Geometry: A High School Course

405 Pages·1988·10.33 MB·English

by  Serge Lang

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"