The Pythagorean Theorem - Educational Outreach

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We have achieved five-digit accuracy in ten seconds.Electronic calculators are indeed wonderful, but how wouldone obtain this answer before an age of technology?Granted, Heron’s formula was available in 1890, but theformula needed numbers to work, numbers phrased interms of decimal equivalents for those wary of roots. Toproduce these decimal equivalents would require themanual, laborious extraction of the three squareroots 370 , 116 , and 74 , not to mention extraction thefinal root in Heron’s Formula itself. Thus, a decimalapproach was probably not a very good fireside option in1890. One would have to inject a dose of cleverness, a lostart in today’s brute-force electronic world.Lloyd’s original solution entails a masterfuldecomposition of three right triangles. In Figure 4.1, thearea of the triangular lake is the area of ABF , which wedenote as .ABFANow: ABF ABD AFE FBC R EFCD .AAAAAFor ABDABAB, we have that22AB DA 922370 DB1722 370 Notice Lloyd was able to cleverly construct a right trianglewith two perpendicular sides (each of integral length)producing a hypotenuse of the needed length 370 . Thiswas not all he did!For AFEAFAF, we have that22 EA 522 7 EF22 74 AF 74138
Finally, for FBCFBFB22 CF 422 CB1022 116 FB 116Thus, our three triangles ABD , AFE , and FBC formthe boundary of the Triangular Lake. <strong>The</strong> area of theTriangular Lake directly followsAAABF ABF 12AABD AFE ( 9)(17) 112(5)(7) 2AAFBC RAEFCD (4)(10) (4)(7) 11<strong>The</strong> truth of Lloyd’s problem statement is now evident: tothose of a mathematical turn, the number 11 is a verypositive and definitive answer not sullied by an irrationaldecimal expansion.One might ask if it is possible to ‘grind through’Heron’s formula and arrive at 11 using the square roots asis. Obviously, due to the algebraic complexity, Lloyd wascounting on the puzzler to give up on this more brute-forcedirect approach and resort to some sort of cleverness.However, it is possible to grind! Below is the computationalsequence, an algebraic nightmare indeed.Note: I happen to agree with Lloyd’s ‘forcing to cleverness’ in that Ihave given students a similar computational exercise for years. Inthis exercise requiring logarithm use, electronic calculators aredeliberately rendered useless due to overflow or underflow ofderived numerical quantities. Students must resort to ‘old fashion’clever use of logarithms in order to complete the computationsinvolving extreme numbers..1 : a 370, b 2a b c : s 2116,& c 74370 116 274139
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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
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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 138 and 139: 4) Pearls of Fun and WonderPearls o
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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