5. Proofs of Euclid's Tests for Congruent Triangles 381 §6. Isometries as Compositions of Reflections. . . . . . .. 384 Index ......................... . 391 Introduction The present book is intended as a text for the geometry course in sec ondary schools. Several features distinguish it from currently available texts. Choice of topics. We do not think that the purpose of the basic geo metry course should be to do geometry a certain way, i.e. should one fol low Euclid, should one not follow Euclid, should one do geometry the transformational way, should one do geometry without coordinates, etc.? We have tried to present the topics of geometry whichever way seems most appropriate. The most famous organization of geometrical material was that of Euclid, who was most active around 300 B.c. Euclid assembled and en hanced the work of many mathematicians before him, like Apollonius, Hippocrates, Eudoxus. His resulting textbook, The Elements, was used virtually unchanged for 2,000 years, making it the most famous school book in history. Many new ideas have been added to the body of knowledge about geometry since Euclid's time. There is no reason a priori to avoid these ideas, as there is no reason to push them excessively if inappropriate. For certain topics (e.g. in Chapters 1, 5, 6, 7), Euclid's way is efficient and clear. The material in Chapters 3 and 4 on Pythagoras' theorem also follows Euclid to a large extent, but here we believe that there is an opportunity to expose the student early to coordinates, which are especially important when considering distances, or making measure ments as applications of the Pythagoras theorem, relating to real life situations. The use of coordinates in such a context does not affect the logical structure of Euclid's proofs for simple theorems involving basic geometric figures like triangles, rectangles, regular polygons, etc. X INTRODUCTION An additional benefit of including some sections on coordinates is that algebraic skills are maintained in a natural way throughout the year principally devoted to geometry. Coordinates also allow for practical computations not possible otherwise. We feel that students who are subjected to a secondary school pro gram during which each year is too highly compartmentalized (e.g. a year of geometry from which all algebra has disappeared) are seriously disadvantaged in their later use of mathematics. Experienced teachers will notice at once the omissions of items tradi tionally included in the high school geometry course, which we regard as having iittle signific,ance. Some may say that such items are fun and interesting. Possibly. But there are topics which are equally, or even more, fun and interesting, and which in addition are of fundamental importance. Among these are the discussion of changes in area and volume under dilation, the proofs of the standard volume formulas, vectors, the dot product and its connec tion with perpendicularity, transformations. The dot product, which is rarely if ever mentioned at the high school level, deserves being included at the earliest possible stage. It provides a beautiful and basic relation between geometry and algebra, in that it can be used to interpret per pendicularity extremely efficiently in terms of coordinates. See, for in stance, how Theorem 10-2 establishes the connection between the Euclidean type of symmetry and the corresponding property of the dot product for perpendicularity. The proofs of the standard volume formulas by means of dilations and other transformations (including shearing) serve, among others, the purpose of developing the student's spatial geometric intuition in a par ticularly significant way. One benefit is a natural extension to higher di mensional space. The standard transformations like rotations, reflections, and transla tions seem fundamental enough and pertinent enough to be mentioned. These different points of view are not antagonistic to each other. On the contrary, we believe that the present text achieves a coherence which never seems forced, and which we hope will seem most natural to stu dents who come to study geometry for the first time. The inclusion of these topics relates the course to the mathematics that precedes and follows. We have tried to bring out clearly all the im portant points which are used in subsequent mathematics, and which are usually drowned in a mass of uninteresting trivia. It is an almost univer sal tendency for elementary texts and elementary courses to torture top ics to death. Generally speaking, we hope to induce teachers to leave well enough alone. Proofs. We believe that most young people have a natural sense of reasoning. One of the objectives of this course, like the "standard"

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.