The Pythagorean Theorem

Transcription

1 CHPTER 9 The Pythgoren Theorem CHPTER 9 OBJECTIVES But serving up n ction, suggesting the dynmic in the sttic, hs become hobby of mine....the flowing on tht motionless plne holds my ttention to such degree tht my preference is to try nd mke it into cycle. M. C. ESCHER Wterfll, M. C. Escher, Cordon rt B. V. Brn Hollnd. ll rights reserved. OBJECTIVES In this chpter you will discover the Pythgoren Theorem, one of the most importnt concepts in mthemtics use the Pythgoren Theorem to clculte the distnce between ny two points use conjectures relted to the Pythgoren Theorem to solve problems Understnd the Pythgoren Theorem more deeply Discover the Converse of the Pythgoren Theorem Prctice working with rdicl epressions Discover reltionships mong the lengths of the sides of tringle nd mong the lengths of the sides of tringle pply the Pythgoren Theorem nd its converse Discover nd pply the Pythgoren reltionship on coordinte plne (the distnce formul) Derive the eqution of circle from the distnce formul Prctice using geometry tools Develop reding comprehension, problem-solving skills, nd coopertive behvior Lern new vocbulry Escher hs cleverly used right ngles to form his rtwork known s Wterfll. The picture contins three uses of the impossible tribr creted by British mthemticin Roger Penrose (b 9) in 954. In 94 Swedish rtist Oscr Reutersvrd (b 95), fther of impossible figures, hd creted n impossible tribr tht consisted of tringulr rrngement of cubes. The shpes topping the towers in Escher s work re, on the left, compound of three cubes nd, on the right, stelltion of the rhombic dodechedron. [sk] Wht impossible things do you see? [Wter seems to be trveling up n incline, yet it is running mill wheel.] Which surfces pper to be horizontl? Verticl? Sloped? There re three impossible tribrs in the picture; where re they? [They ll hve flowing wter long two sides; twice one of the brs is replced by the wterfll, nd once one br is replced by group of four columns.] Penrose tribr CHPTER 9 The Pythgoren Theorem 46

2 LESSON 9. PLNNING LESSON OUTLINE One dy: 5 min Investigtion 5 min Shring 0 min Emples 5 min Closing nd Eercises LESSON 9. I m not young enough to know everything. OSCR WILDE The Theorem of Pythgors In right tringle, the side opposite the right ngle is clled the hypotenuse. The other two sides re clled legs. In the figure t right, nd b represent the lengths of the legs, nd c represents the length of the hypotenuse. In right tringle, the side opposite the right ngle is clled the hypotenuse, here with length c. c b The other two sides re legs, here with lengths nd b. MTERILS construction tools scissors Pythgoren Theorem (W) for One step Dissection of Squres (W), optionl Sketchpd demonstrtion Three Tringles, optionl FUNKY WINKERBEN by Btiuk. Reprinted with specil permission of North meric Syndicte. There is specil reltionship between the lengths of the legs nd the length of the hypotenuse. This reltionship is known tody s the Pythgoren Theorem. Investigtion The Three Sides of Right Tringle TECHING Mny students my lredy know the Pythgoren Theorem s b c. In this lesson they review wht the letters stnd for nd discover proofs showing why the reltionship holds for ll right tringles. INTRODUCTION Direct students ttention to Improving Your Visul Thinking Skills on pge 454. sk wht they cn conclude bout right tringles, nd help them stte the Pythgoren Theorem using res of squres nd the terms hypotenuse nd legs. You will need scissors compss strightedge ptty pper Step Step Step Step 4 The puzzle in this investigtion is intended to help you recll the Pythgoren Theorem. It uses dissection, which mens you will cut prt one or more geometric figures nd mke the pieces fit into nother figure. Construct sclene right tringle in the middle of your pper. Lbel the hypotenuse c nd the legs nd b. Construct squre on ech side of the tringle. To locte the center of the squre on the longer leg, drw its digonls. Lbel the center O. Through point O, construct line j perpendiculr to the hypotenuse nd line k perpendiculr to line j. Line k is prllel to the hypotenuse. Lines j nd k divide the squre on the longer leg into four prts. Cut out the squre on the shorter leg nd the four prts of the squre on the longer leg. rrnge them to ectly cover the squre on the hypotenuse. c b j O k Guiding the Investigtion One step Hnd out copy of the Pythgoren Theorem worksheet to ech group. Chllenge students to cut up one or both of the smller squres nd ssemble the pieces on top of the lrgest squre. s you circulte, you might remind students of the problem-solving technique of 46 CHPTER 9 The Pythgoren Theorem trying specil cse first here, n isosceles right tringle. s needed, point out tht good pieces might be formed if they drw lines through the smller squres prllel to edges of the lrgest squre. [Lnguge] dissection is the result of seprting something into pieces. Step Using the Dissection of Squres worksheets or the Sketchpd demonstrtion will speed the investigtion, but the use of mny different tringles drwn by the students strengthens the inductive conclusion. The constructions re quicker with ptty pper thn with compss nd strightedge. It is lso esy to crete severl emples using geometry softwre. Step s needed, remind students tht the legs re the sides other thn the hypotenuse, so the longer leg is not the hypotenuse. Suggest tht students minimize clutter by mking these digonls very light or by drwing only the portion ner the center of the squre. Step 4 sk students to tke cre in drwing nd cutting out pieces so they will fit together well. Students my wnt to tpe the pieces together.

