The Pythagorean Theorem - Educational Outreach

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3w 2h s w 5 s w 0.38196s&5 1s w 0.61804s2 The Golden Ratio, symbolized by the Greek letter Phi ( ), isthe reciprocal of5 1 5 1, given by 1. 6180422.BBAababCA0363600360108a 72aa0C072bDFigure 4.15: Two Golden TrianglesAs stated, the Golden Ratio permeates mathematicsand science to the same extent as the PythagoreanTheorem. Figure 4.15 depicts two appearances in atriangular context, making them suitable inclusions for thisbook.TriangleABCthe vertical side has lengthon the left is a right triangle whereab equal to the geometricmean of the hypotenuse a and the horizontal side b .158
By the Pythagorean Theorem, we have:aa22 b2 ( ab bab) 0 15 a b 2 aa b b22This last result can be summarized as follows.If one short side of a right triangle is the geometric mean ofthe hypotenuse and the remaining short side, then the ratioof the hypotenuse to the remaining short side is the GoldenRatio .In triangle ABD to the right, all threetriangles ABD , ABC , and ACD are isosceles. Inaddition, the two triangles ABD and ACD are similar: ABD ACD . We have by proportionality rules:a b a a b22a ab b 0 a bThus, in Figure 4.15, ABD has been sectioned as tocreate the Golden Ratio between the slant height andbase for the two similar triangles ABC and ACD .Figure 4.16 aptly displays the inherent, unlabeled beautyof the Golden Ratio when applied to our two triangles.159
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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
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In Table 2.3, we briefly summarize
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We can formally state this similari
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What we will do in Section 3.2 is p
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There are three Pythagorean-triple
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Thus, if we multiply any given prim
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The last equality shows that the in
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A Pythagorean Quartet is a set of f
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3.5) Pythagoras and the Three Means
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yxy MHxab2ab2 y x FC y ab2ab 2ab y
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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
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 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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