Geometry with Trigonometry

282 Pages·2015·4.66 MB·English

by  Patrick D Barry

Geometry with Trigonometry
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Geometry with Trigonometry
2nd Edition
Patrick D. Barry
Professor Emeritus of Mathematics,
School of Mathematical Sciences,
University College, Cork.
“All things stand by proportion.”
George Puttenham (1529–1590)
“Mathematics possesses not only truth, but supreme beauty – a
beauty cold and austere like that of sculpture, and capable of stern
perfection, such as only great art can show.”
Bertrand Russell in The Principles of Mathematics (1872–1970)
Woodhead Publishing is an imprint of Elsevier
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First Published by Horwood Publishing Limited, 2001
Reprinted by Woodhead Publishing Limited, 2014 © P. D. Barry, 2001, 2014.
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Notices
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ISBN: 978-0-12-805066-8
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To my late wife Fran, my children Conor, Una and Brian, and my several
grandchildren
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Contents
About the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xvii
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
1 Preliminaries 1
1.1 Historicalnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Noteondeductivereasoning. . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Euclid’sTheElements . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.3 Postulatesandcommonnotions . . . . . . . . . . . . . . . . . . . . 6
1.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.5 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.6 Quantitiesormagnitudes . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Ourapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.1 Typeofcourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.2 Needforpreparation . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Revisionofgeometricalconcepts . . . . . . . . . . . . . . . . . . . 10
1.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.2 Thebasicshapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.3 Distance;degree-measure ofanangle . . . . . . . . . . . . . . . . . 15
1.5.4 Ourtreatmentofcongruence . . . . . . . . . . . . . . . . . . . . . 17
1.5.5 Parallellines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Pre-requisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6.1 Setnotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6.2 Classicalalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6.3 Otheralgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6.4 Distinctivepropertyofrealnumbersamongfields . . . . . . . . . . 20
2 Basicshapesofgeometry 21
2.1 Lines,segmentsandhalf-lines . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 Plane,points,lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.2 Naturalorderonaline . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.3 Reciprocalorders . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.4 Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.5 Half-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Openandclosedhalf-planes . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 Convexsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2 Openhalf-planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.3 Closedhalf-planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Angle-supports, interiorandexteriorregions,angles . . . . . . . . . 28
viii Contents
2.3.1 Angle-supports, interiorregions . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Exteriorregions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.3 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Trianglesandconvexquadrilaterals . . . . . . . . . . . . . . . . . . 30
2.4.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.2 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.3 Pasch’sproperty,1882 . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.4 Convexquadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Distance;degree-measureofanangle 35
3.1 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1 Axiomfordistance . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 Derivedpropertiesofdistance . . . . . . . . . . . . . . . . . . . . . 36
3.2 Mid-points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Aratioresult . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Thecross-bartheorem . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Degree-measureofangles . . . . . . . . . . . . . . . . . . . . . . . 40
3.5.1 Axiomfordegree-measure . . . . . . . . . . . . . . . . . . . . . . 40
3.5.2 Derivedpropertiesofdegree-measure . . . . . . . . . . . . . . . . . 41
3.6 Mid-lineofanangle-support . . . . . . . . . . . . . . . . . . . . . 44
3.6.1 Right-angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6.2 Perpendicularlines . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6.3 Mid-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.7 Degree-measureofreflexangles. . . . . . . . . . . . . . . . . . . . 46
3.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Congruenceoftriangles;parallellines 49
4.1 Principlesofcongruence . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.1 Congruenceoftriangles . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Alternateangles,parallellines . . . . . . . . . . . . . . . . . . . . . 53
4.2.1 Alternateangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.2 Parallellines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Propertiesoftrianglesandhalf-planes . . . . . . . . . . . . . . . . 55
4.3.1 Side-anglerelationships; thetriangleinequality . . . . . . . . . . . 55
4.3.2 Propertiesofparallelism . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.3 Droppingaperpendicular . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.4 Projectionandaxialsymmetry . . . . . . . . . . . . . . . . . . . . 57
Contents ix
5 Theparallelaxiom;Euclideangeometry 61
5.1 Theparallelaxiom . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.1 Uniquenessofaparallelline . . . . . . . . . . . . . . . . . . . . . 61
5.2 Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.1 Parallelogramsandrectangles . . . . . . . . . . . . . . . . . . . . . 62
5.2.2 Sumofmeasuresofwedge-angles ofatriangle . . . . . . . . . . . . 63
5.3 Ratioresultsfortriangles . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.1 Linesparalleltooneside-lineofatriangle . . . . . . . . . . . . . . 64
5.3.2 Similartriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.4 Pythagoras’theorem,c.550B.C. . . . . . . . . . . . . . . . . . . . 69
5.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.5 Mid-linesandtriangles . . . . . . . . . . . . . . . . . . . . . . . . 70
5.5.1 Harmonicranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.6 Areaoftriangles,andconvexquadrilaterals andpolygons . . . . . . 72
5.6.1 Areaofatriangle . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.6.2 Areaofaconvexquadrilateral . . . . . . . . . . . . . . . . . . . . . 75
5.6.3 Areaofaconvexpolygon . . . . . . . . . . . . . . . . . . . . . . . 75
6 Cartesiancoordinates;applications 81
6.1 Frameofreference,Cartesiancoordinates . . . . . . . . . . . . . . 81
6.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Algebraicnoteonlinearequations . . . . . . . . . . . . . . . . . . 85
6.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3 Cartesianequationofaline . . . . . . . . . . . . . . . . . . . . . . 86
6.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.4 Parametricequationsofaline . . . . . . . . . . . . . . . . . . . . . 89
6.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.5 Perpendicularityandparallelismoflines . . . . . . . . . . . . . . . 92
6.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.6 Projectionandaxialsymmetry . . . . . . . . . . . . . . . . . . . . 93
6.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.6.2 Formulaforareaofatriangle . . . . . . . . . . . . . . . . . . . . . 94
6.6.3 Inequalities forclosedhalf-planes . . . . . . . . . . . . . . . . . . . 95
6.7 Coordinatetreatmentofharmonicranges . . . . . . . . . . . . . . . 95
6.7.1 Newparametrisationofaline . . . . . . . . . . . . . . . . . . . . . 95
6.7.2 Interchangeofpairsofpoints . . . . . . . . . . . . . . . . . . . . . 97
6.7.3 Distancesfrommid-point . . . . . . . . . . . . . . . . . . . . . . . 98
6.7.4 Distancesfromend-point . . . . . . . . . . . . . . . . . . . . . . . 98
6.7.5 Constructionforaharmonicrange . . . . . . . . . . . . . . . . . . 99
7 Circles;theirbasicproperties 103
7.1 Intersectionofalineandacircle . . . . . . . . . . . . . . . . . . .103
7.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103
7.2 Propertiesofcircles . . . . . . . . . . . . . . . . . . . . . . . . . .105