3 b b c NCTM STNDRDS c CONTENT Number lgebr c Geometry Mesurement Dt/Probbility c b Step 5 Stte the Pythgoren Theorem. SHRING IDES The Pythgoren Theorem C-8 You might mke trnsprency of the Dissection of Squres In right tringle, the sum of the squres of the lengths of the legs equls the worksheets for students to use squre of the length of the hypotenuse. If nd b re the lengths of the legs, in presenting their ides. nd c is the length of the hypotenuse, then?. b c sk bout symmetry in the dissected squre on the hypotenuse. The method of the Investigtion gives 4-fold rottionl History symmetry. [sk] Wht if the tringle isn t Pythgors of Smos (c B.C.E.), right tringle? Do you think depicted in this sttue, is often described s the first pure mthemticin. Smos ws there s still reltionship mong principl commercil center of Greece nd is the lengths of the sides? You need locted on the islnd of Smos in the egen not nswer this question now; Se. The ncient town of Smos now lies in it s ddressed lter. [Link] The ruins, s shown in the photo t right. Pythgoren Theorem is specil Mysteriously, none of Pythgors s writings still eist, nd we know very little bout his life. He cse of the Lw of Cosines founded mthemticl society in Croton, in c b b cos C, wht is now Itly, whose members discovered irrtionl numbers nd the five where C is the ngle opposite regulr solids. They proved wht is now clled the Pythgoren Theorem, side c; when mc 90, we lthough it ws discovered nd used 000 yers erlier by the Chinese nd Bbylonins. Some mth historins believe tht the ncient Egyptins lso hve cos C 0. used specil cse of this property to construct right ngles. b theorem is conjecture tht hs been proved. Demonstrtions like the one in the investigtion re the first step towrd proving the Pythgoren Theorem. Believe it or not, there re more thn 00 proofs of the Pythgoren Theorem. Elish Scott Loomis s Pythgoren Proposition, first published in 97, contins originl proofs by Pythgors, Euclid, nd even Leonrdo d Vinci nd U. S. President Jmes Grfield. One well-known proof of the Pythgoren Theorem is included below. You will complete nother proof s n eercise. Prgrph Proof: The Pythgoren Theorem You need to show tht b equls c for the right tringles in the figure t left. The re of the entire squre is b or b b. The re of ny tringle is b, so the sum of the res of the four tringles is b. The re of the qudrilterl in the center is b b b, or b. If the qudrilterl in the center is squre then its re lso equls c.you now need to show tht it is squre. You know tht ll the sides hve length c, but you lso need to show tht the ngles re right ngles. The two cute ngles in the right tringle, long with ny ngle of the qudrilterl, dd up to 80. The cute ngles in right tringle dd up to 90. Therefore the qudrilterl ngle mesures 90 nd the qudrilterl is squre. If it is squre with side length c, then its re is c.so, b c, which proves the Pythgoren Theorem. PROCESS Problem Solving Resoning Communiction Connections Representtion LESSON OBJECTIVES Understnd the Pythgoren Theorem more deeply Prctice using geometry tools Lern new vocbulry [sk] Wht is the longest side of right tringle? Is it the sme s the longest leg? [The hypotenuse, not the longest leg, is the longest side.] If you sk why the longest side is lwys the hypotenuse nd wht cn be sid bout the longer of the two legs, you cn review the Tringle Inequlity Conjecture nd the Side-ngle Inequlity Conjecture. [sk] Wht is theorem? [It s conjecture tht hs been proved deductively within deductive system.] So fr in this course no iom system hs been developed, so there re no rel theorems; this conjecture is clled theorem in this book becuse it hs been proved within n iom system nd it s so well known by tht nme. If students re not very fmilir with the Pythgoren Theorem, sk how this theorem bout res of squres might be used to clculte lengths. Direct students ttention to the two emples. LESSON 9. The Theorem of Pythgors 46

4 EXMPLE fter working through the emple, sk students wht squre roots re nd how to find them using their clcultors. Remind them tht mny of the squre roots they ll find with their clcultors re pproimtions. [sk] Wht re some numbers whose squre roots re whole numbers? [4, 9, 6, 5, 6, nd so on] Students will begin to recognize more emples of perfect squres s they work through the chpter. The eqution h 75 is not perfect model for this problem becuse it hs two solutions, one positive nd one negtive. The negtive solution is ignored becuse ll distnces in geometry re positive. EXMPLE B In Emple, the book mentions tht 9.4 is n pproimtion of 75. Point out tht in Emple B the clcultion is ect. See whether students recognize 56 s perfect squre. gin, the negtive squre root is being ignored. Process tet (the reson for ech lgebric step) is included with ech step in Emple. In Emple B, it hs been left up to students to figure out wht lgebric steps re being crried out. [sk] Why does [this step] follow from the previous step? ssessing Progress You cn ssess students understnding of right tringle, squre, digonl, nd perpendiculr. lso wtch for their bility to follow instructions nd to work together in groups. See how well they relize when ect nswers re pproprite nd when they need to use pproimtions. cute tringle Obtuse tringle Closing the Lesson EXMPLE Solution EXMPLE B Solution The Pythgoren Theorem works for right tringles, but does it work for ll tringles? quick check demonstrtes tht it doesn t hold for other tringles. b 45 c 5 The Pythgoren Theorem is clim bout res of squres built on the sides of right tringle: The re of the squre on the hypotenuse is the sum of the res of the squres on the legs. Its primry pplictions re in finding the length of one side of right tringle in which the lengths of the other two sides re known. The theorem cn be proved by dissection. b 45 c 45 b c b c b c For n interctive version of this sketch, visit Let s look t few emples to see how you cn use the Pythgoren Theorem to find the distnce between two points. How high up on the wll will 0-foot ldder touch if the foot of the ldder is plced 5 feet from the wll? The ldder is the hypotenuse of right tringle, so b c. 5 h 0 Substitute. 5 h 400 Multiply. h 75 Subtrct 5 from both sides. h Tke the squre root of ech side. The top of the ldder will touch the wll bout 9.4 feet up from the ground. Notice tht the ect nswer in Emple is 75. However,this is prcticl ppliction, so you need to clculte the pproimte nswer. Find the re of the rectngulr rug if the width is feet nd the digonl mesures 0 feet. Use the Pythgoren Theorem to find the length. b c L 0 44 L 400 L 56 L 56 L 6 The length is 6 feet. The re of the rectngle is 6, or 9 squre feet. L b 45 c ft 0 ft h 5 ft ft 464 CHPTER 9 The Pythgoren Theorem

