The Pythagorean Theorem - Educational Outreach

We have achieved five-digit accuracy in ten seconds.Electronic calculators are indeed wonderful, but how wouldone obtain this answer before an age of technology?Granted, Heron’s formula was available in 1890, but theformula needed numbers to work, numbers phrased interms of decimal equivalents for those wary of roots. Toproduce these decimal equivalents would require themanual, laborious extraction of the three squareroots 370 , 116 , and 74 , not to mention extraction thefinal root in Heron’s Formula itself. Thus, a decimalapproach was probably not a very good fireside option in1890. One would have to inject a dose of cleverness, a lostart in today’s brute-force electronic world.Lloyd’s original solution entails a masterfuldecomposition of three right triangles. In Figure 4.1, thearea of the triangular lake is the area of ABF , which wedenote as .ABFANow: ABF ABD AFE FBC R EFCD .AAAAAFor ABDABAB, we have that22AB DA 922370 DB1722 370 Notice Lloyd was able to cleverly construct a right trianglewith two perpendicular sides (each of integral length)producing a hypotenuse of the needed length 370 . Thiswas not all he did!For AFEAFAF, we have that22 EA 522 7 EF22 74 AF 74138