The Pythagorean Theorem - WPAFBSTEM

So what does poetry have to do with mathematics?Any mathematics text can be likened to a poetry text. In it,the author is interspersing two languages: a language ofqualification (English in the case of this book) and alanguage of quantification (the universal language ofalgebra). The way these two languages are interspersed isvery similar to that of the poetry text. When we aredescribing, we use English prose interspersed with anillustrative phrase or two of algebra. When it is time to doan extensive derivation or problem-solving activity—usingthe concise algebraic language—then the whole page (or twoor three pages!) may consist of nothing but algebra. Algebrathen becomes the alternate language of choice used tounfold the idea or solution. The Pythagorean Theorem followsthis general pattern, which is illustrated below by adiscussion of the well-known quadratic formula. Let ax bx c 0 be a quadratic equation writtenthe standard form as shown with a 0 . Then2inax 2 bx c 0 has two solutions (including complex andmultiple) given by the formula below, called the quadraticformula.x b b 2 4ac.2aTo solve a quadratic equation, using the quadratic formula,one needs to apply the following four steps considered to bea solution process.1. Rewrite the quadratic equation in standard form.2. Identifythe two coefficientsand constantterm a, b,&c .3. Apply the formula and solve for the two x values.4. Check your two answers in the original equation.To illustrate this four-step process, we will solve thequadratic equation 2 x2 13 x 7 .14

1 : 2 x 2 13 x 7 2 x 2 13 x 7 0****2 : a 2, b 13, c 7****3 : x ( 13) ( 13) 2 4(2)( 7) 2(2)13 169 56 413 225 13 15x 44x { 12 ,7}x ****4 : This step is left to the reader. Taking a look at the text between the two happy-facesymbols , we first see the usual mixture of algebra andprose common to math texts. The quadratic formula itself,being a major algebraic result, is presented first as a standalone result. If an associated process, such as solving aquadratic equation, is best described by a sequence ofenumerated steps, the steps will be presented in indented,enumerated fashion as shown. Appendix B provides adetailed list of all mathematical symbols used in this bookalong with explanations.Regarding other formats, italicized 9-font text isused throughout the book to convey special cautionarynotes to the reader, items of historical or personal interest,etc. Rather than footnote these items, I have chosen toplace them within the text exactly at the place where theyaugment the overall discussion.15

Lastly, throughout the book, the reader will notice a threesquared triangular figure at the bottom of the page. Onesuch figure signifies a section end; two, a chapter end; andthree, the book end.CreditsNo book such as this is an individual effort. Manypeople have inspired it: from concept to completion.Likewise, many people have made it so from drafting topublishing. I shall list just a few and their contributions.Elisha Loomis, I never knew you except throughyour words in The Pythagorean Proposition; but thank youfor propelling me to fashion an every-person’s updatesuitable for a new millennium. To those great Americans ofmy youth—President John F. Kennedy, John Glenn, NeilArmstrong, and the like—thank you all for inspiring anentire generation to think and dream of bigger things thanthemselves.To my two editors, Curtis and Stephanie Sparks,thank you for helping the raw material achieve fullpublication. This has truly been a family affair.To my wife Carolyn, the Heart of it All, what can Isay. You have been my constant and loving partner forsome 40 years now. You gave me the space to complete thisproject and rejoiced with me in its completion. As always,we are a proud team!John C. SparksOctober 2008Xenia, Ohio16

1) Consider the Squares“If it was good enough for old Pythagoras,UnknownIt is good enough for me.”How did the Pythagorean Theorem come to be atheorem? Having not been trained as mathematicalhistorian, I shall leave the answer to that question to thosewho have. What I do offer in Chapter 1 is a speculative,logical sequence of how the Pythagorean Theorem mighthave been originally discovered and then extended to itspresent form. Mind you, the following idealized accountdescribes a discovery process much too smooth to haveactually occurred through time. Human inventiveness inreality always has entailed plenty of dead ends and falsestarts. Nevertheless, in this chapter, I will play the role ofthe proverbial Monday-morning quarterback and execute aperfect play sequence as one modern-day teacher sees it.Figure 1.1: The Circle, Square, and Equilateral TriangleOf all regular, planar geometric figures, the squareranks in the top three for elegant simplicity, the other twobeing the circle and equilateral triangle, Figure 1.1. Allthree figures would be relatively easy to draw by our distantancestors: either freehand or, more precisely, with a stakeand fixed length of rope.17