5 EXERCISES In Eercises, find ech missing length. ll mesurements re in centimeters. Give pproimte nswers ccurte to the nerest tenth of centimeter..? cm. c? 9. cm.? 5 c 5 5. cm 8 6 BUILDING UNDERSTNDING fter students hve solved some eercises, you my wnt to hve severl groups report on how they solved them. SSIGNING HOMEWORK 4. d? 0 cm 5. s? 6 cm 6. c? 8 s d s c 8.5 cm Essentil 6 Portfolio 7, 8 Journl 7, 8 Group 8 Review 9 7. b? 4 cm 8.?.6 cm 9. The bse is circle.? 40 cm Helping with the Eercises 0. s?.5 cm. r? s 5 5 b 8 7 cm y. bsebll infield is squre, ech side mesuring 90 feet. To the nerest foot, wht is the distnce from home plte to second bse? 7 ft. The digonl of squre mesures meters. Wht is the re of the squre? 5 m 4. Wht is the length of the digonl of squre whose re is 64 cm?. cm 5. The lengths of the three sides of right tringle re consecutive integers. Find them., 4, 5 6. rectngulr grden 6 meters wide hs digonl mesuring 0 meters. Find the perimeter of the grden. 8 m.5 (0, 0) r.9.5 (5, ) 9 4 Second bse Home plte [lert] s needed, remind students tht the hypotenuse is lwys the longest side nd is opposite the right ngle. Eercise 4 Students might wonder why the upper tringle is right tringle. [sk] Wht kind of figure is the qudrilterl? Do you see congruent tringles? Eercise 7 [lert] Some students my wnt to use the mesure 8, which is not needed in the clcultion. Eercise 0 If students re hving difficulty, sk wht shpe the qudrilterl is. Eercise This is good preprtion for work with the unit circle in trigonometry. Eercises 6 Students my find it helpful to drw nd lbel pictures. Eercise s n etension, you might hve students mesure the lengths of the distnces between the bses on locl bsebll field. LESSON 9. The Theorem of Pythgors 465

6 7. The re of the lrge squre is 4 re of tringle re of smll squre. c 4 b (b ) c b b b c b 7. One very fmous proof of the Pythgoren Theorem is by the Hindu mthemticin Bhskr. It is often clled the Behold proof becuse, s the story goes, Bhskr drew the digrm t right nd offered no verbl rgument other thn to eclim, Behold. Use lgebr to fill in the steps, eplining why this digrm proves the Pythgoren Theorem. History b b Bhskr (4 85, Indi) ws one of the first mthemticins to gin thorough understnding of number systems nd how to solve equtions, severl centuries before Europen mthemticins. He wrote si books on mthemtics nd stronomy, nd led the stronomicl observtory t Ujjin. Eercise 8 This problem illustrtes specil cse in which SS is congruence shortcut. Tht is, when the non-included ngle is right ngle, there is only one tringle tht cn be formed by SS. This shortcut is sometimes clled HL (Hypotenuse-Leg). Students cn prove HL using the thinking process used in the solution to this eercise. Given two right tringles with congruent, corresponding hypotenuses with length c nd corresponding legs with length, use the Pythgoren Theorem to show tht the other two corresponding legs re congruent; the tringles re congruent by SSS. Students will prove the HL Theorem in Lesson.7. Mrk the unnmed ngles s shown in the figure below. By the Liner Pir Conjecture, p p 60. By I, m q. By the Tringle Sum Conjecture, q p n 80. Substitute m q nd p 60 to get m 60 n 80. m n 0. c Is BC XYZ? Eplin your resoning. Smple nswer: Yes, BC XYZ by SSS. Both tringles re right tringles, so you cn use the Pythgoren Theorem to find tht CB ZY cm. Review 9. The two qudrilterls re squres Give the verte rrngement for the Find. cm -uniform tesselltion. 6 / cm. Eplin why m n Clculte ech lettered ngle, mesure, or rc. EF is dimeter; nd re tngents. m 0 n 6 cm 4 cm 5 cm d C B v s X u f 4 cm F n g E e 5 cm t h c Z r b 06 Y 58 m 0 p q n., b 74, c 06, d 6, e 90, f 74, g 74, h 74, n 74, r, s 74, t 74, u, v 74 Or use the Eterior ngle Conjecture q n 0. By I q m. Substituting, m n CHPTER 9 The Pythgoren Theorem

7 CRETING GEOMETRY FLIP BOOK Hve you ever fnned the pges of flip book nd wtched the pictures seem to move? Ech pge shows picture slightly different from the previous one. Flip books re bsic to nimtion technique. For more informtion bout flip books, see These five frmes strt off the photo series titled The Horse in Motion, by photogrpher, innovtor, nd motion picture pioneer Edwerd Muybridge (80 904). Here re two dissections tht you cn nimte to demonstrte the Pythgoren Theorem. (You used nother dissection in the Investigtion The Three Sides of Right Tringle.) b You could lso nimte these drwings to demonstrte re formuls. [Contet] Edwerd Muybridge, the mn who creted The Horse in Motion, ws born in Englnd but moved to the United Sttes s boy. s prt of scientific reserch to improve techniques of horsercing, he devised n ingenious method of setting up dozen or more cmers to go off in rpid sequence. He ws leder in the erly dys of photogrphy, improving on eisting methods, including the rt of film developing. He invented precursor to the movie projector bsed on the ide of zoetrope, slitted drum with pictures on the inside tht is spun to show the illusion of motion when the pictures re wtched through the slits. EXTENSIONS. Hve students reserch nd try one of numerous other dissections tht demonstrte the Pythgoren Theorem. B. Use Tke nother Look ctivity,, or on pges Choose one of the nimtions mentioned bove nd crete flip book tht demonstrtes it. Be redy to eplin how your flip book demonstrtes the formul you chose. Here re some prcticl tips. Drw your figures in the sme position on ech pge so they don t jump round when the pges re flipped. Use grph pper or trcing pper to help. The smller the chnge from picture to picture, nd the more pictures there re, the smoother the motion will be. Lbel ech picture so tht it s cler how the process works. Supporting the Suggest tht students use smll grph pper tblet to help keep the nonmoving fetures in the sme position from pge to pge. They cn glue trcing ppers of the flip book onto crds for firmer flip. OUTCOMES Movement in the flip book is smooth. The student cn eplin the dissection used. The student dds other nimtion, for emple, drwing hnd flipping the pges of flip book, thus creting flip book of flip book! n nimtion is creted using geometry softwre. LESSON 9. The Theorem of Pythgors 467

