THE PYTHAGOREAN THEOREM

Transcription

1 THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this theorem to find missing sides of right tringles s well s the distne etween two points. Finlly, we will provide proofs of the Pythgoren Theorem tht pply to vriety of lerning styles. Let us now turn to the lssi illustrtion of the Pythgoren Theorem, nd perform some mesurements s n investigtion. (The digrms nd mesurements from this illustrtion hve een reted using The Geometer s Skethpd softwre.) First, we rete right tringle ABC. The onvention for nming sides is tht side is loted ross from A, side is loted ross from B, nd side is loted ross from C. We then find the length of its sides, rounded to the nerest hundredth. B =.41 m = 4.00 m = 4.66 m C A Next, we onstrut squres on the three sides of ABC. We then mesure the re of eh of the squres (to the nerest hundredth), s shown elow. Wht do you notie? B Are of Squre t side = 5.80 m Are of squre t side = m Are of squre t side = 1.76 m C A

2 Notie tht the sum of the res of the two smller squres equls the re of the lrgest squre; this leds us to the more forml wording of the Pythgoren Theorem: The Pythgoren Theorem In right tringle, the sum of the squres of the lengths of the legs equls the squre of the length of the hypotenuse. In other words, if nd represent the lengths of the legs of right tringle, nd represents the length of the hypotenuse, the Pythgoren Theorem sttes tht: + = Exmples Find x. Write ll nswers in simplest rdil form. 1. Solution: The lengths of the legs re 6 nd 8, nd the x 8 length of the hypotenuse is x, so = x = x 100 = x 6 10 = x x = 10. Solution: 11 The lengths of the legs re 7 nd x, nd the x length of the hypotenuse is 11, so 7 + x = x = 11 7 x = 7 x = 7 x = 36 x = 6

3 3. Solution: 1 The lengths of the legs re x nd 3 5, nd 3 5 the length of the hypotenuse is 1, so x ( 3 5) 1 ( ) x + = x + = x = 144 x + 45 = 144 x = 99 x = 99 x = 9 11 x = 3 11 Exerises Find x. Write ll nswers in simplest rdil form x 5. x x

4 The Pythgoren Theorem nd the Distne Formul The Pythgoren Theorem n e used to find the distne etween two points, s shown elow. Exmples 1. Use the Pythgoren Theorem to find the distne etween the points A(, 3) nd B(7, 10). Write your nswer in simplest rdil form.. Use the Pythgoren Theorem to find the distne etween the points A(-3, 4) nd B(5, -6). Write your nswer in simplest rdil form. Solution to #1: We first plot the points A(, 3) nd B(7, 10) on the oordinte plne. We wnt to find the distne AB. Next, we drw right tringle ABC tht hs hypotenuse AB, s shown elow y A (, 3) B (7, 10) C (7, 3) x We n esily find the lengths of the legs of the tringle: AC = 5 (We n quikly ount the units or tke the solute vlue of the differene of the x-oordintes: 7 = 5, or equivlently 7 = 5.)

5 BC = 7 (We n quikly ount the units or tke the solute vlue of the differene of the y-oordintes: 10 3 = 7, or equivlently 3 10 = 7.) We n now use the Pythgoren Theorem: = ( AB) = ( AB) 74 = ( AB) AB = 74 (Note: This squre root n not e simplified ny further. For dditionl informtion on simplifying squre roots, refer to the Irrtionl Numers tutoril in the ppendix ) Solution to #: We first plot the points A(-3, 4) nd B(5, -6) on the oordinte plne. We wnt to find the distne AB. Next, we drw right tringle ABC tht hs hypotenuse AB, s shown elow. A (-3, 4) C (-3, -6) -7 B (5, -6) y x

