Plane Geometry - Bruce E. Shapiro

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192 SECTION 38. THE PYTHAGOREAN THEOREMwhile a later sutra by Katyayana (c 200 BCE) is more general 3A rope stretched along the length of the diagonal produces anarea which the vertical and horizontal sides make together.It has inundated common culture as well. Gilbert and Sullivan gave theMajor-General “many cheerful facts about the square of the hypotenuse”while the scarecrow in the Wizard of Oz mistakenly quotes it as:The sum of the square roots of any two sides of an isoscelestriangle is equal to the square root of the remaining side. Oh,joy! Oh, rapture! I’ve got a brain! 4Even Homer Simpson has had a word to say about it, and at least onepresident of the United States (James Garfield) is credited with finding anew proof. 5Theorem 38.1 (Pythagorean Theorem) Let △ABC be a right trianglewith right angle at vertex C. Then if a, b, and c are the sides oppositevertices A,B, and C, respectively,a 2 + b 2 = c 2Figure 38.1: Notation for the Pythagorean Theorem.Proof. Drop a perpendicular from C to ←→ AB and call its foot D. Label thesegments as shown in figure 38.2.By lemma 33.5, A ∗ D ∗ B.By the angle sum theorem 180 = m(∠A) + m(∠B) + m(∠C), hence since3 Mac Tutor History of Math, http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html.4 Internet Movie Data Base (IMDB), http://www.imdb.com/title/tt0032138/quotes?qt04099235 See Venema Problem 9.12.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 38. THE PYTHAGOREAN THEOREM 193m(∠C) = 90,m(∠A) + m(∠B) = 90Also by the angle sum theorem,m(∠A) + m(∠ACD) = 90m(∠B) + m(∠BCD) = 90Hencem(∠A) = m(∠BCD)m(∠B) = (∠ACD)Thus△ABC ∼ △ACD ∼ △CBDBy the fundamental theorem of similar triangles,HenceAdding,pb = b c and q a = a cb 2 = pc and a 2 = qca 2 + b 2 = pc + qc = c(p + q) = c 2Figure 38.2: Notation for proof of the Pythagorean Theorem.Corollary 38.2 Then h = √ pq (see figure 38.2).Proof. (Exercise.)Corollary 38.3 a = √ qc and b = √ pc.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.
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Foundations of GeometryLecture Note
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ivCONTENTS30 Triangles in Neutral G
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2 SECTION 1. PREFACEare enrolled in
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4 SECTION 1. PREFACEre-learning it
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6 SECTION 2. NCTM3. To guide in the
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8 SECTION 2. NCTMTechnology. Techno
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10 SECTION 2. NCTM« CC BY-NC-ND 3.
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12 SECTION 3. CA STANDARDIn 2005 Ca
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14 SECTION 3. CA STANDARDThe Califo
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16 SECTION 3. CA STANDARDCalifornia
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18 SECTION 4. COMMON COREAt all lev
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20 SECTION 5. LOGIC AND PROOFour th
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22 SECTION 5. LOGIC AND PROOFRene D
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24 SECTION 5. LOGIC AND PROOF“I
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26 SECTION 6. REAL NUMBERSThe inter
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28 SECTION 6. REAL NUMBERS“The Su
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30 SECTION 7. EUCLID’S ELEMENTSEu
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32 SECTION 7. EUCLID’S ELEMENTSFi
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34 SECTION 8. HILBERT’S AXIOMSiom
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36 SECTION 8. HILBERT’S AXIOMS«
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38 SECTION 9. BIRKHOFF/MACLANE AXIO
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40 SECTION 9. BIRKHOFF/MACLANE AXIO
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42 SECTION 10. THE SMSG AXIOMSP is
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44 SECTION 10. THE SMSG AXIOMS« CC
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46 SECTION 11. THE UCSMP AXIOMSin s
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48 SECTION 11. THE UCSMP AXIOMS« C
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50 SECTION 12. VENEMA’S AXIOMS3.
