The Pythagorean Theorem and Its Converse

Transcription

1 - Wht You ll Lern To use the Pthgoren Theorem To use the onverse of the Pthgoren Theorem... nd Wh To find the distne etween two doks on lke, s in Emple The Pthgoren Theorem nd Its onverse liforni ontent Stndrds GEOM.0 Students use the Pthgoren theorem to determine distne nd find missing lengths of sides of right tringles. Introdue.. ±. m m. ± in. Skills Hndook, p. 7 Squre the lengths of the sides of eh tringle. Wht do ou notie?.. 6 d m 0 d in. in. m m -. Pln liforni ontent Stndrds GEOM.0 Students use the Pthgoren theorem to determine distne nd find missing lengths of sides of right tringles. liforni Mth kground The Pthgoren Theorem d 6 ± 0 m ± (") New Voulr Pthgoren triple The well-known right tringle reltionship lled the Pthgoren Theorem is nmed for Pthgors, Greek mthemtiin who lived in the sith entur.. We now know tht the lonins, Egptins, nd hinese were wre of this reltionship efore its disover Pthgors. There re mn proofs of the Pthgoren Theorem. You will see one proof in Eerise nd others lter in the ook. lthough severl nient ultures postulted the Pthgoren Theorem nd used it to mesure distnes, the first proof of it ws ttriuted Eulid to Pthgors. The distne formul is oordinte form of the Pthgoren Theorem, whih is the foundtion of ll trigonometri funtions. More Mth kground: p. Lesson Plnning nd Resoures Ke onepts Theorem - Pthgoren Theorem In right tringle, the sum of the squres of the lengths of the legs is equl to the squre of the length of the hpotenuse. + = See p. E for list of the resoures tht support this lesson. D, Online, or Trnsprenies ell Ringer Prtie Voulr Tip Legs Hpotenuse Pthgoren triple is set of nonzero whole numers,, nd tht stisf the eqution + =. Here re some ommon Pthgoren triples.,,,,,, 7 7,, hek Skills You ll Need Use student pge, trnspren, or PowerPoint. liforni Stndrds Dil Review Use trnspren 7. If ou multipl eh numer in Pthgoren triple the sme nonzero whole numer, the three numers tht result lso form Pthgoren triple. Lesson - The Pthgoren Theorem nd Its onverse 7 Speil Needs L s ou red the Pthgoren Theorem together with the lss, point out how muh esier it is when stted lgerill rther thn in words. Hve students prtie reiting + =. elow Level L efore the lesson, list the squres of whole numers less thn. lso review how to simplif rdil epression. 7

2 . Teh Guided Instrution Tehing Tip Let students know tht Pthgoren triples often pper on stndrdized tests. Error Prevention Some students m ssume tht the legs re lws the known quntities. Point out tht is lws the hpotenuse when ppling the formul + = to right tringle. D E D E D E D E D E D E Test-Tking Tip Memorizing the ommon Pthgoren triples, like those t the ottom of p. 7, n help ou solve prolems more quikl. Pthgoren Triples Find the length of the hpotenuse of #. Do the lengths of the sides of # form Pthgoren triple? + = Use the Pthgoren Theorem. + = Sustitute for nd for = Simplif. = = 9 Tke the squre root. The length of the hpotenuse is 9. The lengths of the sides,,, nd 9, form Pthgoren triple euse the re whole numers tht stisf + =. right tringle hs hpotenuse of length nd leg of length 0. Find the length of the other leg. Do the lengths of the sides form Pthgoren triple? "; no In some ses, ou will write the length of side in simplest rdil form. Tehnolog Tip Students m wonder wh the re sked to use lultor in some eerises ut not in other similr eerises. Tell them tht rel-world pplitions tpill require deiml nswers. Point