Right Triangles and Trigonometry. Right Triangles and Trigonometry

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1 Right Tringles nd Trigonometr hpter Overview nd Pcing PING (ds) Regulr lock sic/ sic/ verge dvnced verge dvnced Geometric Men (pp. ) Find the geometric men etween two numers. Solve prolems involving reltionships etween prts of right tringle nd the ltitude to its hpotenuse. The Pthgoren Theorem nd Its onverse (pp. 9 ) (with 7- (with 7-0. Preview: Phsicll eplore side-length reltionships in the Pthgoren Theorem. Preview) Preview) Use the Pthgoren Theorem. Use the converse of the Pthgoren Theorem. LESSON OJETIVES Yer-long pcing: pges T0 T. Specil Right Tringles (pp. 7 ) Use properties of tringles. Use properties of tringles. Trigonometr (pp. 70) Find trigonometric rtios using right tringles. Solve prolems using trigonometric rtios. ngle of Elevtion nd epression (pp. 7 7) 0. Solve prolems involving ngles of elevtion. Solve prolems involving ngles of depression. The Lw of Sines (pp. 77 ) optionl (with 7- optionl (with 7- Use the Lw of Sines to solve tringles. Follow-Up) Follow-Up) Solve prolems using the Lw of Sines. Follow-Up: Use geometr softwre to investigte the Lw of Sines. The Lw of osines (pp. 9) optionl optionl Use the Lw of osines to solve tringles. Solve prolems using the Lw of osines. Follow-Up: Use trigonometric rtios to identif trigonometric identities. Stud Guide nd Prctice Test (pp. 9 97) Stndrdized Test Prctice (pp. 9 99) hpter ssessment TOTL 0 7 n electronic version of this chpter is ville on StudentWorks TM. This ckpck solution -ROM llows students instnt ccess to the Student Edition, lesson worksheet pges, nd we resources. 0 hpter 7 Right Tringles nd Trigonometr

2 Timesving Tools hpter Resource Mnger ll-in-one Plnner nd Resource enter See pges T nd T. Stud Guide nd Intervention HPTER 7 RESOURE MSTERS Prctice (Skills nd verge) Reding to Lern Mthemtics Enrichment ssessment Prerequisite Skills Workook pplictions* -Minute heck Trnsprencies, 7-7- Interctive hlkord GeomPSS: Tutoril Plus (lessons) Mterils , G (Preview: ptt pper, ruler) S grid pper, scissors , 09 S 7-7- pper, scissors, metric ruler, protrctor G , , 0 *Ke to revitions: G Grphing lcultor nd omputer Msters S School-to-reer Msters hpter 7 Right Tringles nd Trigonometr 0

3 Mthemticl onnections nd ckground ontinuit of Instruction Prior Knowledge In previous courses, students evluted rdicl epressions nd equtions involving frctions. The Pthgoren Theorem is reviewed in hpter. In hpter, students solve proportions. hpter introduces students to the ngle Sum Theorem. Geometric Men The geometric men etween two numers is the squre root of their product. For two positive numers nd, the geometric men is the positive numer for which the proportion : : is true. This proportion is equivlent to. The geometric men hs prticulr ppliction to right tringle. If n ltitude is drwn from the verte of the right ngle of right tringle to its hpotenuse, then the two tringles formed re similr to the given tringle nd to ech other. The mesure of this ltitude is the geometric men etween the mesures of the two segments of the hpotenuse. Moreover, the mesure of leg of the tringle is the geometric men etween the mesures of the hpotenuse nd the segment of the hpotenuse djcent to tht leg. This hpter This chpter provides students with n introduction to trigonometr. Students lern how to use the geometric men to solve prolems involving side length. The solve prolems using the Pthgoren Theorem nd its converse. Trigonometric rtios re defined nd then used to solve right tringle prolems. Students lso use the Lw of Sines nd the Lw of osines to solve non-right tringles. The Pthgoren Theorem nd Its onverse The Pthgoren Theorem sttes tht in right tringle, the sum of the squres of the mesures of the legs equls the squre of the mesure of the hpotenuse. The converse of the Pthgoren Theorem is useful in determining whether given mesures re those of right tringle. If the sum of the squres of the mesures of two sides of tringle equls the squre of the mesure of the longest side, then the tringle is right tringle. Pthgoren triple is group of three whole numers tht stisf the eqution c, where c is the gretest numer. Future onnections Students gin sic knowledge of trigonometric rtios from this chpter. This knowledge will help them in higher-level mth courses, including pre-clculus nd clculus. rchitects, surveors, nd civil engineers use trigonometric rtios in their work. Specil Right Tringles tringle is the onl tpe of isosceles right tringle. One of its specil properties is tht the hpotenuse is times s long s leg, so the rtio of the sides is :: tringle lso hs specil properties. The mesures of the sides re,, nd, giving the sides rtio of ::. Knowing these properties cn sve ou vlule time when ou re solving prolems involving specil right tringles. 0 hpter 7 Right Tringles nd Trigonometr

