The Pythagorean Theorem - Educational Outreach

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We close this section with a complete restatement of thePythagorean Theorem as found in Chapter 2, but now withthe inclusion of the converse relationshipA2 B2 C2 900. Euclid’s subtle proof of thePythagorean Converse follows (Book 1 of The Elements,Proposition 48).The Pythagorean Theorem andPythagorean ConverseSuppose we have a triangle with side lengthsand angles labeled as shown below.Then: CA B A2 B2 C2Figure 2.12 on the next page shows Euclid’s originalconstruction used to prove the Pythagorean Converse. Theshaded triangle conforms to the hypothesisA2 B2 C2whereby design. From the intentional design,0one is to show or deduce that 90 in orderto prove the converse.Note: the ancients and even some of my former public-schoolteachers would have said ‘by construction’ instead of ‘by design.’However, the year is 2008, not 1958, and the word design seems tobe a superior conveyor of the intended meaning.42
XB'A090 CBFigure 2.12: Euclid’s Converse DiagramEuclid’s first step was to construct a line segment oflength B ' where B' B . Then, this line segment was joinedas shown to the shaded triangle in such a fashion that thecorner angle that mirrors was indeed a right angle ofmeasure—again by design! Euclid then added a second lineof unknown length X in order to complete a companiontriangle with common vertical side as shown in Figure2.12. Euclid finally used a formal verbally-descriptive logicstream quite similar to the annotated algebraic logic streambelow in order to complete his proof.Algebraic Logic1: A2 B2 C 22:B B'3: B'A 9042 2 2: A ( B') X5: XXX222 A CX C A222 B0 ( B')22Verbal Annotation1] By hypothesis2] By design3] By design4] Pythagorean Theorem5] Properties of algebraicequality90043
Page 1: The PythagoreanTheoremCrown Jewel o
Page 4: The Pythagorean TheoremCrown Jewel
Page 7 and 8: Table of ContentsList of Tables and
Page 9 and 10: List of Tables and FiguresTablesNum
Page 11 and 12: Figures…continuedNumber and Title
Page 13 and 14: List of Proofs and DevelopmentsSect
Page 15 and 16: PrefaceThe Pythagorean Theorem has
Page 17 and 18: 1 : 2x2x2****22 13x 7 13x 7 0 : a
Page 19 and 20: 1) Consider the Squares“If it was
Page 21 and 22: s090045 0ss45450450090Figure 1.3: O
Page 23 and 24: Notice that the right-triangle prop
Page 25 and 26: The middle square (minus the donut
Page 27 and 28: Our proof in Chapter 1 has been by
Page 29 and 30: 2) Four Thousand Years of Discovery
Page 31 and 32: The proof Pythagoras is thought to
Page 33 and 34: cabcFigure 2.4: Annotated Square wi
Page 35 and 36: Figure 2.6 is the diagram for a sec
Page 37 and 38: 1x:b2:Area(CBA)3:Area( ABC)Area(ABC
Page 39 and 40: Euclid’s proof of the Pythagorean
Page 41 and 42: First we establish that the two tri
Page 43: 1 2y b b: y b c c2:A3:A4:Aunshaded
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 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
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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
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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
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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
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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
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The four-step solution follows. All
Page 150 and 151:
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
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
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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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