For this reason, I would think that the square would be oneof the earliest geometrics objects examined.Note: Even in my own early-sixties high-school days, string, chalk,and chalk-studded compasses were used to draw ‘precise’geometric figures on the blackboard. Whether or not this ranks mewith the ancients is a matter for the reader to decide.So, how might an ancient mathematician study asimple square? Four things immediately come to mymodern mind: translate it (move the position in planarspace), rotate it, duplicate it, and partition it into twotriangles by insertion of a diagonal as shown in Figure e 1.2: Four Ways to Contemplate a SquareI personally would consider the partitioning of the square tobe the most interesting operation of the four in that I havegenerated two triangles, two new geometric objects, fromone square. The two right-isosceles triangles so generatedare congruent—perfect copies of each other—as shown onthe next page in Figure 1.3 with annotated side lengthss and angle measurements in degrees.18

s90 0450 045ss450 04590 0sFigure 1.3: One Square to Two TrianglesFor this explorer, the partitioning of the square into perfecttriangular replicates would be a fascination starting pointfor further exploration. Continuing with our speculativejourney, one could imagine the replication of a partitionedsquare with perhaps a little decorative shading as shown inFigure 1.4. Moreover, let us not replicate just once, butfour times.1 23 45 Figure 1.4: One Possible Path to Discovery19

Now, continue to translate and rotate the four replicated,shaded playing piece pieces as if working a jigsaw puzzle.After spending some trial-and-error time—perhaps a fewhours, perhaps several years—we stop to ponder afascinating composite pattern when it finally meanders intoview, Step 5 in Figure 1.4.Note: I have always found it very amusing to see a concise andlogical textbook sequence [e.g. the five steps shown on the previouspage] presented in such a way that a student is left to believe thatthis is how the sequence actually happened in a historical context.Recall that Thomas Edison had four-thousand failures before finallysucceeding with the light bulb. Mathematicians are no less prone todead ends and frustrations!Since the sum of any two acute angles in any one of0the right triangles is again 90 , the lighter-shaded figurebounded by the four darker triangles (resulting from Step 4)is a square with area double that of the original square.Further rearrangement in Step 5 reveals the fundamentalPythagorean sum-of-squares pattern when the threesquares are used to enclose an empty triangular areacongruent to each of the eight original right-isoscelestriangles.Of the two triangle properties for each littletriangle—the fact that each was right or the fact that eachwas isosceles—which was the key for the sum of the twosmaller areas to be equal to the one larger area? Or, wereboth properties needed? To explore this question, we willstart by eliminating one of the properties, isosceles; in orderto see if this magical sum-of-squares pattern still holds.Figure 1.5 is a general right triangle where the threeinterior angles and side lengths are labeled.A C BFigure 1.5: General Right Triangle20

Notice that the right-triangle property implies that the sumof the two acute interior angles equals the right angle asproved below. 180 0 & 90 0 90 0 Thus, for any right triangle, the sum of the two acute anglesequals the remaining right angle orin terms of Figure 1.5 as .90 0 ; eloquently statedContinuing our exploration, let’s replicate thegeneral right triangle in Figure 1.5 eight times, dropping allalgebraic annotations. Two triangles will then be fusedtogether in order to form a rectangle, which is shaded viathe same shading scheme in Figure 1.4. Figure 1.6 showsthe result, four bi-shaded rectangles mimicking the four bishaded squares in Figure 1.4.Figure 1.6: Four Bi-Shaded Congruent RectanglesWith our new playing pieces, we rotate as before, finallyarriving at the pattern shown in Figure 1.7.21