4 Trigonometr rtio of the lengths of the sides of right tringle is clled trigonometric rtio. The three most common trigonometric rtios re sine, cosine, nd tngent, revited sin, cos, nd tn. Sine of is the mesure of the leg opposite divided the mesure of the hpotenuse. osine of is the mesure of the leg djcent divided the mesure of the hpotenuse. Tngent of is the mesure of the leg opposite the ngle divided the mesure of the leg djcent the ngle. The vlue of this rtio does not depend on the size of the tringle or the mesures of the sides. Trigonometric rtios re used to find missing mesures of right tringle. You onl need to know the mesures of two sides or the mesure of one side nd one cute ngle. The inverse of ech trigonometric rtio ields the ngle mesure. The inverses re written sin, cos, nd tn. ngle of Elevtion nd epression n ngle of elevtion is the ngle etween the line of sight nd the horizontl when n oserver looks upwrd. n ngle of depression is the ngle etween the line of sight nd the horizontl when n oserver looks downwrd. Trigonometric rtios cn e used to solve prolems involving ngles of elevtion nd depression. ngles of elevtion nd depression to two different ojects cn e used to find the difference etween those ojects. The Lw of Sines In trigonometr, the Lw of Sines cn e used to find missing mesures of tringles tht re not right tringles. Let e n tringle with,, nd c representing the mesures of the sides opposite the ngles with mesures,, nd, respectivel. Then sin sin sin. The Lw of Sines cn e used c to solve tringle. This mens finding the mesure of ever side nd ngle. The Lw of Sines cn e used to solve tringle if ou know the mesures of two ngles nd n side of tringle, or if ou know the mesures of two sides nd n ngle opposite one of these sides of the tringle. There is one cse in which using the Lw of Sines to solve tringle will ield n miguous solution. If ou know the mesures of two sides nd n ngle opposite one of the sides (SS) nd the ngle is cute, it is possile to find two different tringles. One tringle will e n cute tringle, nd the other will e otuse. The Lw of osines The Lw of osines llows ou to solve tringle in some situtions when the Lw of Sines cnnot e used. This occurs when ou know the mesures of two sides nd the included ngle (SS) or three sides (SSS). Let e n tringle with,, nd c representing the mesures of sides opposite ngles with mesures,, nd, respectivel. Then the following equtions re true: c c cos c c cos c cos hpter 7 Right Tringles nd Trigonometr 0

