The Pythagorean Theorem - Educational Outreach

Notice how quickly and easily our result is obtained oncealgebraic is used to augment the geometric picture. Simplyput, algebra coupled with geometry is superior to geometryalone in quantifying and tracking the diverse and subtlerelationships between geometric whole and the assortedpieces. Hence, throughout the remainder of the book,analytic geometry will be used to help prove and developresults as much as possible.Since the larger square in Figure 2.4 is dissectedinto five smaller pieces, we will say that this is a DissectionOrder V (DRV) proof. It is a good proof in that all threecritical dimensions—a, b, c—and only these dimensions areused to verify the result. This proof is the proof mostcommonly used when the Pythagorean Theorem is firstintroduced. As we have seen, the origins of this proof can betraced to Pythagoras himself.We can convey the proof in simpler fashion bysimply showing the square-within-the-square diagram(Figure 2.5) and the associated algebraic developmentbelow unencumbered by commentary. Here forward, thiswill be our standard way of presenting smaller and moreobvious proofs and/or developments.abcFigure 2.5: Algebraic Form of the First Proof1:A ( a b)2:(a b) a2set22 c2 2ab b& A c 4(212 c22ab) 4(12ab) 2ab a2 b2 c232