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52 SECTION 12. VENEMA’S AXIOMS«
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54 SECTION 13. INCIDENCE GEOMETRYFi
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56 SECTION 13. INCIDENCE GEOMETRYEx
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58 SECTION 13. INCIDENCE GEOMETRYFi
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60 SECTION 13. INCIDENCE GEOMETRY«
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62 SECTION 14. BETWEENNESSFigure 14
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64 SECTION 14. BETWEENNESSTheorem 1
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66 SECTION 14. BETWEENNESS(a) To sh
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68 SECTION 14. BETWEENNESSThe follo
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70 SECTION 14. BETWEENNESSExample 1
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72 SECTION 14. BETWEENNESSorf(A) >
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74 SECTION 14. BETWEENNESS« CC BY-
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76 SECTION 15. THE PLANE SEPARATION
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78 SECTION 15. THE PLANE SEPARATION
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80 SECTION 16. ANGLESFigure 16.1: T
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82 SECTION 16. ANGLESCorollary 16.6
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84 SECTION 16. ANGLESFigure 16.6: I
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86 SECTION 16. ANGLESD cannot be in
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88 SECTION 17. THE CROSSBAR THEOREM
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90 SECTION 17. THE CROSSBAR THEOREM
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92 SECTION 18. LINEAR PAIRSFigure 1
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94 SECTION 18. LINEAR PAIRSγ + ∠
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96 SECTION 19. ANGLE BISECTORSwith
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98 SECTION 19. ANGLE BISECTORSAngle
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100 SECTION 20. THE CONTINUITY AXIO
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102 SECTION 20. THE CONTINUITY AXIO
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104 SECTION 21. SIDE-ANGLE-SIDEBirk
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106 SECTION 21. SIDE-ANGLE-SIDEFigu
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108 SECTION 22. NEUTRAL GEOMETRYEll
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110 SECTION 22. NEUTRAL GEOMETRY«
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112 SECTION 23. ANGLE-SIDE-ANGLEFig
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114 SECTION 23. ANGLE-SIDE-ANGLE«
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116 SECTION 24. EXTERIOR ANGLESTheo
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118 SECTION 24. EXTERIOR ANGLES« C
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120 SECTION 25. ANGLE-ANGLE-SIDEBy
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122 SECTION 26. SIDE-SIDE-SIDECase
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124 SECTION 26. SIDE-SIDE-SIDEBy si
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126 SECTION 27. SCALENE AND TRIANGL
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128 SECTION 27. SCALENE AND TRIANGL
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130 SECTION 28. CHARACTERIZATION OF
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132 SECTION 28. CHARACTERIZATION OF
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134 SECTION 29. TRANSVERSALSFigure
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136 SECTION 29. TRANSVERSALSCorolla
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138 SECTION 30. TRIANGLES IN NEUTRA
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140 SECTION 30. TRIANGLES IN NEUTRA
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Page 192 and 193: 188 SECTION 37. AREADefinition 37.5
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242 SECTION 44. ESTIMATING πso tha
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244 SECTION 44. ESTIMATING πn π n
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246 SECTION 44. ESTIMATING πTable
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248 SECTION 44. ESTIMATING πnums =
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250 SECTION 44. ESTIMATING π« CC
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252 SECTION 45. EUCLIDEAN CONSTRUCT
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264 SECTION 46. HYPERBOLIC GEOMETRY
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272SECTION 47.PERPENDICULAR LINES I
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274SECTION 47.PERPENDICULAR LINES I
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276 SECTION 48. PARALLEL LINES IN H
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284 SECTION 49. TRIANGLES IN HYPERB
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290 SECTION 50. AREA IN HYPERBOLIC
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298 SECTION 51. THE POINCARE DISK M
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306 SECTION 52. ARC LENGTHAs we see
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308 SECTION 52. ARC LENGTHProof. Su
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310 SECTION 52. ARC LENGTHThus the
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312 SECTION 52. ARC LENGTHMeasuring
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314 SECTION 53. SPHERICAL GEOMETRYW
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316 SECTION 53. SPHERICAL GEOMETRYF
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318 SECTION 53. SPHERICAL GEOMETRYP
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320 APPENDIX A. SYMBOLS USEDAppendi
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322 APPENDIX A. SYMBOLS USEDTechnol
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Plane Geometry - Bruce E. Shapiro

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