5 nd ssessment Ke to revitions: TWE = Techer Wrpround Edition; RM = hpter Resource Msters Tpe Student Edition Techer Resources Technolog/Internet SSESSMENT INTERVENTION Ongoing Prerequisite Skills, pp.,,,, 70, 7, Prctice Quiz, p. Prctice Quiz, p. Mied Review Error nlsis Stndrdized Test Prctice Open-Ended ssessment hpter ssessment pp.,,, 70, 7,, 90 Find the Error, pp.,, 0 ommon Misconceptions, p. 7 pp.,,, 70, 7, 7, 7,, 90, 9, 9, 99 Writing in Mth, pp.,,, 9, 7,, 9 Open Ended, pp.,, 0, 7, 7, 0, 7 Stndrdized Test, p. 99 Stud Guide, pp. 9 9 Prctice Test, p. 97 -Minute heck Trnsprencies Prerequisite Skills Workook, pp., 7,,, Quizzes, RM pp Mid-hpter Test, RM p. 09 Stud Guide nd Intervention, RM pp., 7,, 9 70, 7 7,, 7 umultive Review, RM p. 0 Find the Error, TWE pp.,, 0 Unlocking Misconceptions, TWE p. 9 Tips for New Techers, TWE p. TWE pp Stndrdized Test Prctice, RM pp. Modeling: TWE pp., 7 Speking: TWE pp., 70 Writing: TWE pp.,, 90 Open-Ended ssessment, RM p. 0 Multiple-hoice Tests (Forms,, ), RM pp. 9 9 Free-Response Tests (Forms,, ), RM pp Voculr Test/Review, RM p. 0 GeomPSS: Tutoril Plus, Lessons,,, nd self_check_quiz etr_emples Stndrdized Test Prctice -ROM stndrdized_test EmView Pro (see elow) MindJogger Videoquizzes voculr_review chpter_test Geometr Lesson 7- Geometric Men 7- Yerl ProgressPro Skill Lesson The Pthgoren Theorem nd Its onverse 7- Specil Right Tringles 7- Trigonometr 7- ngles of Elevtion nd epression 7- The Lws of Sines 7-7 The Lw of osines For more informtion on Yerl ProgressPro, see p. 7. EmView Pro Use the networkle EmView Pro to: rete multiple versions of tests. rete modified tests for Inclusion students. Edit eisting questions nd dd our own questions. Use uilt-in stte curriculum correltions to crete tests ligned with stte stndrds. ppl rt to our test from progrm nk of rtwork. For more informtion on Intervention nd ssessment, see pp. T T. 0E hpter 7 Right Tringles nd Trigonometr

6 Reding nd Writing in Mthemtics Glencoe Geometr provides numerous opportunities to incorporte reding nd writing into the mthemtics clssroom. Student Edition Foldles Stud Orgnizer, p. oncept heck questions require students to verlize nd write out wht the hve lerned in the lesson. (pp.,, 0, 7, 7, 0, 7) Writing in Mth questions in ever lesson, pp.,,, 9, 7,, 9 Reding Stud Tip, p. WeQuest, pp., 90 Techer Wrpround Edition Foldles Stud Orgnizer, pp., 9 Stud Noteook suggestions, pp., 9,, 0, 7, 7, 0, Modeling ctivities, pp., 7 Speking ctivities, pp., 70 Writing ctivities, pp.,, 90 ifferentited Instruction (Verl/Linguistic), p. ELL Resources, pp. 0,,,,, 7,,, 9, 9 dditionl Resources Voculr uilder worksheets require students to define nd give emples for ke voculr terms s the progress through the chpter. (hpter 7 Resource Msters, pp. vii-viii) Proof uilder helps students lern nd understnd theorems nd postultes from the chpter. (hpter 7 Resource Msters, pp. i ) Reding to Lern Mthemtics mster for ech lesson (hpter 7 Resource Msters, pp.,, 7, 7, 79,, 9) Voculr PuzzleMker softwre cretes crossword, jumle, nd word serch puzzles using voculr lists tht ou cn customize. Teching Mthemtics with Foldles provides suggestions for promoting cognition nd lnguge. Reding Strtegies for the Mthemtics lssroom WeQuest nd Project Resources For more informtion on Reding nd Writing in Mthemtics, see pp. T T7. Venn digrm cn help students understnd the commonlities nd differences in two or more concepts. Show students the emple elow tht compres nd contrsts the Lw of Sines nd the Lw of osines. While studing hpter 7, hve students work in groups to mke Venn digrms for specil right tringles (Lesson 7-) nd for ngles of elevtion nd depression (Lesson 7-). Lw of Sines Lw of osines sin sin sin c used to solve n = + c c cos = + c c cos c = + cos hpter 7 Right Tringles nd Trigonometr 0F

