The Pythagorean Theorem - Educational Outreach

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3.2) Pythagorean Triples and Triangles53c m2 n2b m2 n24a 2mnFigure 3.2: Pythagorean TriplesA Pythagorean Triple is a set of three positiveintegers ( a,b,c)satisfying the classical Pythagoreanrelationshipa2 2 2 b c .Furthermore, a Pythagorean Triangle is simply a righttriangle having side lengths corresponding to the integers ina Pythagorean Triple. Figure 3.2 shows the earliest andsmallest such triple, ( 3,4,5), known to the Egyptians some4000 years ago in the context of a construction device forlaying out right angles (Section 2.1). The triple ( 6,8,10)wasalso used by early builders to lay out right triangles.Pythagorean triples were studied in their own right by theBabylonians and the Greeks. Even today, PythagoreanTriples are a continuing and wonderful treasure chest forprofessionals and amateurs alike as they explore variousnumerical relationships and oddities using the methods ofnumber theory, in particular Diophantine analysis (thestudy of algebraic equations whose answers can only bepositive integers). We will not even attempt to open thetreasure chest in this volume, but simply point the readerto the great and thorough references by Beiler andSierpinski where the treasures are revealed in full glory.90
What we will do in Section 3.2 is present theformulas for a well-known method for generatingPythagorean Triples. This method provides one of thetraditional starting points for further explorations ofPythagorean Triangles and Triples.Theorem: Let m n 0 be two positive integers. Then a setof Pythagorean Triples is given by the formula:( a,b,c) (2mn,m22 n , m2 n2)Proof:aaaa2222 b b b b22222 2 2 2 (2mn) ( m n ) 2 2 4 2 2 4mn m 2mn n4 2 2 4 m 2mn n 2 2 2 2 ( m n ) c 4From the formulas for the three Pythagorean Triples, wecan immediately develop expressions for both the perimeterP and area A of the associated Pythagorean Triangle.A 12(2mn)(mP 2mn m222 n ) mn(m n2 m291 n22 n2) 2m(m n)One example of an elementary exploration is to find allPythagorean Triangles whose area numerically equals theperimeter. To do so, first setA P mn(m2 nn(m n) 2 2) 2m(m n)m 3, n 1& m 3, n 2For m 3,n 1, we have that A P 24 .For m 3,n 2 , we have that A P 30 .
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The PythagoreanTheoremCrown Jewel o
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The Pythagorean TheoremCrown Jewel
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Table of ContentsList of Tables and
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List of Tables and FiguresTablesNum
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Figures…continuedNumber and Title
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List of Proofs and DevelopmentsSect
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PrefaceThe Pythagorean Theorem has
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1 : 2x2x2****22 13x 7 13x 7 0 : a
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1) Consider the Squares“If it was
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s090045 0ss45450450090Figure 1.3: O
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Notice that the right-triangle prop
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The middle square (minus the donut
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Our proof in Chapter 1 has been by
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2) Four Thousand Years of Discovery
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The proof Pythagoras is thought to
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cabcFigure 2.4: Annotated Square wi
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Figure 2.6 is the diagram for a sec
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1x:b2:Area(CBA)3:Area( ABC)Area(ABC
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Euclid’s proof of the Pythagorean
Page 41 and 42: First we establish that the two tri
Page 43 and 44: 1 2y b b: y b c c2:A3:A4:Aunshaded
Page 45 and 46: XB'A090 CBFigure 2.12: Euclid’s C
Page 47 and 48: 2.3) Liu Hui Packs the SquaresLiu H
Page 49 and 50: One of my favorites was Boxel, a ga
Page 51 and 52: 2.4) Kurrah Transforms the Bride’
Page 53 and 54: Note: as is the occasional custom i
Page 55 and 56: 2.5) Bhaskara Unleashes the Power o
Page 57 and 58: 2.6) Leonardo da Vinci’s Magnific
Page 59 and 60: In Figure 2.24, we enlarge Step 5 a
Page 61 and 62: But, this is precisely the nature o
Page 63 and 64: 2.8) Henry Perigal’s TombstoneHen
Page 65 and 66: In the century following Perigal, b
Page 67 and 68: Returning to Henry Perigal, Figure
Page 69 and 70: Figure 2.32 is President Garfield
Page 71 and 72: Even though the Cartesian coordinat
Page 73 and 74: B 2← γ →α y βx C-xC 2Figure
Page 75 and 76: Being polynomial in form, the funct
Page 77 and 78: Thus, the critical point ( C 2 ,0)i
Page 79 and 80: Sincexcpwas chosen on an arbitrary
Page 81 and 82: ∂F/∂x = ∂F/∂y = 0x(C - x) 2
Page 83 and 84: Define a new function21 A2 A3 2xcp
Page 85 and 86: 2.11) Shear, Shape, and AreaOur las
Page 87 and 88: Step 1: Cut the big square into two
Page 89 and 90: In Table 2.3, we briefly summarize
Page 91: We can formally state this similari
Page 95 and 96: There are three Pythagorean-triple
Page 97 and 98: Thus, if we multiply any given prim
Page 99 and 100: The last equality shows that the in
Page 101 and 102: A Pythagorean Quartet is a set of f
Page 103 and 104: 3.5) Pythagoras and the Three Means
Page 105 and 106: yxy MHxab2ab2 y x FC y ab2ab 2ab y
Page 107 and 108: Hero was the founder of the Higher
Page 109 and 110: 5 : Solve for area using the formul
Page 111 and 112: 4 : Construct the parallelogram wit
Page 113 and 114: We already have introduced Thabit i
Page 115 and 116: Some historians believe Stewart’s
Page 117 and 118: In pre-computer days, both all four
Page 119 and 120: One example of a periodic process i
Page 121 and 122: The six trigonometric functions are
Page 123 and 124: The following four relationships ar
Page 125 and 126: As with any set of identities, the
Page 127 and 128: Figure 3.17 can be used as a jumpin
Page 129 and 130: The remaining two pillars are the L
Page 131 and 132: a2Law of Cosines c2 b2 2bccos()The
Page 133 and 134: An example of a Diophantine equatio
Page 136 and 137: However, Euler’s Conjecture did n
Page 138 and 139: 4) Pearls of Fun and WonderPearls o
Page 140 and 141: We have achieved five-digit accurac
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140212213116)74370116)(74370(370)74
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X X X X O X X O X XO X X O O OO X O
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4.3) Earth, Moon, Sun, and StarsIn
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The four-step solution follows. All
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Figure 4.9 depicts Eratosthenes’
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---High above the earth, we see the
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The commonly accepted value is abou
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1BC : tan( BAC)BABC BAtan(BAC)0BC
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BD ACtan( 21 )Even for the closest
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3w 2h s w 5 s w 0.38196s&5 1s
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Figure 4.16: Triangular Phi---The h
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What remains to be done is to devel
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3 : h32 4 h222 2 24C 2 h33162 2 2
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Epilogue: The Crown and the Jewels
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Not only has it endured, but the Py
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A] Greek AlphabetGREEK LETTERUpper
Page 174 and 175:
C] Geometric FoundationsThe Paralle
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8. Congruent Triangles: Two triangl
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3. Rectangle: A bh : P 2b 2h, b &
Page 180 and 181:
Recreational Mathematics11. Pickove
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GGolden RatioDefined and algebraica
Page 184 and 185:
Puzzles (cont)Pythagorean Magic Squ
Page 186 and 187:
Pythagorean Theorem ProofsAlgebraic
Page 188:
Trigonometry (cont)Eratosthenes mea
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The Pythagorean Theorem - Educational Outreach

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