8 LESSON 9. PLNNING LESSON OUTLINE One dy: 0 min Investigtion 0 min Shring 5 min Closing 0 min Eercises MTERILS rulers string (two -meter strings per group) pper clips (si per group) protrctors ptty pper TECHING The Converse of the Pythgoren Theorem cn lso be proved. Guiding the Investigtion One step Pose this problem: Wht cn you sy bout tringles in which the side lengths stisfy the eqution b c? s you circulte, encourge groups to try vrious triples from the list in the student book or to mke up their own lengths, perhps not integers. s needed, remind students tht the converse of true sttement might be flse. The Pythgoren triples re rrnged in fmilies. [sk] How re the triples in column relted? [They re multiples of the first triple in tht column.] Wht would be the net triple in the second column? [5, 6, 9] Why is -4-5 or 5-- clled primitive Pythgoren triple? [The numbers in the triple hve no common fctor.] LESSON 9. ny time you see someone more successful thn you re, they re doing something you ren t. MLCOLM X You will need string ruler pper clips piece of ptty pper Step Step Step The Converse of the Pythgoren Theorem In Lesson 9., you sw tht if tringle is right tringle, then the squre of the length of its hypotenuse is equl to the sum of the squres of the lengths of the two legs. Wht bout the converse? If, y, nd z re the lengths of the three sides of tringle nd they stisfy the Pythgoren eqution, b c,must the tringle be right tringle? Let s find out. Investigtion Is the Converse True? You might let the clss do this investigtion outside, with the students themselves cting s rope stretchers of long ropes. Step Suggest tht leving ecess string t both ends will mke it esier to tie the ends together in Step. Three positive integers tht work in the Pythgoren eqution re clled Pythgoren triples. For emple, is Pythgoren triple becuse Here re nine sets of Pythgoren triples Select one set of Pythgoren triples from the list bove. Mrk off four points,, B, C, nd D, on string to crete three consecutive lengths from your set of triples. 8 cm 5 cm 7 cm B C D Loop three pper clips onto the string. Tie the ends together so tht points nd D meet. Three group members should ech pull pper clip t point, B, or C to stretch the string tight. LESSON OBJECTIVES 6 7 Discover the Converse of the Pythgoren Theorem Lern new vocbulry Develop reding comprehension, problem-solving skills, nd coopertive behvior 468 CHPTER 9 The Pythgoren Theorem

9 Step 4 right tringle Step 4 With your pper, check the lrgest ngle. Wht type of tringle is formed? Step 5 Select nother set of triples from the list. Repet Steps 4 with your new lengths. Step 6 Compre results in your group. Stte your results s your net conjecture. Converse of the Pythgoren Theorem Step 4 Students cn lso use corner of piece of pper (or of some other object such s crpenter s squre) to verify right ngle. C-8 If the lengths of the three sides of tringle stisfy the Pythgoren eqution,?. is right tringle then the tringle This ncient Bbylonin tblet, clled Plimpton, dtes sometime between 900 nd 600 B.C.E. It suggests severl dvnced Pythgoren triples, such s SHRING IDES Hve students show their work with vriety of lengths. Let the clss discuss mesurement errors. Elicit the ide tht slight error in the mesurement of length cn result in mesurbly different ngle. Keep sking students whether they think the converse is lwys true. gree on sttement of the conjecture without resolving the question of truth. [sk] Will the conjecture hold if you use different mesurement units so tht the triples chnge, possibly to non-integers? History Some historins believe Egyptin rope stretchers used the Converse of the Pythgoren Theorem to help reestblish lnd boundries fter the yerly flooding of the Nile nd to help construct the pyrmids. Some ncient tombs show workers crrying ropes tied with eqully spced knots. For emple, eqully spced knots would divide the rope into equl lengths. If one person held knots nd together, nd two others held the rope t knots 4 nd 8 nd stretched it tight, they could hve creted -4-5 right tringle. [sk] Suppose one tringle hs sides whose lengths re Pythgoren triple nd nother tringle hs sides whose lengths re multiple of tht Pythgoren triple. How re the tringles relted? [They re similr.] Students cn try drwing such tringles nd looking for ptterns, but you need not nswer the question yet. It foreshdows the ides of similrity in Chpter. lso sk whether students who believe the conjecture is true cn prove it deductively. Then hve the clss red the proof outline in the student book. NCTM STNDRDS CONTENT PROCESS Number Problem Solving lgebr Resoning Geometry Communiction Mesurement Connections Dt/Probbility MKING THE CONNECTION Pythgors himself is believed to hve studied in Egypt, nd he my hve lerned the tringle reltionship there. lthough some historins discount the rope stretchers tle, there s no doubt tht Egyptin mthemticins knew the reltionship. Representtion LESSON 9. The Converse of the Pythgoren Theorem 469