7 Notes Hve students red over the list of ojectives nd mke list of n words with which the re not fmilir. Point out to students tht this is onl one of mn resons wh ech ojective is importnt. Others re provided in the introduction to ech lesson. Lessons 7-, 7-, nd 7- Solve prolems using the geometric men, the Pthgoren Theorem, nd its converse. Lessons 7- nd 7- Use trigonometric rtios to solve right tringle prolems. Lessons 7- nd 7-7 Solve tringles using the Lw of Sines nd the Lw of osines. Trigonometr is used to find the mesures of the sides nd ngles of tringles. These rtios re frequentl used in rel-world pplictions such s rchitecture, vition, nd surveing. You will lern how surveors use trigonometr in Lesson 7-. Ke Voculr geometric men (p. ) Pthgoren triple (p. ) trigonometric rtio (p. ) Lw of Sines (p. 77) Lw of osines (p. ) NTM Locl Lesson Stndrds Ojectives 7-,,,, 9, 0 7-, Preview 7-,,, 9, 0 7-,,, 9, 0 7-,,, 9, 0 7-,,, 9, 0 7-,,, 9, 0 7-, Follow-Up 7-7,,, 9, 0 7-7, Follow-Up Ke to NTM Stndrds: =Numer & Opertions, =lger, =Geometr, =Mesurement, =t nlsis & Proilit, =Prolem Solving, 7=Resoning & Proof, =ommuniction, 9=onnections, 0=Representtion 0 hpter 7 Right Tringles nd Trigonometr o emmrich/the Imge Works Voculr uilder ELL The Ke Voculr list introduces students to some of the min voculr terms included in this chpter. For more thorough voculr list with pronuncitions of new words, give students the Voculr uilder worksheets found on pges vii nd viii of the hpter 7 Resource Msters. Encourge them to complete the definition of ech term s the progress through the chpter. You m suggest tht the dd these sheets to their stud noteooks for future reference when studing for the hpter 7 test. 0 hpter 7 Right Tringles nd Trigonometr