Figure 1.7: A Square Donut within a SquareThat the rotated interior quadrilateral—the ‘square donut’—is indeed a square is easily shown. Each interior cornerangle associated with the interior quadrilateral is part of a180 0 . The two acute angles0flanking the interior corner angle sum to 90 since thesethree-angle group that totalsare the two different acute angles associated with the righttriangle. Thus, simple subtraction gives the measure of any0one of the four interior corners as 90 . The four sides ofthe quadrilateral are equal in length since they are simplyfour replicates of the hypotenuse of our basic right triangle.Therefore, the interior quadrilateral is indeed a squaregenerated from our basic triangle and its hypotenuse.Suppose we remove the four lightly shaded playingpieces and lay them aside as shown in Figure 1.8.Figure 1.8: The Square within the Square is Still There22

The middle square (minus the donut hole) is still plainlyvisible and nothing has changed with respect to size ororientation. Moreover, in doing so, we have freed up fourplaying pieces, which can be used for further explorations.If we use the four lighter pieces to experiment withdifferent ways of filling the outline generated by the fourdarker pieces, an amazing discover will eventually manifestitself—again, perhaps after a few hours of fiddling andtwiddling or, perhaps after several years—Figure 1.9.Note: To reiterate, Thomas Edison tried 4000 different light-bulbfilaments before discovering the right material for such anapplication.Figure 1.9: A Discovery Comes into ViewThat the ancient discovery is undeniable is plain fromFigure 1.10 on the next page, which includes yet anotherpattern and, for comparison, the original square shown inFigure 1.7 comprised of all eight playing pieces. The 12thcentury Indian mathematician Bhaskara was alleged tohave simply said, “Behold!” when showing these diagramsto students. Decoding Bhaskara’s terseness, one can createfour different, equivalent-area square patterns using eightcongruent playing pieces. Three of the patterns use half ofthe playing pieces and one uses the full set. Of the threepatterns using half the pieces, the sum of the areas for thetwo smaller squares equals the area of the rotated square inthe middle as shown in the final pattern with the threeoutlined squares.23

Figure 1.10: Behold!Phrasing Bhaskara’s “proclamation” in modern algebraicterms, we would state the following:The Pythagorean TheoremSuppose we have a right triangle with side lengthsand angles labeled as shown below.AC B and A2 B 2 C 2Then24

Our proof in Chapter 1 has been by visual inspection andconsideration of various arrangements of eight triangularplaying pieces. I can imagine our mathematically mindedancestors doing much the same thing some three to fourthousand years ago when this theorem was first discoveredand utilized in a mostly pre-algebraic world.To conclude this chapter, we need to address oneloose end. Suppose we have a non-right triangle. Does thePythagorean Theorem still hold? The answer is aresounding no, but we will hold off proving what is knownastheconverseofthePythagoreanTheorem,A 2 B 2 C 2 , until Chapter 2. However, wewill close Chapter 1 by visually exploring two extreme caseswhere non-right angles definitely imply thatA2 B 2 C 2 .Pivot Point to Perfection BelowFigure 1.11: Extreme Differences VersusPythagorean Perfection25

In Figure 1.11, the lightly shaded squares in the upperdiagram form two equal sides for two radically differentisosceles triangles. One isosceles triangle has a large centralobtuse angle and the other isosceles triangle has a smallcentral acute angle. For both triangles, the darker shadedsquare is formed from the remaining side. It is obvious tothe eye that two light areas do not sum to a dark area nomatter which triangle is under consideration. By way ofcontrast, compare the upper diagram to the lower diagramwhere an additional rotation of the lightly shaded squarescreates two central right angles and the associatedPythagorean perfection.Euclid Alone Has Looked on Beauty BareEuclid alone has looked on Beauty bare.Let all who prate of Beauty hold their peace,And lay them prone upon the earth and ceaseTo ponder on themselves, the while they stareAt nothing, intricately drawn nowhereIn shapes of shifting lineage; let geeseGabble and hiss, but heroes seek releaseFrom dusty bondage into luminous air.O blinding hour, O holy, terrible day,When first the shaft into his vision shownOf light anatomized! Euclid aloneHas looked on Beauty bare. Fortunate theyWho, though once and then but far away,Have heard her massive sandal set on stone.Edna St. Vincent Millay26

