Proving the Pythagorean Theorem


 Jocelin Chapman
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1 Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or derivtion. This will now e derived here. Contents 1 The Tringle 2 2 The Preliminries 2 3 The Constrution 3 4 The Proof 4 1
2 1 The Tringle The Pythgoren Theorem pplies only to right ngled tringles; i.e. tringles in whih one of the three interior ngles hs mesure of 90 or π 2 rdins. The longest side, whih is opposite this right ngle, hs length, while the other two sides hve lengths nd. 1 We lel the two ngles whih re not right ngles θ nd φ s in the following digrm: φ θ Note tht, s usul, ox hs een used in the orner to denote the right ngle. With this nottion, Pythgors first proved tht = 2. 2 The Preliminries There re few prerequisite fts tht need to e lid out in order to follow this proof. They re s follows: The sum of the three interior ngles of ny tringle (whih hs een drwn on flt surfe) is onstnt equl to two right ngles in whtever your preferred ngulr mesure hppens to e. (i.e. 180, π rdins, et.) If you mesure the ngle of stright line, s though it were tully two lines onneted k to k, this ngle is lso equl to two right ngles. The re of squre is the length of one side multiplied y itself. 2 The re of tringle is 1 2h, where the se is one side, nd the height h is perpendiulr distne etween the se nd the orner opposite the se. In right ngled tringle, leled with sides, nd s ove, this re redues to While some individul tehers insist on the onvention tht e the shortest side, there is no mthemtil reson to mke this demnd. It is purely mtter of esthetis. 2 This is extly why rising x to exponent 2, x 2, is referred to s squring x. 2
3 When squring inomil suh s +, one gets ross terms involving the produt of nd. Written out expliitly, ( + ) 2 = ( + ) ( + ) = ( + )+ ( + ) = = where the 2 ross term hs een written lst in define of onvention for resons tht will soon e ler. 3 The Constrution The following onstrution is fr from the only wy to derive nd prove the Pythgoren Theorem. It is not even the one whih Pythgors himself used. (When he did his work, lger ws not ommonly epted mthemtil tool mong his peers, nd geometry ws gretly preferred. Alger will e used here.) The onstrution egins y reting n ext opy of our tringle, nd tthing it to the originl tringle in prtiulr wy, suh tht side in the originl is prllel to side of the opy, nd suh tht the opy hs een rotted y right ngle ounter lokwise efore onneting the orners. The result looks like this: φ θ ω φ θ Let us refully exmine the ngles within this onstrution, inluding the newly leled ngle ω. As ll three ngles within our originl tringle dd up 3
4 to two right ngles, nd s one orner is right ngle lredy, we n onlude tht θ nd φ must, together, lso dd up to right ngle. If we look t the orner where our two tringles meet, we see tht (y design) we hve stright line running long our originl side nd our new side. Thus, on the left side of tht joining, the ngles θ, φ nd ω must dd up to two right ngles. We know tht θ nd φ lone dd up to right ngle, so we must onlude tht ω lone, whih is the ngle etween the two sides of length, is lso right ngle. We duplite our tringle twie more, nd put ll four tringles together in similr mnner to form this finl onstrution: 4 The Proof We re now in position to derive nd prove the Pythgoren theorem. Our ove onstrution n e viewed in two different wys. On one hnd, it is gint squre, with sides of length +. On the other hnd, it is smller squre 3 with sides length surrounded y four right ngle tringles with short sides nd. We now lulte the re of this shpe in two different wys. Treting the onstrution s gint squre gives us n re A equl to A = ( + ) 2 = We know this is squre, s we hve shown tht the ngle etween the sides of length, originlly leled ω, is tully right ngle. 4
5 while treting it s smll squre surrounded y tringles gives us n re equl to ( ) 1 A = = Now, s this is the sme shpe represented in two wys, we n equte the expressions for re nd find ourselves left with or = = 2 fter we nel the ommon 2. This is, indeed, the Pythgoren Theorem tht is often tught y rote. We now hve one wy to prove tht it is, indeed, true. 5
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