10 Proof sk students to critique this outline in order to deepen their understnding of it. s they discuss it, monitor their fcil epressions nd try to include students who seem to hve ides but ren t speking up. You my wnt to let the discussion led to filling in detils s clss nd then writing up good model proof. Or, if your students re firly comfortble with proof, you cn chllenge them to write up the detils s homework. ssessing Progress You cn check how well students cn mesure lengths nd ngles, eperiment systemticlly, nd follow deductive proof. Closing the Lesson The Converse of the Pythgoren Theorem is true nd cn be used to determine right ngles. common proof ctully uses the Pythgoren Theorem itself. BUILDING UNDERSTNDING These eercises help students prctice both the Pythgoren Theorem nd its converse. SSIGNING HOMEWORK Essentil 7, 9 5, 9, 0 Performnce ssessment 8 Portfolio 6 Journl 0 Group 7, 8 Review 5 Helping with the Eercises Eercise This is good plce to point out the power of recognizing Pythgoren multiples. The triple is bsed on the Pythgoren primitive 5--, so sides of those lengths form right tringle. EXERCISES In Eercises 6, use the Converse of the Pythgoren Theorem to determine whether ech tringle is right tringle.. yes. yes. no 8 4. no 5. no 6. no The proof of the Converse of the Pythgoren Theorem is very interesting becuse it is one of the few instnces where the originl theorem is used to prove the converse. Let s tke look. One proof is strted for you below. You will finish it s n eercise. Proof: Converse of the Pythgoren Theorem Conjecture: If the lengths of the three sides of tringle work in the Pythgoren eqution, then the tringle is right tringle. Given:, b, c re the lengths of the sides of BC nd b c Show: BC is right tringle Pln: Begin by constructing second tringle, right tringle DEF (with F right ngle), with legs of lengths nd b nd hypotenuse of length. The pln is to show tht c, so tht the tringles re congruent. Then show tht C nd F re congruent. Once you show tht C is right ngle, then BC is right tringle nd the proof is complete. 0 0 In Eercises 7 nd 8, use the Converse of the Pythgoren Theorem to solve ech problem. 7. Is tringle with sides mesuring 9 feet, feet, nd 8 feet right tringle? no 8. window frme tht seems rectngulr hs height 408 cm, length 06 cm, nd one digonl with length 55 cm. Is the window frme relly rectngulr? Eplin. No, the given lengths re not Pythgoren triple. Eercise 7 Students justifictions might cite Pythgoren multiples. The numbers 9,, nd 8 hve common fctor of, so primitive would be This is not Pythgoren triple; the fmilir -4-5 triple sys tht if the two legs hve lengths nd 4, the hypotenuse must hve length 5 to give right tringle B E 6 5 C F.7.4. Eercise 8 fter students decide tht the ngle isn t right, you might sk whether students could hve known the window frme ws rectngulr if the ngle hd turned out to be right. Hving one right ngle is not sufficient condition for qudrilterl to be rectngle. c b b D 470 CHPTER 9 The Pythgoren Theorem

11 .8 In Eercises 9, find y. y 4 units 9. Both qudrilterls 0. y. y 7. m re squres. y 5 cm ( 7, y). The lengths of the three sides of right. Find the re of right tringle with tringle re consecutive even integers. hypotenuse length 7 cm nd one leg Find them. 6, 8, 0 length 5 cm. 60 cm 4. How high on building will 5-foot ldder 5. The congruent sides of n isosceles tringle touch if the foot of the ldder is 5 feet from mesure 6 cm, nd the bse mesures 8 cm. the building? 4. ft Find the re. 7.9 cm 6. Find the mount of fencing in liner feet needed for the perimeter of rectngulr lot with digonl length 9 m nd side length 6 m. 0 m 7. rectngulr piece of crdbord fits snugly on digonl in this bo.. Wht is the re of the crdbord rectngle? 44 cm b. Wht is the length of the digonl of the crdbord rectngle? 74.8 cm 8. Look bck t the strt of the proof of the Converse of the Pythgoren Theorem. Copy the conjecture, the given, the show, the pln, nd the two digrms. Use the pln to complete the proof. 9. Wht s wrong with this picture? 0. Eplin why BC is right tringle..75 cm Review 5 cm y 5 cm cm 4.5 cm Smple nswer: The numbers given stisfy the Pythgoren Theorem, so the tringle is right tringle; but the right ngle should be inscribed in n rc of 80. Thus the tringle is not right tringle.. Identify the point of concurrency from the construction mrks. centroid 8. Becuse DEF is right tringle, b. By substitution, c nd c. Therefore, EFD BC by SSS nd C F by CPCTC. Hence, C is right ngle nd BC is right tringle. 5 6 m B 40 cm m D 9 m C 5 m y 60 cm 8 m 0 cm Smple nswer: BD 6 7; BC BD 9 08; then B BC C ( ), so BC is right tringle by the Converse of the Pythgoren Theorem. Eercise 9 Here is nother sitution in which Pythgoren triples cn be pplied. For the tringle with y s the hypotenuse, the legs hve lengths 5 cm nd 0 cm. Ech length hs common fctor of 5 cm. Becuse -4-5 is common Pythgoren primitive, the hypotenuse must hve length 5(5 cm) 5 cm. Eercise 0 This eercise provides groundwork for Chpter work with the unit circle in trigonometry. Eercises 6 Encourge students to drw pictures. Eercise Students might miss the condition tht the integers re even. If pproprite, [sk] If is the first even integer, how would the second integer be described? [ ] Is there only one triple of consecutive even integers? [Hlfofny such triple is triple of consecutive integers, sy,,, nd. The lgebric eqution ( ) ( ) hs only two solutions: 0 nd 4.] Eercise 5 [sk] In n isosceles tringle, wht does n ltitude from the verte ngle to the bse do to the bse? [It is the perpendiculr bisector of the bse; it is the sme s the medin.] Eercise 6 [Lnguge] Liner feet refers to the length of the fence, wheres squre feet mesures re. Eercise 7 [lert] Some students will hve difficulty visulizing the bo. They might crete three-dimensionl model. Students re being grdully introduced to the Pythgoren Theorem in three dimensions. s in Eercises nd 6, ectness of nswers will vry. The nswers given ssume tht mesurements given in the eercises re ect nd thus nswers to the nerest squre cm or nerest tenth of cm re resonble. In rel life the ectness of n nswer depends on how it will be used nd further knowledge of given mesurements. LESSON 9. The Converse of the Pythgoren Theorem 47