6 We n esily find the lengths of the legs of the tringle: BC = 8 (We n quikly ount the units or find the solute vlue of the differene of the x-oordintes: 5 ( 3) = 8, or equivlently ( 3) 5 = 8.) AC = 10 (We n quikly ount the units or find the solute vlue of the differene of the y-oordintes; ( 6) 4 = 10, or equivlently 4 ( 6) = 10.) We n now use the Pythgoren Theorem: = ( AB) = ( AB) 164 = ( AB) AB = 164 AB = 4 41 AB = 41 (Note: For dditionl informtion on simplifying squre roots, refer to the Irrtionl Numers tutoril in the ppendix ) Now tht we hve used the Pythgoren Theorem, we will solve the sme prolems using the distne formul insted. The distne formul is shown elow. (An informl derivtion will e shown lter in this setion.) The Distne Formul Given two points Ax ( 1, y 1) nd B( x, y ), the distne d etween A nd B is ( ) ( ) 1 1 d = x x + y y Exmples 1. Use the distne formul to find the distne etween the points A(, 3) nd B(7, 10). Write your nswer in simplest rdil form.. Use the distne formul to find the distne etween the points A(-3, 4) nd B(5, -6). Write your nswer in simplest rdil form.

7 Solution to #1: ( ) ( ) 1 1 d = x x + y y ( 7 ) ( 10 3) d = + d = d = d = 74 Solution to #: ( 1) ( 1) ( 5 ( 3) ) ( 6 4) d = x x + y y d = + ( ) 8 10 d = + d = d = 164 = 4 41 = 41 Notie how the numers in this solutions ove orrespond to those in the previous set of exmples where we used the Pythgoren Theorem diretly. We will now show n informl derivtion of the distne formul. x y x represents the length of one leg of the tringle. 1 y represents the length of the other leg of the tringle. 1 Sine d represents the hypotenuse of the tringle, we n use the Pythgoren Theorem nd otin the following formul: ( 1 ) ( 1 ) x x + y y = d ( ) ( ) = x x y y d ( ) ( ) 1 1 d = x x + y y Exmples 1. Given the points A(3, 4) nd B(7, 1), ) Use the Pythgoren Theorem to find the distne from A to B. Write your nswer in simplest rdil form. ) Verify your nswer in prt () y using the distne formul.

8 . Given the points A(-1, 7) nd B(-4, -), ) Use the Pythgoren Theorem to find the distne from A to B. Write your nswer in simplest rdil form. ) Verify your nswer in prt () y using the distne formul.

9 Proofs of the Pythgoren Theorem We will now present four proofs of the Pythgoren Theorem. These prtiulr proofs hve een hosen euse s group, they ommodte vriety of lerning styles. The lerning styles re desried elow. Visul Visul lerners lern y seeing. They lern est from digrms, hrts, detiled written explntions, nd y tking detiled notes themselves. Auditory Auditory lerners lern y listening. When deling with written informtion, they lern muh etter when they n her the informtion eing explined to them. Ttile/Kinestheti Ttile/Kinestheti lerners lern y doing. They lern est when they n prtiipte in hnds-on tivities nd demonstrtions. After working through eh proof in detil, we will disuss the lerning style(s) tht pply most to tht prtiulr proof. Proof #1 The following proof is sed on the proof y Pythgors, ut we will first set up the proof in detil y sking the reder to help onstrut some digrms. We egin with the lssi illustrtion of the Pythgoren Theorem. We wish to show tht + =, or in other words, (the re of the squre t side ) + (the re of the squre t side ) = (the re of the squre t side ) B C A

10 We will now exmine the omined re of eh of the following groups of polygons. Group 1: The squre t side, the squre t side, nd four opies of the right tringle (with legs nd ). Group : The squre t side nd four opies of the right tringle (with legs nd ).

11 Proof #1: Ativity 1. Cut out the six piees from Group 1 elow.. Rerrnge the six piees nd ple them in suh wy tht they fit extly into the following squre of side length +.

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13 3. Cut out the five piees from Group elow. 4. Rerrnge the six piees nd ple them in suh wy tht they fit extly into the following squre of side length +. Note: An extr opy of this tivity n e found on the next two pges (sine this one will e ll slied up fter you omplete the tivity). The solution to the tivity n e found on the pge fter tht No peeking until you hve figured it out yourself!!!