out tht rdils re et, so the re preferred when eerises re of purel mthemtil nture. right tringle hs legs of length 6 nd 0. Find the length of the hpotenuse. Do the lengths of the sides form Pthgoren triple? ; es Find the vlue of. Leve our nswer in simplest rdil form. D, Online, or Trnsprenies dditionl Emples " 0 sell dimond is squre with 90-ft sides. Home plte nd seond se re t opposite verties of the squre. out how fr is home plte from seond se? out 7 ft Using Simplest Rdil Form lger Find the vlue of. Leve our nswer in simplest rdil form (pge 90). + = Pthgoren Theorem + = Sustitute. 6 + = 00 Simplif. = 6 =!6 =!6() =! Sutrt 6 from eh side. Tke the squre root. Simplif. The hpotenuse of right tringle hs length. One leg hs length 6. Find the length of the other leg. Leve our nswer in simplest rdil form. 6" Rel-World onnetion The Prks Deprtment rents pddle ots t doks ner eh entrne to the prk. To the nerest meter, how fr is it to pddle from one dok to the other? + = Pthgoren Theorem + 0 = Sustitute.,000 = Simplif. =!,000 = 0.66 It is 0 m from one dok to the other. Tke the squre root. Use lultor. ritil Thinking When ou wnt to know how fr ou hve to pddle ot, wh is n pproimte nswer more useful thn n nswer in simplest rdil form? You wnt to know the nerest whole numer vlue, whih m not e pprent in rdil epression. hpter Right Tringles nd Trigonometr 0 m m dvned Lerners L Hve students desrie how tringle whose sides form Pthgoren triple nd tringle whose sides re multiple of tht triple re relted. Students should reognize tht the re similr tringles. English Lerners EL Review the term onverse, using the Pthgoren Theorem nd its onverse s n emple. Then hve students write the Pthgoren Theorem s ionditionl sttement.

3 The onverse of the Pthgoren Theorem You n use the onverse of the Pthgoren Theorem to determine whether tringle is right tringle. You will prove Theorem - in Eerise. Ke onepts Theorem - onverse of the Pthgoren Theorem If the squre of the length of one side of tringle is equl to the sum of the squres of the lengths of the other two sides, then the tringle is right tringle. Guided Instrution Tehnolog Tip Hve students use geometr softwre to eplore nd demonstrte the theorems If +, the tringle is otuse nd If +, the tringle is ute. Diret students to keep nd onstnt while mnipulting ltering the ngle opposite. For: Pthgoren tivit Use: Intertive Tetook, - Is It Right Tringle? Is this tringle right tringle? Sustitute the gretest length for Simplif. 7 = 7 = +, so the tringle is right tringle. tringle hs sides of lengths 6,, nd 0. Is the tringle right tringle? no You n lso use the squres of the lengths of the sides of tringle to find whether the tringle is ute or otuse. The following two theorems tell how. Error Prevention Remind students tht must e the longest side of the tringle for the omprison of nd + to give vlid tringle lssifition. lso, students should use the Tringle Inequlit Theorem to hek tht + so tht the side lengths form tringle. D, Online, or Trnsprenies dditionl Emples Is this tringle right tringle? Ke onepts Theorem - If the squre of the length of the longest side of tringle is greter thn the sum of the squres of the lengths of the other two sides, the tringle is otuse. If. +, the tringle is otuse. Theorem - If the squre of the length of the longest side of tringle is less thn the sum of the squres of the lengths of the other two sides, the tringle is ute. If, +, the tringle is ute. no 7 m 6 m m The numers represent the lengths of the sides of tringle. lssif eh tringle s ute, otuse, or right..,, right. 