12 Eercise This eercise cn be pproched through finding tht the complement of is 90, through using the properties of n isosceles right tringle, or by considering the limit s secnt line through C rottes to become the tngent line (nd the intercepted rc becomes the rc with mesure ). Eercise If students re hving difficulty, sk how they might count tringles systemticlly. One pproch is to consider where the verte ngle might go. Symmetry helps. Eercise 4 s needed, focus students ttention on the number of squre fces being dded t ech step. EXTENSIONS. Pose this problem: If the sum of the squres of the lengths of the two shorter sides of tringle is less thn the squre of the length of the longest side, wht cn you conjecture bout the ngle opposite the longest side? If the sum of the squres of the lengths of the two shorter sides of tringle is greter thn the squre of the length of the longest side, wht cn you conjecture bout the ngle opposite the longest side? [If the sum is less, the ngle is obtuse. If the sum is greter, the ngle is cute.] B. Hve students use geometry softwre to drw tringles. Hve them lbel nd mesure ngles nd sides, clculte squres of the lengths of the sides or construct squres on the sides nd mesure their re, nd drg vertices until the sum of the squres of the lengths of the two smllest sides equls the squre of the length of the lrgest side. Students should find right ngle. See pge 775 for nswers to Eercises nd Line CF is tngent to circle D t C.. Wht is the probbility of rndomly selecting The rc mesure of CE is. Eplin three points tht form n isosceles tringle why. from the 0 points in this isometric grid? 0 E..8 D C 4. If the pttern of blocks continues, wht will be the surfce re of the 50th solid in the pttern? 790 squre units 5. Sketch the solid shown, but with the two blue cubes removed nd the red cube moved to cover the visible fce of the green cube. IMPROVING YOUR LGEBR SKILLS lgebric Sequences III F The outlines of stcked cubes crete visul impct in this untitled module unit sculpture by conceptul rtist Sol Lewitt. Find the net three terms of this lgebric sequence. 9,9 8 y,6 7 y,84 6 y, 6 5 y 4, 6 4 y 5,84 y 6,?,?,? IMPROVING LGEBR SKILLS Students might think of number combintions, Pscl s tringle, or symmetry. Or they might see this pttern: fter the first term, ech coefficient cn be determined by multiplying the previous coefficient by the eponent on in the previous term nd then dividing by the number of the term (counting from zero). For emple, the term tht includes 6 is the third term, so its coefficient is 7( 6) 84. The net three terms in the sequence re 6 y 7,9y 8,nd y CHPTER 9 The Pythgoren Theorem

13 LGEBR SKILLS USING USING YOUR YOUR LGEBR SKILLS 8USING YOUR LGEBR SKILLS 8 USING YO EXMPLE Solution NCTM STNDRDS CONTENT Number lgebr Geometry Mesurement Dt/Probbility Rdicl Epressions When you work with the Pythgoren Theorem, you often get rdicl epressions, such s 50. Until now you my hve left these epressions s rdicls, or you my hve found deciml pproimtion using clcultor. Some rdicl epressions cn be simplified. To simplify squre root mens to tke the squre root of ny perfect-squre fctors of the number under the rdicl sign. Let s look t n emple. Simplify 50. One wy to simplify squre root is to look for perfect-squre fctors. The lrgest perfect-squre fctor of 50 is nother pproch is to fctor the number s fr s possible with prime fctors. Write 50 s set of prime fctors. 5 is perfect squre, so you cn tke its squre root. Look for ny squre fctors (fctors tht pper twice) Squring nd tking the squre root re inverse opertions they undo ech other. So, 5 equls 5. You might rgue tht 5 doesn t look ny simpler thn 50. However,in the dys before clcultors with squre root buttons, mthemticins used pper-ndpencil lgorithms to find pproimte vlues of squre roots. Working with the smllest possible number under the rdicl mde the lgorithms esier to use. Giving n ect nswer to problem involving squre root is importnt in number of situtions. Some ptterns re esier to discover with simplified squre roots thn with deciml pproimtions. Stndrdized tests often epress nswers in simplified form. nd when you multiply rdicl epressions, you often hve to simplify the nswer. PROCESS Problem Solving Resoning Communiction Connections Representtion LESSON OBJECTIVES Lern to simplify squre roots Lern to multiply rdicl epressions USING YOUR LGEBR SKILLS 8 PLNNING LESSON OUTLINE One dy or prtil dy: 5 min Emples 0 min Eercises MTERILS none TECHING Being ble to work with rdicl epressions will help students lter see ptterns in specil right tringles. One step Drw right tringle with legs lbeled with lengths 4 nd 8. sk wht multiple of 4 the length of the hypotenuse is. Students might divide the hypotenuse into four pieces nd look t squres of ech piece. Or they might rewrite 80 s product of 4 nd squre root. During Shring, sk bout working bckwrd, s in Emple B. INTRODUCTION [Lnguge] The symbol for the nonnegtive squre root,, is n emple of rdicl. Roots to other powers re lso clled rdicls. EXMPLE You might give definition such s Prime fctors re fctors tht cn t be broken down into smller fctors. Emphsize tht 5 equls 50 ectly; it is not n pproimtion. USING YOUR LGEBR SKILLS 8 Rdicl Epressions 47

14 EXMPLE B You my wnt to mention tht the commuttive property of multipliction llows you to rewrite 6 5 s 5 6. SHRING IDES You might sk the clss to critique this shortcut for Emple s shortcut for ll cses: 5 5 immeditely becomes 5 becuse you cn move ny fctor tht ppers twice inside the rdicl to one ppernce outside the rdicl. Keep sking for justifiction, using the generl rules bout squre roots of products nd of squres. sk whether the reltionship b b cn be represented by lengths, remembering tht squre roots re often represented by sides of squres. This question foreshdows work with re rtios in Lesson.5. ssessing Progress You cn ssess students previous understnding of rdicl epressions s well s their bility to see nd generlize ptterns. Closing the Lesson Two rules re useful in rewriting squre roots nd in multiplying rdicl epressions: The (positive) squre root of product is the product of the squre roots, nd the squre root of the squre of number is the (bsolute vlue of) the number. LGEBR SKILLS 8 USING YOUR LGEBR SKILLS 8 USING YOUR LGEBR SKILLS 8 USING Y EXMPLE B Solution EXERCISES Multiply 6 by 5. To multiply rdicl epressions, ssocite nd multiply the quntities outside the rdicl sign, nd ssocite nd multiply the quntities inside the rdicl sign In Eercises 5, epress ech product in its simplest form In Eercises 6 0, epress ech squre root in its simplest form Wht is the net term in the pttern? 95, 548, 800, 808, 607, IMPROVING YOUR VISUL THINKING SKILLS Folding Cubes II Ech cube hs designs on three fces. When unfolded, which figure t right could it become?.. B. C. D... B. C. D BUILDING UNDERSTNDING Though students re tught to simplify rdicl epressions, they re not tught to rtionlize the denomintors. You might decide to tech tht lso. SSIGNING HOMEWORK Essentil Group odds 0 evens Helping with the Eercises Eercise Squring nd tking the squre root re opposite opertions. Eercise 9 It is possible to fctor 85 s 5 7, but neither fctor is perfect squre. Eercise If students re mystified, refer them to Eercises 6 0. IMPROVING VISUL THINKING SKILLS If students re hving difficulty, point out tht good problem-solving technique is to eliminte s mny choices s possible. For emple, cn be eliminted becuse the bse of the green fce is touching the red fce, unlike its position in the originl cube.. D. C 474 CHPTER 9 The Pythgoren Theorem

