Proving the Pythagorean Theorem


 Alfred Berry
 2 years ago
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1 Proving the Pythgoren Theorem Proposition 47 of Book I of Eulid s Elements is the most fmous of ll Eulid s propositions. Disovered long efore Eulid, the Pythgoren Theorem is known y every high shool geometry student: In rightngled tringles the squre on the side sutending the right ngle is equl to the squres on the sides ontining the right ngle. Our more modern resttements of the theorem re more lgeri in nture: In right tringle whose sides hve length nd, nd whose hypotenuse hs length, the reltionship = 2 holds. There re literlly hundreds of proofs of the Pythgoren Theorem. Your tsk, with your prtners, is to present one proof of the Pythgoren Theorem to the lss. The proofs re very visul, nd they ll omine lger nd geometry in some wy. Unlike the Euliden propositions you reently presented, however, these proofs re not quite omplete, nd you will need to supply few detils. Your jo will e to explin your proof to the lss so tht we ll understnd your digrm nd how the Pythgoren Theorem rises from it. Some Suggestions for your Presenttions 1. Introdue yourselves t the strt of your presenttion. Strt with single, leled right tringle, nd uild your digrm from it. Tht will help us to see where your piture omes from. 2. Mke sure to justify your sttements. It doesn t need to e quite s forml s the Euliden propositions, ut we wnt to know, for exmple, how you know ertin squre is squre, or why n re is wht it is. Espeilly mke sure to tell us where in your proof you use the ft tht your tringle is right tringle. 3. Feel free to drw pitures on the ord s you go. You re welome to use trnsprenies or PowerPoint if you wnt. But do mke sure tht you lerly explin your steps slowly nd in detil. 4. Meet with your prtners outside of lss one or twie to go over the proposition. Prtie your delivery! 5. Emil me or see me if you re stuk on your proposition or if you hve other onerns. Don t wit!
2 Pythgors Proof #1 Strting with our given tringle, mke three more opies of the tringle, nd ssemle them to onstrut the following figure: Let A 1 e the re of the entire figure s shown. Let A 2 e the sum of the res of the two squres with sides of length nd, respetively. Let A 3 e the re of the squre with side of length. Then A 1 A 2 =2(½)= A 1 A 3 =2(½)= Thus A 1 A 2 =A 1 A 3 nd so A 2 =A 3. Therefore = 2. In your presenttion, mke sure to explin: 1. where you use the ft tht the given tringle hs right ngle (there my e more thn one ple where you use this ft.) 2. how you know your squres re relly squres. 3. how you figure out the res involved.
3 Pythgors Proof #2 (This proof ws first pulished y Jmes Grfield, our 20 th U.S. President.) Strting with our given tringle, mke seond opy of it lying on its side nd onstrut the following figure: Let A 1 e the sum of the res of the three tringles. Let A 2 e the re of the trpezoid. Then A 1 =½ ()+ ½ ( 2 )+ ½ ()= ½ ( 2 +2) A 2 =(+)(+)/2=[(+) 2 ]/2=( )/2 Now A 1 =A 2 So ( 2 +2)/2= ( )/2 nd therefore 2 = In your presenttion, mke sure to explin: 1. where you use the ft tht the given tringle hs right ngle (there my e more thn one ple where you use this ft.) 2. how you know tht the entire figure is relly trpezoid nd the middle tringle is relly right tringle. 3. how you figure out the res involved, espeilly the trpezoid.
4 Pythgors Proof #3 Strting with our given tringle, mke 3 more opies of it nd ssemle the four tringles into the following figure: The inside figure is squre with sides of length . The outside figure is squre with sides of length. We n write the re of the outside squre two different wys whih re equl. So 2 =4(½)+() 2 = = In your presenttion, mke sure to explin: 1. where you use the ft tht the given tringle hs right ngle (there my e more thn one ple where you use this ft.) 2. how you know tht the squres in your figure re truly squres. 3. how you figure out the res involved.
5 Pythgors Proof #4 Strting with our given tringle, mke 3 more opies of it nd ssemle the four tringles into the following figure: The inside figure is squre with sides of length. The outside figure is squre with sides of length +. We n write the re of the outside squre two different wys whih re equl. So (+) 2 =4(½)+ 2 So = 2+ 2, nd therefore = 2. In your presenttion, mke sure to explin: 1. where you use the ft tht the given tringle hs right ngle (there my e more thn one ple where you use this ft.) 2. how you know tht the squres in your figure re truly squres. 3. how you figure out the res involved.
6 Pythgors Proof #5 (This proof ws first given y Leonrdo d Vini) Strting with our given tringle, uild squres on the sides of length nd. Then onstrut squre with side length s shown nd then omplete the figure y dding one more opy of the given tringle s shown: Let A 1 e the sum of the res of the two squres with sides of lengths nd, respetively, together with the two tringles tht shre sides with these squres. Let A 2 e the sum of the res of the squre with side lengths, together with the two tringles tht shre sides with tht squre. In oth A 1 nd A 2, the dshed segment divides the figure into two ongruent prts tht hve re ½(+) 2 ½()= ½( ). Then A 1 =A 2. Also, A 1 = nd A 2 = 2 +. Therefore = 2. In your presenttion, mke sure to explin: 1. where you use the ft tht the given tringle hs right ngle (there my e more thn one ple where you use this ft.) 2. how you figure out the res involved.
7 Pythgors Proof #6 Strting with our given tringle, turn it on end nd drop perpendiulr from the right ngle to the hypotenuse: x x The perpendiulr divides the right tringle into two tringles tht re similr to eh other nd to the originl tringle. Lel the segments s in the piture. Then, sine orresponding sides re proportionl, we hve = nd = x x Cross multiplying gives us 2 =x nd 2 =(x). Adding these two equtions gives us =x+(x)= 2 In your presenttion, mke sure to explin: 1. where you use the ft tht the given tringle hs right ngle (there my e more thn one ple where you use this ft.) 2. how you know ll the tringles re similr (s well s wht similr mens). 3. whih orresponding sides give us the frtions in the equtions ove.
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