0,, otuse Rel-World onnetion The horizontl re is kept short to ensure n ute ngle t the top of the ldder, so the ldder won t ollpse. lssifing Tringles s ute, Otuse, or Right lssif the tringle whose side lengths re 6,, nd s ute, otuse, or right ompre to ±. Sustitute the gretest length for Sine. +, the tringle is otuse. tringle hs sides of lengths 7,, nd 9. lssif the tringle its ngles. ute Lesson - The Pthgoren Theorem nd Its onverse 9 Resoures Dil Notetking Guide - L Dil Notetking Guide - dpted Version L losure The re of is ft. Find nd. Leve our nswer in simplest rdil form. ft ft " ft; " ft 9

4 . Prtie ssignment Guide -7, 7-9,, -9, - -6, 0,,, 0-7 hllenge - Multiple hoie Prtie 9, 60 Mied Review 6-69 Homework Quik hek To hek students understnding of ke skills nd onepts, go over Eerises 6,, 0, 6, 0. Eerises, These eerises ntiipte the speil right tringle reltionships in Lesson -. sk: Wht is the rtio : : in eh tringle? :: " Eerises 6 In onl some of the eerises do the first two lengths represent nd. Remind students to ompre the sum of the squres of the two smller lengths with the squre of the gretest length. Guided Prolem Solving Enrihment Retehing dpted Prtie Prtie Nme lss Dte Prtie - The Pthgoren Theorem nd Its onverse Find the vlue of eh vrile. Leve our nswers in simplest rdil form Find the length of eh hpotenuse. Use our lultor, nd round our nswers to the nerest whole numer. 7.. in m 0 z 7 60 m 9 in. The numers represent the lengths of the sides of tringle. lssif eh tringle s ute, otuse, or right. m 60 ft 6 m 0 ft L L L L L Person Edution, In., pulishing s Person Prentie Hll. ll rights reserved. EXERISES Stndrds Prtie Prtie Emple GO for Help Emple (pge ) Emple (pge ) Emple (pge ) Emple (pge 9) Emple (pge 9) hpter Right Tringles nd Trigonometr For more eerises, see Etr Skill, Word Prolem, nd Proof Prtie. lger Find the vlue of Does eh set of numers form Pthgoren triple? Eplin. 7.,, 6. 0,, 6 9.,, no; ± u 6 es; 0 ± 6 es; ± lger Find the vlue of. Leve our nswer in simplest rdil form. 0. ". ".. "9. ". " Home Mintenne pinter lens -ft ldder ginst house. The se of the ldder is ft from the house.. ft. To the nerest tenth of foot, how high on the house does the ldder reh?. The ldder in prt () rehes too high on the house. how muh should the pinter move the ldder s se w from the house to lower the top ft? out. ft 7. wlkw forms the digonl of squre plground. The wlkw is m long. To the nerest tenth of meter, how long is side of the plground? 7.0 m Is eh tringle right tringle? Eplin no; 9 ± u no; 9 ± u es; ± 6 6 The lengths of the sides of tringle re given. lssif eh tringle s ute, right, or otuse..,, 6 ute. 0., 0., 0.6 otuse.,, ute.!,, otuse. 0, 0, 0 right 6.!,!7, ute 7 " GEOM , 9, 0.,, 0., 00, 0. 7,,.,, 6 6.,,

5 ppl Your Skills 0. nswers m vr. Smple: Hve three people hold the rope units, units, nd units prt in the shpe of tringle. GO for Help For guide to solving Eerise, see p... PQ ( ) ± ( ) Visit: PHShool.om We ode: e-00 Proof lger Find the vlue of. Leve our nswer in simplest rdil form " " 0. Writing Eh er in n nient lnd, lrge river overflowed its nks, often destroing oundr mrkers. The rol surveors used rope with knots t equl intervls to help reonstrut oundries. Eplin how surveor ould use this rope to form right ngle. (Hint: Use the Pthgoren triple,,.). Multiple hoie Whih tringle is not right tringle? Emroider You wnt to emroider squre design. You hve n emroider hoop with 6 in. dimeter. Find