15 LESSON 9. In n isosceles tringle, the sum of the squre roots of the two equl sides is equl to the squre root of the third side. THE SCRECROW IN THE 99 FILM THE WIZRD OF OZ NCTM STNDRDS CONTENT Number lgebr Geometry Mesurement Dt/Probbility Step Step Step Step 4 Two Specil Right Tringles In this lesson you will use the Pythgoren Theorem to discover some reltionships between the sides of two specil right tringles. One of these specil tringles is n isosceles right tringle, lso clled tringle. Ech isosceles right tringle is hlf squre, so they show up often in mthemtics nd engineering. In the net investigtion, you will look for shortcut for finding the length of n unknown side in tringle. Investigtion Isosceles Right Tringles Sketch n isosceles right tringle. Lbel the legs l nd the hypotenuse h. Pick ny integer for l, the length of the legs. Use the Pythgoren Theorem to find h. Simplify the squre root. Repet Step with severl different vlues for l. Shre results with your group. Do you see ny pttern in the reltionship between l nd h? Stte your net conjecture in terms of length l. Isosceles Right Tringle Conjecture In n isosceles right tringle, if the legs hve length l, then the hypotenuse hs length?. l PROCESS Problem Solving Resoning Communiction Connections Representtion LESSON OBJECTIVES? 45 n isosceles right tringle h l?? Prctice simplifying squre roots l? C-84 Discover reltionships mong the lengths of the sides of tringle nd tringle Develop problem-solving skills nd coopertive behvior LESSON 9. PLNNING LESSON OUTLINE One dy: 5 min Investigtion 0 min Shring 5 min Closing 5 min Eercises MTERILS squre nd isometric dot pper TECHING Two specil right tringles occur so often in rel life (nd on college entrnce ems nd chievement tests) tht it s good to know the rtios of their side lengths. One step sk students to drw on dot pper right tringles in which the two legs re congruent nd right tringles in which the hypotenuse hs length twice tht of one of the legs. They should look for the length of the third sides of these tringles nd mesure the ngles. Guiding Investigtion Step If students re hving difficulty seeing pttern, suggest tht they systemticlly try consecutive integers nd mke tble of the results. LESSON 9. Two Specil Right Tringles 475

16 Guiding Investigtion Steps Students cn use ptty pper to nswer some of the questions in Steps nd without doing the mesurements in Step. They might lso recll tht every ltitude is medin. SHRING IDES fter the clss reches consensus bout wht conjectures to record in their notebooks, sk how to restte the conjectures using rtios. [For tringle, the side lengths hve the rtio ::; for tringle, they hve the rtio ::.] sk whether these rtios cn be represented geometriclly, remembering tht squre roots re often shown s sides of squres. Students cn drw the tringles on dot pper (or isometric dot pper) nd clculte the res of pproprite squres. [sk] How do you know which ngle is the 0 ngle in tringle? [By the Side-ngle Inequlity Conjecture, it s the ngle opposite the shortest side.] [sk] How cn these specil tringles be constructed with strightedge nd compss? [Students cn construct perpendiculr lines nd then ly out equl segments from the intersection point to get tringle. Now tht they know the Tringle Conjecture, they cn construct one leg nd hypotenuse to determine the third side.] Wonder loud whether the Tringle Conjecture cn be proved for ll tringles, even if none of the lengths re integers. fter students hve mde suggestions, direct their ttention to the proof in the student book, sking them to critique nd rewrite the resoning in order to understnd it better tringle Step 60 ; 0 ; 90 Step Step yes, SS, S, Step or S Step Step Yes, CPCTC. C D; yes ll tringles re similr. Step 4 Step 5 You cn lso demonstrte this property on geobord or grph pper, s shown t right. The other specil right tringle is tringle. If you fold n equilterl tringle long one of its ltitudes, the tringles you get re tringles tringle is hlf n equilterl tringle, so it lso shows up often in mthemtics nd engineering. Let s see if there is shortcut for finding the lengths of its sides. Investigtion 0º-60º-90º Tringles Let s strt by using little deductive thinking to find the reltionships in tringles. Tringle BC is equilterl, nd CD is n ltitude. Wht re m nd mb? Wht re mcd nd mbcd? Wht re mdc nd mbdc? Is DC BDC? Why? ssessing Progress ssess students bility to generte isosceles right tringles nd equilterl tringles, to simplify squre roots, to mesure ngles, to find ptterns, nd to follow deductive proof. lso check their understnding of ltitudes of isosceles tringles, SS, nd the Pythgoren Theorem. Is D BD? Why? How do C nd D compre? In tringle, will this reltionship between the hypotenuse nd the shorter leg lwys hold true? Eplin. Sketch tringle. Choose ny integer for the length of the shorter leg. Use the reltionship from Step nd the Pythgoren Theorem to find the length of the other leg. Simplify the squre root. Repet Step 4 with severl different vlues 7 for the length of the shorter leg. Shre results with your group. Wht is the reltionship between the lengths of the two legs? You should notice pttern in your nswers. Step 6 Stte your net conjecture in terms of the length of the shorter leg, Tringle Conjecture In tringle, if the shorter leg hs length, then the longer leg hs length? nd the hypotenuse hs length?. Closing the Lesson C D B? 4?? 4 C-85 In tringle, if the legs hve length l, then the hypotenuse hs length l. In tringle, if the leg opposite the 0 ngle hs length, then the hypotenuse hs length nd the other leg hs length. 476 CHPTER 9 The Pythgoren Theorem

