The Pythagorean Theorem - Educational Outreach

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Table 3.2 lists all possible Pythagorean Triples with longside c 100 generated via the m & n formulas on theprevious page. Shaded are the two solutions where A P .M N A=2MN B=M 2 -N 2 C=M 2 +N 2 T P A2 1 4 3 5 PT 12 63 1 6 8 10 C 24 243 2 12 5 13 PT 30 304 1 8 15 17 P 40 604 2 16 12 20 C 48 964 3 24 7 25 PT 56 845 1 10 24 26 C 60 1205 2 20 21 29 P 70 210*5 3 30 16 34 C 80 2405 4 40 9 41 PT 90 1806 1 12 35 37 P 84 210*6 2 24 32 40 C 96 3846 3 36 27 45 C 108 4866 4 48 20 52 C 120 4806 5 60 11 61 PT 132 3307 1 14 48 50 C 112 3367 2 28 45 53 P 126 6307 3 42 40 58 C 140 840*7 4 56 33 65 P 154 9247 5 70 24 74 C 168 840*7 6 84 13 85 PT 182 5468 1 16 63 65 P 144 5048 2 32 60 68 C 160 9608 3 48 55 73 P 176 13208 4 64 48 80 C 192 15368 5 80 39 89 P 208 15609 1 18 80 82 C 180 7209 2 36 77 85 P 198 13869 3 54 72 90 C 216 19449 4 72 65 97 P 234 2340Table 3.2: Pythagorean Triples with c
There are three Pythagorean-triple types: primitive (P),primitive twin (PT), and composite (C). The definitions foreach are as follows.1. Primitive: A Pythagorean Triple ( a , b,c)where there is nocommon factor for all three positive integers a , b,&c .2. Primitive twin: A Pythagorean Triple ( a , b,c)where thelongest leg differs from the hypotenuse by one.3. Composite: A Pythagorean Triple ( a , b,c)where there is acommon factor for all three positive integers a , b,&c .The definition for primitive twin can help us find anassociated m & n condition for identifying the same. Ifa 2mn is the longest leg, thenc a 1mm22 n2 2mn n( m n)2 2mn12 1 1 m n 1Examining Table 3.2 confirms the last equality; primitivetwins occur whenever m n 1. As a quick exercise, weinvite the reader to confirm that the two cases m, n 8, 7and m , n 9, 8 also produce Pythagorean twins withc 100 .b m2 n2Ifis the longest leg [such as the case form 7 & n 2 ], thenc b 1m22n n221 m2 n21..93
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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
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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 and 92: We can formally state this similari
Page 93: What we will do in Section 3.2 is p
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
Page 142 and 143: 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
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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 &
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Recreational Mathematics11. Pickove
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GGolden RatioDefined and algebraica
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Puzzles (cont)Pythagorean Magic Squ
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Pythagorean Theorem ProofsAlgebraic
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Trigonometry (cont)Eratosthenes mea
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The Pythagorean Theorem - Educational Outreach

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