the lrgest vlue of so tht the entire squre will fit in the hoop. Round to the nerest tenth.. in.. In prllelogrm RSTW, RS = 7, ST =, nd RT =. Is RSTW retngle? Eplin. Yes; 7 ±, so lrst is rt. l.. oordinte Geometr You n use the Pthgoren Theorem to prove the Distne Q(, ) Formul. Let points P(, ) nd Q(, ) e the endpoints of the hpotenuse of right tringle.. Write n lgeri epression to omplete eh of the following: PR = j nd QR = j.» ;». the Pthgoren Theorem, P(, ) O R(, ) PQ = PR + QR. Rewrite this sttement sustituting the lgeri epressions ou found for PR nd QR in prt ().. omplete the proof tking the squre root of eh side of the eqution tht ou wrote in prt (). PQ "( ) ( ). onstrutions Eplin how to onstrut segment of length!n, where n is n positive integer, nd ou re given segment of length. (Hint: See the digrm.) See mrgin. Find third whole numer so tht the three numers form Pthgoren triple. 6., 9 7., 0., 9., Error Prevention Eerise Students m think the tringle with side lengths, ", nd is right tringle. Point out tht there is no right ngle smol in the lrge tringle. Students must use the Pthgoren Theorem twie, first to find the side of the smllest tringle, nd then to find the hpotenuse of the tringle with se 6. Eerise Show students how to use Pthgoren triples to hek for right tringles. For nswer hoie, the n multipl eh side 0 to get sides of 6, nd 0. The should reognize this s multiple of,, tringle. Similrl, dividing eh side in nswer hoie ", students n reognize tht the tringle nnot e right tringle. Eerise Some students m e unfmilir with the terms emroider nd emroider hoop. sk volunteer to ring emroider mterils nd n emroider hoop to lss nd demonstrte how to use the hoop. Eerise 7 Point out tht the Pthgoren triple,, 0 is twie the triple 7,,. Eplin tht this is n emple of wht is stted in the lst prgrph on pge 7. hllenge students to give onvining rgument tht if,, is Pthgoren triple, then n, n, n is lso Pthgoren triple for n nonzero whole numer n. lso, hve students stte wh two right tringles with sides,, nd n, n, n must e similr. (n) + (n) = n + n = n ( + ) = n = (n) ; SSS ~ Lesson - The Pthgoren Theorem nd Its onverse. nswers m vr. Smple: Using segments of length, onstrut the hp. of the right k formed these segments. Using the hp. found s one leg nd segment of length s the other leg, onstrut the hp. of the k formed those legs. ontinue this proess until onstruting hpotenuse of length "n.

6 . ssess & Reteh. Find the vlue of. 9. Find the vlue of. Leve our nswer in simplest rdil form. D, Online, or Trnsprenies Lesson Quiz ". The town of Elen is mi north nd mi west of Holerg. trin runs on stright trk etween the two towns. How mn miles does it over? Round our nswer to the nerest whole numer. mi. The lengths of the sides of tringle re m, m, nd 0 m. Is it ute, right, or otuse? otuse lterntive ssessment Hve students use the Pthgoren Theorem to find the length of the digonl of their noteook pper nd eplin in writing how the Pthgoren Theorem ws used. Then hve them mesure the digonl to onfirm the length found using the Pthgoren Theorem.. r nd q. So = r nd = q. + = r + q = (r + q) = Rel-World Reserh Edwin Hule (9 9), here guiding telesope in 9, led to the ig ng Theor of the formtion of the universe.. nswers m vr. Smple: n 6;,, 7. (n) + (n ) = n + n n + = n + n + = (n + ) P onnetion hllenge z O Q hpter Right Tringles nd Trigonometr Find integers j nd k so tht () the two given integers nd j represent the lengths of the sides of n ute tringle nd () the two given integers nd k represent the lengths of the sides of n otuse tringle nswers m vr. Smples re given. 