17 EXERCISES You cn use lgebr to verify tht the conjecture will hold true for ny tringle. Proof: Tringle Conjecture b Strt with the Pythgoren Theorem. 4 b Squre. b Subtrct from both sides. b Tke the squre root of both sides. lthough you investigted only integer vlues, the proof shows tht ny number, even non-integer, cn be used for. You cn lso demonstrte this property for integer vlues on isometric dot pper. 4 6 In Eercises 8, use your new conjectures to find the unknown lengths. ll mesurements re in centimeters..? 7 cm. b? cm.?, b? 0 cm, 5 cm You will need 0 Construction tools for Eercises 9 nd 0 b BUILDING UNDERSTNDING The eercises give prctice with the two specil right tringles. SSIGNING HOMEWORK Essentil Performnce ssessment 8 Portfolio 7 Group 6 Review 9 Helping with the Eercises Eercise If students ren t sure wht to do, suggest tht in tringle it often helps to locte the shortest side first. [sk] Where is the 0-degree ngle? b b 4. c?, d? 5. e?, f? 6. Wht is the perimeter of 0 cm, 0 cm 4 cm, 7 cm squre SQRE? E 7 cm R f 8 d c e 0 S Q LESSON 9. Two Specil Right Tringles 477

18 Eercise 7 This is nother emple of using the Pythgoren Theorem in three dimensions. [sk] How would you generlize the Pythgoren Theorem to three dimensions? [In right rectngulr prism, the spce digonl (d) cn be found from the three dimensions of the prism (, b, c): d b c. Eercise 0 In situtions in which you wnt to find the coordintes of point, it s often useful to drw segments whose lengths re those coordintes. [Link] Students will work with the unit circle in trigonometry.. possible nswer:. possible nswer: 6 7. The solid is cube. 8. g?, h? 9. Wht is the re of d? cm 50 cm, 00 cm the tringle? 6 cm E D cm H d B F 0. Find the coordintes of P.. Wht s wrong with this picture? y tringle must hve, sides whose lengths re multiples of P (?,?),, nd. The tringle shown 8 5 does not reflect this rule. 45 (0, ) G C (, 0) h 0. Sketch nd lbel figure to demonstrte tht 7 is equivlent to. (Use isometric dot pper to id your sketch.). Sketch nd lbel figure to demonstrte tht is equivlent to 4. (Use squre dot pper or grph pper.) 4. In equilterl tringle BC, E, BF, nd CD re ll ngle bisectors, medins, nd ltitudes simultneously. These three segments divide the equilterl tringle into si overlpping tringles nd si smller, non-overlpping tringles.. One of the overlpping tringles is CDB. Nme the other five tringles tht re congruent to it. CD, EC, EB, BF, BFC b. One of the non-overlpping tringles is MD. Nme the other five tringles congruent to it. MDB, MEB, MEC, MFC, MF g F M C D 8 E B 4 4 Eercise 6 sk students how to find specil tringle in picture of this sitution. Eercise 8 Students using geometry softwre might discover tht letting the hypotenuses (not the right ngles) coincide produces slightly lrger tringle Use lgebr nd deductive resoning to show tht the Isosceles Right Tringle Conjecture holds true for ny isosceles right tringle. Use the figure t right. 6. Find the re of n equilterl tringle whose sides mesure 6 meters. 69 m 7. n equilterl tringle hs n ltitude tht mesures 6 meters. Find the re of the tringle to the nerest squre meter. 90 m 8. Sketch the lrgest tringle tht fits in tringle. Wht is the rtio of the re of the tringle to the re of the tringle? 5. c Strt with the Pythgoren Theorem. c c Combine like terms. Tke the squre root of both sides. c CHPTER 9 The Pythgoren Theorem

19 Review Construction In Eercises 9 nd 0, choose either ptty pper or compss nd strightedge nd perform the constructions. 9. Given the segment with length below, construct segments with lengths,, nd Construct n isosceles right tringle with legs of length, construct tringle with legs of lengths nd, nd construct right tringle with legs of lengths nd Mini-Investigtion Drw right tringle with sides of lengths 6 cm, 8 cm, nd 0 cm. Locte the midpoint of ech side. Construct semicircle on ech side with the midpoints of the sides s centers. Find the re of ech semicircle. Wht reltionship do you notice mong the three res? 9.. The Jiuzhng sunshu is n ncient Chinese mthemtics tet of 46 problems. Some solutions use the gou gu, the Chinese nme for wht we cll the Pythgoren Theorem. The gou gu reds gou gu (in). Here is gou gu problem trnslted from the ninth chpter of Jiuzhng. rope hngs from the top of pole with three chih of it lying on the ground. When it is tightly stretched so tht its end just touches the ground, it is eight chih from the bse of the pole. How long is the rope? 7 6. chih.6. Eplin why m m The lterl surfce re of the cone below is unwrpped into sector. Wht is the ngle t the verte of the sector? 80 IMPROVING YOUR VISUL THINKING SKILLS Mudville Monsters l 7 cm, r 6 cm The strting members of the Mudville Monsters footbll tem nd their coch, Osgood Gipper, hve been invited to compete in the Smllville Punt, Pss, nd Kick Competition. To get there, they must cross the deep Smllville River. The only wy cross is with smll bot owned by two very smll Smllville footbll plyers. The bot holds just one Monster visitor or the two Smllville plyers. The Smllville plyers gree to help the Mudville plyers cross if the visitors gree to py $5 ech time the bot crosses the river. If the Monsters hve totl of $00 mong them, do they hve enough money to get ll plyers nd the coch to the other side of the river? l r l? Eercise 0 This mini-investigtion foreshdows re rtios in Lesson res: 4.5 cm,8 cm,.5 cm , tht is, the sum of the res of the semicircles on the two legs is equl to the re of the semicircle on the hypotenuse.. Etend the rys tht form the right ngle. m4 m5 80 by the Liner Pir Conjecture, nd it s given tht m5 90. m4 90. m m m4 m m m m 90. m m by I. m m EXTENSION Use Tke nother Look ctivity 4 or 5 on pge IMPROVING VISUL THINKING SKILLS Ech Monster visitor must go cross lone. It mkes no sense for one to return, so before ech Monster crosses, the two Smllville plyers must cross nd leve one on the other side to bring the bot bck on the net trip. This will require round trips for the plyers nd the coch. The tem hs enough money for 0 one-wy trips. Even without seeing how to rrnge the Smllville plyers to return the bot t lest times, the Monsters don t hve enough money. LESSON 9. Two Specil Right Tringles 479