8 h. 7 Right Tringles Prerequisite Skills To e successful in this chpter, ou ll need to mster these skills nd e le to ppl them in prolem-solving situtions. Review these skills efore eginning hpter 7. For Lesson 7- Proportions Solve ech proportion. Round to the nerest hundredth, if necessr. (For review, see Lesson -.). c... e 0 f 0. z e, f., z 0.7 For Lesson 7- Pthgoren Theorem Find the mesure of the hpotenuse of ech right tringle hving legs with the given mesures. Round to the nerest hundredth, if necessr. (For review, see Lesson -.). nd. nd 0 7. nd.. nd 7 0. For Lessons 7- nd 7- Rdicl Epressions Simplif ech epression. (For review, see pges 7 nd 7.) For Lessons 7- through 7-7 ngle Sum Theorem Find. (For review, see Lesson -.). 9.. ( + ) 0 Fold Fold ech sheet long the digonl from the corner of one end to. inches w from the corner of the other end. Right Tringles nd Trigonometr Mke this Foldle to help ou orgnize our notes. egin with seven sheets of grid pper.. in. Stck Stck the sheets, nd fold the rectngulr prt in hlf. This section provides review of the sic concepts needed efore eginning hpter 7. Pge references re included for dditionl student help. dditionl review is provided in the Prerequisite Skills Workook, pges, 7,,,. Prerequisite Skills in the Getting Red for the Net Lesson section t the end of ech eercise set review skill needed in the net lesson. For Prerequisite Lesson Skill 7- Using the Pthgoren Theorem, p. 7- Simplifing Rdicl Epressions, p. 7- Solving Equtions Tht Involve Frctions, p. 7- ngles Formed Prllel Lines nd Trnsversl, p Solving Proportions, p Evluting Epressions, p. Stple Stple the sheets in three plces. Lel Lel ech sheet with lesson numer, nd the rectngulr prt with the chpter title. 7- Reding nd Writing our Foldle. s ou red nd stud the chpter, write notes, define terms, nd solve prolems in hpter 7 Right Tringles nd Trigonometr For more informtion out Foldles, see Teching Mthemtics with Foldles. TM Journls nd Visuls Use this Foldle journl for writing out right tringles nd trigonometr. fter students mke their tringle-shped journl, hve them lel the front of ech tringle to correspond to the seven lessons in this chpter. Under the ts of their Foldle, students tke notes, define terms, solve prolems, nd write emples of how lws re used. On the front of ech section, sk students to design visul (grph, digrm, picture, chrt) tht presents the informtion introduced in the lesson in concise, es-to-stud formt. Encourge students to lel their visuls nd write cptions. hpter 7 Right Tringles nd Trigonometr

9 Lesson Notes Geometric Men Focus -Minute heck Trnsprenc 7- Use s quiz or review of hpter. Mthemticl ckground notes re ville for this lesson on p. 0. cn the geometric men e used to view pintings? sk students: Wht hppens if ou view the pinting t distnce tht is less thn the geometric men of the two distnces descried ove? You re too close to the pinting nd cnnot see ll of it t once. Your ees must move left, right, up, nd down to see ll the detils of the pinting. Nme some other scenrios where ou would e est positioned t geometric men to view something. wtching progrm on television or movie in thetre, viewing sports event, seeing production on stge, using computer, skimming ook or mgzine, etc. Voculr geometric men Stud Tip Mens nd Etremes In the eqution, the two s in represent the mens, nd nd represent the etremes of the proportion. Find the geometric men etween two numers. Solve prolems involving reltionships etween prts of right tringle nd the ltitude to its hpotenuse. cn the geometric men e used to view pintings? When ou look t pinting, ou should stnd t distnce tht llows ou to see ll of the detils in the pinting. The distnce tht cretes the est view is the geometric men of the distnce from the top of the pinting to ee level nd the distnce from the ottom of the pinting to ee level. GEOMETRI MEN The geometric men etween two numers is the positive squre root of their product. Emple Geometric Men Find the geometric men etween ech pir of numers.. nd 9 Let represent the geometric men. efinition of geometric men 9 ross products Tke the positive squre root of ech side. Simplif.. nd efinition of geometric men 90 ross products Tke the positive squre root of ech side. Simplif. Use clcultor. Geometric Men For two positive numers nd, the geometric men is the positive numer where the proportion : : is true. This proportion cn e written using frctions s or with cross products s or. Resource Mnger hpter 7 Right Tringles nd Trigonometr Roert renner/photoedit Workook nd Reproducile Msters hpter 7 Resource Msters Stud Guide nd Intervention, pp. Skills Prctice, p. Prctice, p. Reding to Lern Mthemtics, p. Enrichment, p. Prerequisite Skills Workook, pp., Trnsprencies -Minute heck Trnsprenc 7- nswer Ke Trnsprencies Technolog Interctive hlkord