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15 Proof #1: Ativity 1. Cut out the six piees from Group 1 elow.. Rerrnge the six piees nd ple them in suh wy tht they fit extly into the following squre of side length +.

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17 3. Cut out the five piees from Group elow. 4. Rerrnge the five piees nd ple them in suh wy tht they fit extly into the following squre of side length +.

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19 Solution to the Ativity for Proof #1: Arrngement of Group 1 Ojets: There re slight vritions of the rrngement of the ojets from Group 1, ut ll solutions hve the following hrteristis: The squre with side must e pled in one orner, the squre with side must e pled in the opposite orner, nd the four tringles must e pled in pirs to form two retngles to omprise the reminder of the digrm. Arrngement of Group Ojets: We will lern on the next pge why this result leds us to the onlusion of the Pythgoren Theorem

20 Justifition for Proof #1: We first find the re of eh of the individul ojets in Group 1: Are of squre with side : Are of squre with side : 1 Are of one tringle: (There re four suh tringles in the digrm.) We n now find the omined re of the Group 1 ojets y dding the individul res together: 1 Comined re of Group 1 Ojets ( ) = = + + We then find the re of eh of the individul ojets in Group : Are of squre with side : 1 Are of one tringle: (There re four suh tringles in the digrm.) We n now find the omined re of the Group ojets y dding the individul res together: 1 Comined re of Group Ojets ( ) = + 4 = + The omined re of the Group 1 ojets is equl to the omined re of the Group ojets (sine they oth omprise squre of length + ). Therefore, we n set the following quntities equl: Comined re of Group 1 Ojets = Comined re of Group Ojets + + = + We n then sutrt the quntity from oth sides of the eqution (whih is equivlent to removing the four tringles from eh of our digrms). We otin the following result, whih is the Pythgoren Theorem: + = Lerning Style Anlysis: Proof #1 would ppel primrily to the Kinestheti lerner. If presented in leture formt, the explntions would ppel to the Auditory lerner. The digrms nd equtions in this proof would lso ppel to the Visul lerner.

21 Proof # We egin with the lssi illustrtion of the Pythgoren Theorem. We wish to show tht + =, or in other words, (the re of the squre t side ) + (the re of the squre t side ) = (the re of the squre t side ) We will now onstrut two lines through the enter of the squre t side ; one whih is prllel to side, nd the other whih is perpendiulr to side, s shown elow.

22 Proof #: Ativity 1. Cut out the squre t side.. Cut out the squre t side, nd then ut long the thik lines inside this squre to divide it into four piees 3. Rerrnge the five piees (from squres nd ) nd ple them in suh wy tht they fit extly into the squre t side. Note: An extr opy of this tivity n e found on the next pge (sine this one will e ll slied up fter you omplete the tivity). The solution to the tivity n e found on the pge fter tht No peeking until you hve figured it out yourself!!!

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24 Now perform the following tivity: 1. Cut out the squre t side.. Cut out the squre t side, nd then ut long the thik lines inside this squre to divide it into four piees 3. Rerrnge the five piees (from squres nd ) nd ple them in suh wy tht they fit extly into the squre t side.

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26 Solution to Proof # (Perigl s Proof)

27 The previous proof is known s dissetion proof, nd is ttriuted to Henry Perigl, London stokroker who pulished the proof in 1873 (Bogomolny). The lgeri justifition of this proof is lengthy nd will not e shown in this text. Lerning Style Anlysis: Proof # would ppel primrily to the Kinestheti lerner. As with ll of our proofs of the Pythgoren Theorem, the digrms would lso ppel to the Visul lerner. Proof #3 The following proof ws disovered y President Grfield in 1876 (Bogomolny). Grfield s proof is sed on the formul for the re of trpezoid. His digrm is shown elow. If we find the re of the figure y using the formul for the re of trpezoid, h Are of trpezoid = se + se ( ) 1 + Are of trpezoid = + ( ) + + Are of trpezoid = = + + = Now we insted find the re of eh of the individul tringles, nd then dd them up to find the re of the entire figure:

