Pythagoras theorem is one of the most popular theorems. Paper Folding And The Theorem of Pythagoras. Visual Connect in Teaching.

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1 in the lssroom Visul Connet in Tehing Pper Folding And The Theorem of Pythgors Cn unfolding pper ot revel proof of Pythgors theorem? Does mking squre within squre e nything more thn n exerise in geometry t est? Art nd mth ome together in delightful mthemtil exerises desried in this rtile. Sivsnkr Sstry Pythgors theorem is one of the most populr theorems in geometry. Rems of pper hve een used to write different proofs of this theorem ut in this rtile we ut nd fold pper to demonstrte two different proofs. Mke ot nd prove Pythgors' theorem Rememer how hildren flot pper ots in running wter fter hevy rin? There re mny types of ots tht n e mde y folding single pper sheet. Here, we mke the simplest nd most ommon type of pper ot using squre sheet of pper. In se you hve forgotten how to fold ot here re the steps:- 48

2 Step 1 Step 2 Step 3 Step 4 Step 5 Fold k one lyer on one side nd three lyers on the other Step 6 Step 7 Step 8 Pull Pull BOAT Pull out to mke squre gin After step 8 you will hve ot. Wht is its shpe? If you look losely you find reses whih show mny right ngled tringles. Now unfold the ot. Rememer we strted with plin squre pper. Now look t the reses ppering in the unfolded ot. You will see pttern whih is known mthemtilly s tesselltion ( mking tiles ). This prtiulr tesselltion onsists of squres with reses long the digonls whih divide eh squre into two right ngled tringles. The wy we folded the pper ensures tht ll the squres (nd therefore the tringles too) re identil in ll respets. Vol. 1, No. 1, June 2012 At Right Angles 49

3 They look like this: Choose ny right ngled tringle ABC. Here ngle B is right ngle. Shde the squres on sides AC, BC nd AC. Look losely t these squres. All hve reses long the digonls nd re divided into right-ngled tringles. How do we mesure their res? Are need not only e mesured in terms of unit squres. We n lso mesure the re y ounting the numer of identil right ngled tringles ontined in them. The squre upon AB hs 2 right-ngled tringles; so AB 2 = 2 right-ngled tringles. The squre upon BC hs 2 right-ngled tringles; so BC 2 = 2 right-ngled tringles. The squre upon AC hs 4 right ngled tringles; so AC 2 = 4 right-ngled tringles. Hene: AC 2 = AB 2 + BC 2 This is the theorem of Pythgors pplied to tringle ABC. Mke squre within squre nd prove Pythgors Theorem Tke squre sheet of pper. Fold long digonl nd mke shrp rese (Fig. 1). Fold the ottom right orner towrds the digonl, so tht the edge of the sheet lies prllel to the digonl. Mke rese. You will hve folded right ngled tringle (Fig. 2). Fig 1 Fig 2 50

4 Now fold the next side of the squre to the side of the right ngle lredy folded (Fig. 3)nd mke rese. Repet the sme with the remining two orners.(fig. 4 nd fig. 5) Fig 3 Fig 4 Now you hve squre with squre hole in the middle. (Fig. 5). Crese ll the sides nd unfold(fig.6). Let AP =, AQ = nd PQ =. In tringle APQ, ngle PAQ is right ngle. PQRS is squre with side PQ =. Hene re of PQRS = 2. A Q B P Fig 5 Fig 6 R D S C Vol. 1, No. 1, June 2012 At Right Angles 51

5 Q Fold k tringles PAQ, BQR, RCS, SDP inside, s efore. P A B C R Now in the squre PQRS stnding on PQ we hve identil tringles PAQ, BQR, RCS, SDP, nd smll squre ABCD. Squre PQRS = Tringle PAQ + Tringle QBR + Tringle RCS + Tringle SDP + Squre ABCD = squre ABCD D = AB 2 = 2 + ( ) 2. S Hene 2 = nd so 2 = Hene: PQ 2 = AP 2 + AQ 2. This is the theorem of Pythgors pplied to tringle APQ. Sivsnkr Sstry s interests rnge from origmi, kirigmi, pleogrphy nd mteur stronomy to ly modelling, skething nd onsi. He is lso pulished uthor hving written 27 ooks in Knnd on siene nd mthemtis. Mr. Sstry my e ontted vsssstry@gmil.om 52