10 Right Tringles Formed the ltitude Use The Geometer s Sketchpd to drw right tringle XYZ with right ngle Z. rw the ltitude ZW from the right ngle to the hpotenuse. Eplore the reltionships mong the three right tringles formed. Think nd iscuss. Find the mesures of X, XZY, Y, XWZ, XZW, YWZ, nd YZW. See students work.. Wht is the reltionship etween the mesures of X nd YZW? Wht is the reltionship etween the mesures of Y nd XZW? The re equl.. rg point Z to nother position. escrie the reltionship etween the mesures of X nd YZW nd etween the mesures of Y nd XZW. The re equl.. Mke conjecture out XYZ, XZW, nd ZYW. The re similr. Stud Tip ltitudes of Right Tringle The ltitude drwn to the hpotenuse origintes from the right ngle. The other two ltitudes of right tringle re the legs. LTITUE OF TRINGLE onsider right tringle Z XYZ with ltitude WZ drwn from the right ngle Z to the hpotenuse XY. specil reltionship eists for the three right tringles, XYZ, XZW, nd ZYW. X Y Z Y W X The results of the Geometr Softwre Investigtion suggest the following theorem. Theorem 7. If the ltitude is drwn from the verte of the right ngle of right tringle to its hpotenuse, then the two tringles formed re similr to the given tringle nd to ech other. Emple: XYZ XWY YWZ You will prove this theorem in Eercise. X W Y Z W Tech GEOMETRI MEN In-lss Emple Find the geometric men etween ech pir of numers.. nd 0 0. nd 7. In-lss Emple Power Point LTITUE OF TRINGLE Power Point Teching Tip Remind students tht the utomticll discrd the negtive root when finding the ltitude or geometric men ecuse these vlues represent lengths, nd lengths cnnot e negtive. In, nd 7. Find..7 Theorem 7., since XWY YWZ, the corresponding sides re proportionl. Thus, X W YW. Notice tht XW nd ZW re segments of the hpotenuse of the YW ZW lrgest tringle. Theorem 7. The mesure of n ltitude drwn from the verte of the right ngle of right tringle to its hpotenuse is the geometric men etween the mesures of the two segments of the hpotenuse. Emple: YW is the geometric men of XW nd ZW. You will prove this theorem in Eercise. Geometr Softwre Investigtion X W Y Lesson 7- Geometric Men Students should recognize tht if the were given just one ngle mesure other thn the right ngle, the would hve enough informtion to find ll the other ngles represented in the figure. The should lso see tht given one segment length, the could use proportions to find n of the other segment lengths in the figure s well. Z Interctive hlkord PowerPoint Presenttions This -ROM is customizle Microsoft PowerPoint presenttion tht includes: Step--step, dnmic solutions of ech In-lss Emple from the Techer Wrpround Edition dditionl, Tr These eercises for ech emple The -Minute heck Trnsprencies Hot links to Glencoe Online Stud Tools Lesson 7- Geometric Men

11 In-lss Emples Power Point KITES Ms. lspch is constructing kite for her son. She hs to rrnge perpendiculrl two support rods, the shorter of which is 7 inches long. If she hs to plce the short rod 7. inches from one end of the long rod in order to form two right tringles with the kite fric, wht is the length of the long rod? Stud Tip Squre Roots Since these numers represent mesures, ou cn ignore the negtive squre root vlue. Emple ltitude nd Segments of the Hpotenuse In PQR, RS nd QS. Find PS. Let PS. R S PS PS Q S. PS is out.. RS, QS, nd PS ross products Tke the positive squre root of ech side. Use clcultor. Rtios in right tringles cn e used to solve prolems. P R S Q 7 in. 7. in..9 in. Find c nd d in JKL. K L d 0 c J M c 0; d. Emple ltitude nd Length of the Hpotenuse RHITETURE Mr. Mrtinez is designing wlkw tht must pss over n elevted trin. To find the height of the elevted trin, he holds crpenter s squre t ee level nd sights long the edges from the street to the top of the trin. If Mr. Mrtinez s ee level is. feet ove the street nd he is.7 feet from the trin, find the distnce from the street to the top of the trin. Round to the nerest tenth. rw digrm. Let YX e the ltitude drwn from the right ngle of WYZ. W X Y X YX ZX..7 WX. nd YX.7. 7 ZX.ZX 7. ross products ZX.9 ivide ech side.. Mr. Mrtinez estimtes tht the elevted trin is..9 or out 9. feet high.. ft Y.7 ft Z X W The ltitude to the hpotenuse of right tringle determines nother reltionship etween the segments. Theorem 7. If the ltitude is drwn from the verte of the right ngle of right tringle to its hpotenuse, then the mesure of leg of the tringle is the geometric men etween the mesures of the hpotenuse nd the segment of the hpotenuse djcent to tht leg. XZ XY Emple: X Y X W nd X Z ZY Y Z W Z X W Y Z You will prove Theorem 7. in Eercise 7. hpter 7 Right Tringles nd Trigonometr ifferentited Instruction Intrpersonl llow students to sit quietl nd eplore similrities nd differences etween Theorem 7. nd Theorem 7.. Encourge students to use the emples in the ook or crete their own to reinforce the concepts outlined in these two theorems. sk students to think nd write out wh the formuls for geometric men work for right tringle with n ltitude drwn to its hpotenuse. hpter 7 Right Tringles nd Trigonometr