28 1 Rememer tht the formul for the re of tringle is A= h. Tringle #1 Tringle # Tringle #3 A = A = A = 1 3 Adding up the res of the three tringles, we otin Totl re of tringles = A + A + A = + + = We now set the re of the trpezoid equl to the totl re of the tringles: Are of trpezoid = Totl re of tringles + + = We then sutrt from oth sides, nd otin the eqution + = Finlly, we multiply oth sides y, nd otin our desired result: + =. Lerning Style Anlysis: The digrms nd equtions in Proof #3 would ppel to the Visul lerner. If Proof #3 were to e presented in leture formt, the explntions would ppel to the Auditory lerner.

29 Prefe to Proof #4 We need to prefe our first proof with disussion out similrity. First, we drw right tringle ABC. C A B Next, we drw n ltitude from vertex C to the hypotenuse, AB, nd we lel the point of intersetion D. Notie tht the digrm elow ontins three tringles; the originl tringle s well s two smller tringles tht hve een formed y the ltitude. C A D B Let us suppose, for the purpose of illustrtion, tht B mesures 0 o. We then wnt to find the mesures of ll the other ngles in the digrm. (Note tht the digrm my not e drwn to sle.) C A D 0 o B

30 Sine the sum of the mesures of the ngles of tringle is 180 o, we n quikly otin the following ngle mesures. C 0 o 70 o A 70 o D 0 o B We now drw the three tringles seprtely nd lel the mesures of their ngles. C A 70 o 0 o B C C 0 o 70 o A 70 o D D 0 o B If three ngles of one tringle re ongruent to three orresponding ngles of nother tringle, then the tringles re similr. (This is lled AA Similrity, sine it is tully suffiient to just show tht two ngles of one tringle re ongruent to two orresponding ngles of nother tringle -- sine the third ngles must then e ongruent s well.) Sine eh of the ove tringles hve the sme set of ngle mesurements, the three tringles re similr to eh other. We must refully mth up the orresponding ngles s we nme them in the following similrity sttement. (We hve hosen to nme the 70 o ngle first, then the 0 o ngle, then the 90 o ngle.): ABC ACD CBD (The symol mens similr.)

31 The 0 o nd 70 o ngles were used solely for the purpose of illustrtion; the ove exmple n e generlized to ny right tringle, s will e shown in Proof #1. The similr tringles will e used to prove the Pythgoren Theorem. Proof #4 We egin with right tringle ABC. We need to prove tht ( AC) ( BC) ( AB) C + =. A B We then drw n ltitude to the hypotenuse, with point of intersetion D. C A D B There re three different tringles in the digrm ove. Their ongruent ngles re mrked elow. C A C C B A D D B

32 The three tringles re similr y AA Similrity; therefore ABC ACD CBD Sine the tringles re similr, their orresponding sides re proportionl. Using the ft tht ABC ACD, we n sy tht AC AB AD =. AC Cross multiplying, we otin the eqution AC AC = AB AD. (1) Using the ft tht ABC CBD, we n sy tht BC BD =. AB BC Cross multiplying, we otin the eqution BC BC = AB BD. () Adding equtions (1) nd (), we otin AC AC+ BC BC = AB AD+ AB BD ( AC) + ( BC) = AB( AD+ BD) C Sine AD + BD = AB (see digrm t right), ( ) ( ) AC + BC = AB AB, so A D B ( AC) + ( BC) = ( AB) Lerning Style Anlysis: If Proof #4 were to e presented in leture formt, the explntions would ppel to the Auditory lerner. The digrms nd equtions would ppel to the Visul lerner. Online Resoure The following wepge ontins n wrd-winning Jv pplet whih ws written y Jim Morey. This pplet is n exellent visul demonstrtion of the Pythgoren Theorem: gors.html

33 Works Cited Bogomolny, Alexnder. Pythgoren Theorem CTK Softwre, In. <wester.ommnet.edu/ml/online.shtml>