The Pythagorean Theorem - Educational Outreach

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4) Pearls of Fun and WonderPearls of ancient mind and wonder,Time will never pillage, plunder,Or give your soul to the worm—Or worse yet, for nerds to keepWith their mental treasures deepWhere secret squares are cut asunder.So, come good fun, have a turn,Bring your gold, build and learn!April 20054.1) Sam Lloyd’s Triangular LakeSam Lloyd was a famous American creator ofpuzzles, tricks and conundrums who produced most of hismasterpieces in the late 1800s. Many of Lloyd’s puzzleshave survived and actually thrived, having found their wayinto modern puzzle collections. Martin Gardner, of ScientificAmerican fame, has been a tremendous preserver of Lloyd’slegacy to recreational mathematics. Recreationalmathematics, one might classify that phrase as anoxymoron. Yet, back in the late 1800s, Lloyd’s heyday,people actually worked puzzles for evening relaxation, muchlike we moderns watch TV or play video games. The need torelax has always been there; how people fulfill the need ismore a function of the era in which we live and the availabletechnology that enables us to recreate.One of Sam Lloyd’s famous creations is hisTriangular Lake. Lloyd subtlety gives his readers twochoices: solution by sweat and brute force, or solution bycleverness and minimal effort. The clever solution requiresuse of the Pythagorean Theorem. What follows is Lloyd’soriginal statement:“The question I ask our puzzlists is to determine how manyacres there would be in that triangular lake, surrounded asshown in Figure 4.1 by square plots of 370, 116 and 74acres.136
The problem is of peculiar interest to those of a mathematicalturn, in that it gives a positive and definite answer to aproposition, which, according to usual methods produces oneof those ever-decreasing, but never-ending decimal fractions.TriangularLake370AcresAE54F74Acres 116AcresD 7 C 10BFigure 4.1: Triangle Lake and SolutionIn the year 2006, the Triangular Lake does notrequire a great deal of original thought to solve. Nowadays,we would say it is a ‘cookie cutter’ problem and requires nosweat whatsoever! One simply needs to apply Heron’sformula for triangular area, A s( s a)(s b)(s a)using a modern electronic calculator. The answer isavailable after a few keystrokes.1 : a c 2a b c : s 2s 19.304023 : A A 370 19.23538, b 74 8.60233370 116 2s(s a)(s b)(s a)116 10.77033,&7419.30402(0.06864)(8.53369)(10.70169) 11.00037137
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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
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1 2y b b: y b c c2:A3:A4:Aunshaded
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XB'A090 CBFigure 2.12: Euclid’s C
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2.3) Liu Hui Packs the SquaresLiu H
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One of my favorites was Boxel, a ga
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2.4) Kurrah Transforms the Bride’
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Note: as is the occasional custom i
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2.5) Bhaskara Unleashes the Power o
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2.6) Leonardo da Vinci’s Magnific
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In Figure 2.24, we enlarge Step 5 a
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But, this is precisely the nature o
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2.8) Henry Perigal’s TombstoneHen
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In the century following Perigal, b
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Returning to Henry Perigal, Figure
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Figure 2.32 is President Garfield
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Even though the Cartesian coordinat
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B 2← γ →α y βx C-xC 2Figure
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Being polynomial in form, the funct
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Thus, the critical point ( C 2 ,0)i
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Sincexcpwas chosen on an arbitrary
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∂F/∂x = ∂F/∂y = 0x(C - x) 2
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Define a new function21 A2 A3 2xcp
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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 and 94: What we will do in Section 3.2 is p
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 140 and 141: We have achieved five-digit accurac
Page 142 and 143: 140212213116)74370116)(74370(370)74
Page 144 and 145: X X X X O X X O X XO X X O O OO X O
Page 146 and 147: 4.3) Earth, Moon, Sun, and StarsIn
Page 148 and 149: The four-step solution follows. All
Page 150 and 151: Figure 4.9 depicts Eratosthenes’
Page 152 and 153: ---High above the earth, we see the
Page 154 and 155: The commonly accepted value is abou
Page 156 and 157: 1BC : tan( BAC)BABC BAtan(BAC)0BC
Page 158 and 159: BD ACtan( 21 )Even for the closest
Page 160 and 161: 3w 2h s w 5 s w 0.38196s&5 1s
Page 162 and 163: Figure 4.16: Triangular Phi---The h
Page 164 and 165: What remains to be done is to devel
Page 166 and 167: 3 : h32 4 h222 2 24C 2 h33162 2 2
Page 168 and 169: Epilogue: The Crown and the Jewels
Page 170 and 171: Not only has it endured, but the Py
Page 172 and 173: A] Greek AlphabetGREEK LETTERUpper
Page 174 and 175: C] Geometric FoundationsThe Paralle
Page 176 and 177: 8. Congruent Triangles: Two triangl
Page 178 and 179: 3. Rectangle: A bh : P 2b 2h, b &
Page 180 and 181: Recreational Mathematics11. Pickove
Page 182 and 183: 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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