2) Four Thousand Years of DiscoveryConsider old Pythagoras,A Greek of long ago,And all that he did give to us,Three sides whose squares now showIn houses, fields and highways straight;In buildings standing tall;In mighty planes that leave the gate;And, micro-systems small.Yes, all because he got it rightWhen angles equal ninety—One geek (BC), his plain delight—January 2002One world changed aplenty!2.1) Pythagoras and the First ProofPythagoras was not the first in antiquity to knowabout the remarkable theorem that bears his name, but hewas the first to formally prove it using deductive geometryand the first to actively ‘market’ it (using today’s terms)throughout the ancient world. One of the earliest indicatorsshowing knowledge of the relationship between righttriangles and side lengths is a hieroglyphic-style picture,Figure 2.1, of a knotted rope having twelve equally-spacedknots.Figure 2.1: Egyptian Knotted Rope, Circa 2000 BCE27

The rope was shown in a context suggesting its use as aworkman’s tool for creating right angles, done via thefashioning of a 3-4-5 right triangle. Thus, the Egyptianshad a mechanical device for demonstrating the converse ofthe Pythagorean Theorem for the 3-4-5 special case:3 2 4 2 5 2 90 0 .Not only did the Egyptians know of specificinstances of the Pythagorean Theorem, but also theBabylonians and Chinese some 1000 years beforePythagoras definitively institutionalized the general resultcirca 500 BCE. And to be fair to the Egyptians, Pythagorashimself, who was born on the island of Samos in 572 BCE,traveled to Egypt at the age of 23 and spent 21 years thereas a student before returning to Greece. While in Egypt,Pythagoras studied a number of things under the guidanceof Egyptian priests, including geometry. Table 2.1 brieflysummarizes what is known about the Pythagorean Theorembefore Pythagoras.DateCulturePersonEvidenceWorkman’s rope forfashioning a3-4-5 triangleRules for righttriangles written onclay tablets alongwith en geometriccharacterizations ofright angles520BCEGreekPythagorasGeneralized resultand deductivelyprovedTable 2.1: Prior to Pythagoras28

The proof Pythagoras is thought to have actuallyused is shown in Figure 2.2. It is a visual proof in that noalgebraic language is used to support numerically thedeductive argument. In the top diagram, the ancientobserver would note that removing the eight congruent righttriangles, four from each identical master square, brings themagnificent sum-of-squares equality into immediate view. Figure 2.2: The First Proof by Pythagoras29

Figure 2.3 is another original, visual proofattributed to Pythagoras. Modern mathematicians wouldsay that this proof is more ‘elegant’ in that the samedeductive message is conveyed using one less triangle. Eventoday, ‘elegance’ in proof is measured in terms of logicalconciseness coupled with the amount insight provided bythe conciseness. Without any further explanation on mypart, the reader is invited to engage in the mental deductivegymnastics needed to derive the sum-of-squares equalityfrom the diagram below.Figure 2.3: An Alternate Visual Proof by PythagorasNeither of Pythagoras’ two visual proofs requires theuse of an algebraic language as we know it. Algebra in itsmodern form as a precise language of numericalquantification wasn’t fully developed until the Renaissance.The branch of mathematics that utilizes algebra to facilitatethe understanding and development of geometric conceptsis known as analytic geometry. Analytic geometry allows fora deductive elegance unobtainable by the use of visualgeometry alone. Figure 2.4 is the square-within-the-square(as first fashioned by Pythagoras) where the length of eachtriangular side is algebraically annotated just one time.30

bacbcFigure 2.4: Annotated Square within a SquareThe proof to be shown is called a dissection proof due to thefact that the larger square has been dissected

2.1: Algebraic Twin-Triangle Proof* 34 2.2: Euclid’s Windmill Proof* 39 2.2: Algebraic Windmill Proof* 40 2.2: Euclid’s Proof of the Pythagorean Converse 42 2.3: Liu Hui’s Packing Proof* 45 2.3: Stomachion Attributed to Archimedes 48 2.4: Kurrah’s Bride’s Chair 49 2.4: Kurrah’