0., 6; 7., ;. 6, 9 ;., 0 ;. 6, 7 ; 0. 9, ; 6 6., 7 ; , 0 9; Proof. Prove the Pthgoren Theorem. Given: # is right tringle Prove: + = (Hint: egin with proportions suggested Theorem 7- or its orollries.) 9. stronom The Hule Spe Telesope is oriting Erth 600 km ove Erth s 600 km surfe. Erth s rdius is out 670 km. Use the Pthgoren Theorem to find the distne from the telesope to Erth s horizon. Round our nswer to the nerest ten kilometers. 0 km The figures elow re drwn on entimeter grid pper. Find the perimeter of eh shded figure to the nerest tenth. 0. m.. GPS. m.. The nient Greek philosopher Plto used the epressions n, n -, nd n + to produe Pthgoren triples. hoose n integer greter thn. Sustitute for n nd evlute the three epressions. ± 7. Verif tht our nswers to prt () form Pthgoren triple.. Show tht, in generl, (n) (n ) (n ) for n n. See left.. Geometr in Dimensions The o t the right is D retngulr solid. d in.. Use # to find the length d of the digonl of the se. in.. Use #D to find the length d of the digonl of d in. the o. "9 in.. You n generlize the steps in prts () nd (). Use the fts tht + = d nd d "D d + D = d to write one-step formul to find d. d. Use the formul ou wrote to find the length of the longest fishing pole ou n pk in o with dimensions in., in., nd 6 in. in. Geometr in Dimensions Points P(,, z ) nd Q(,, z ) t the left re points in three-dimensionl oordinte sstem. Use the following formul to find PQ. Leve our nswer in simplest rdil form. d = "( ) ( ) (z z ). P(0, 0, 0), Q(,, ) 6. P(0, 0, 0), Q(-,, -6) 7. P(-,, ), Q(,, 7) " "6 "7 q D r not to sle 670 km 7.9 m

7 GEOM.0 GEOM.0 Mied Review GO for Help Lesson -7 Lesson - Lesson - Proof Multiple hoie Prtie. Use the pln nd write proof of Theorem -, the onverse of the Pthgoren Theorem. Given: # with sides of length,, nd where + = Prove: # is right tringle. Pln: Drw right tringle (not #) with legs of lengths nd. Lel the hpotenuse. the Pthgoren Theorem, + =. Use sustitution to ompre the lengths of the sides of our tringle nd #. Then prove the tringles ongruent. See mrgin. For liforni Stndrds nd HSEE Tutorils, visit PHShool.om. We ode: q highw detour ffets ompn s deliver route. The pln showing the old route nd the detour is t the right. How mn etr miles will the truks trvel one the detour is estlished? Determine the vlue of in the figure t the right. 9 For the figure t the right, omplete the proportion j j 7 6. Find the vlues of nd. =, = old shipping route ) In the seond figure, PS isets lrpt. Solve for R eh vrile. Then find RS. 7; P 6. RS = + 9, ST = 7-6; = 7, RS = 7 ; S 6. RS = (7 - ), ST = + ; = 7, RS = 7 T #PQR O#STV. Solve for eh vrile. 66. m&p = w +, m&s = 6w RQ = 0-6, VT = m&t = - 0, m&q = PR = z +, SV = z miles 7. miles 0 0 liforni Resoures Stndrds Mster liforni Dil Review Trnsprenies liforni Review nd Prtie Workook. Drw right kfde with legs DE of length nd EF of length, nd hp. of length. Then ± the Pthgoren Thm. We re given k with sides of length,, nd ±. sust.,, so. Sine ll side lengths of k nd kfde re the sme, k OkFDE SSS. l OlE PT, so ml 90. Therefore, k is right k. Eerise hllenge students to prove the onverse of the Pthgoren Theorem using oordinte methods. Given: + = Prove (using slopes) n = m m n, or n = m m. Proof: = m + n ; = ( m) + n ; = + = m + n + ( m) + n = m + n + m m + + n, so 0 = m + n m, whih simplifies to n = m m. (m, n) O (0, 0) (, 0) nline Visit: PHShool.om, We ode: -00