The Pythagorean Theorem - Educational Outreach

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1BC : tan( BAC)BABC BAtan(BAC)0BC (240,002.5) tan(87 ) BC 4,579,520.5miles20 : BC (240,002.5) tan(89.85 ) BC 91,673,992.7milesOur ‘back-of-the-envelope’ computed value has the sunabout 382 times further away from the earth than themoon. By contrast, the Greek estimate was about 19 times,a serious underestimate. However, even with anunderestimate, it is important to understand that Greekgeometry—again, Pythagorean geometry—was being usedcorrectly. The failing was not having sophisticatedinstrumentation. Knowing the distance to the sun, one caneasily obtain the sun’s diameter, whose disk also spans0approximately 0 .5 .Note: The sun and moon have the same apparent size in the skymaking solar eclipses possible.RSun0 tan(0.25 ) 91,673,992.7milesR 400,005.79milesDsunsun 800,011.58milesUsing Figure 4.11, we invite the reader to calculate the earth-tosundistance using the Pythagorean Theorem and the modern valueof the moon-to-sun distance shown above.---154
CEarth at point C in itsorbit about the sunSunBParallax angle> 1212AlphaCentauriDAEarth at point A, six months laterand diametrically opposite CFigure 4.13: From Sun to Alpha CentauriEarth, Moon, Sun, and Stars, the section title hints at aprogressive journey using ever-increasing stepping stones.The Greeks were limited to the known solar system.Starting in the 1800s, increasingly sophisticatedastronomical instrumentation made determination of stellardistances possible by allowing the measurement of very tinyangles just a few seconds in size, as depicted via the muchmagnified angle in Figure 4.13. From the measurementof tiny angles and the building of huge imaginaryinterstellar triangles, astronomers could ascertaintremendous distances using a Pythagorean-based methodcalled the parallax technique. Figure 4.13 illustrates theuse of the parallax technique to find the distance to ournearest stellar neighbor, Alpha Centauri, about 4 .2ly(lightyears) from the sun.The position of the target star, in this case Alpha Centari, ismeasured from two diametrically opposite points on theearth’s orbit. The difference in angular location against abackdrop of much farther ‘fixed’ stars is called the parallaxangle . Half the parallax angle is then used to compute thedistance to the star by the expression155
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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
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
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 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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