12 Stud Tip Simplifing Rdicls Rememer tht. Since,. For more prctice simplifing rdicls, see pges 7 nd 7. Emple Hpotenuse nd Segment of Hpotenuse Find nd in PQR. PQ nd RQ re legs of right tringle PQR. Use Theorem 7. to write proportion for ech leg nd then solve. PR P Q P Q PS PR RQ R Q SR PS, PQ, PR RS, RQ, PR ross products ross products Tke the squre root. Tke the squre root. Simplif. Simplif. S P. Use clcultor..9 Use clcultor. R Q Prctice/ppl Stud Noteook Hve students dd the definitions/emples of the voculr terms to their Voculr uilder worksheets for hpter 7. include n other item(s) tht the find helpful in mstering the skills in this lesson. oncept heck. Smple nswer: nd 7. In; his proportion shows tht the ltitude is the geometric men of the two segments of the hpotenuse. Guided Prctice GUIE PRTIE KEY Eercises Emples 7, 9 0, nswer. OPEN ENE Find two numers whose geometric men is.. rw nd lel right tringle with n ltitude drwn from the right ngle. From our drwing, eplin the mening of the hpotenuse nd the segment of the hpotenuse djcent to tht leg in Theorem 7.. See mrgin.. FIN THE ERROR RST is right isosceles tringle. Holl nd In re finding the mesure of ltitude SU. Holl In RS S U = SU R T 9.9 = =. =.. Who is correct? Eplin our resoning. Find the geometric men etween ech pir of numers.. 9 nd. nd 9. nd 7. nd.9. Find the mesure of the ltitude drwn to the hpotenuse For leg, is the segment of the hpotenuse tht shres n endpoint. Thus, it is the djcent segment. The sme is true for leg nd segment. R U SU = SU U T 7 = 7 = 9 = 7 G E H R 7 U 7 S F 9.9 Lesson 7- Geometric Men T.9 FIN THE ERROR sk students to nme the hpotenuse of RST, nd point out tht In correctl found the geometric men etween the two segments of the hpotenuse, s the rule sttes in Theorem 7.. out the Eercises Orgniztion Ojective Geometric Men: 0 ltitude of Tringle: Odd/Even ssignments Eercises re structured so tht students prctice the sme concepts whether the re ssigned odd or even prolems. lert! Eercise 0 requires the Internet or other reserch mterils. ssignment Guide sic: odd, odd,, verge: odd,, dvnced: even, (optionl: ) Lesson 7- Geometric Men

13 Stud Stud Guide nd nd Intervention Intervention, p. (shown) nd p. 7- NME TE PERIO Geometric Men Geometric Men The geometric men etween two numers is the squre root of their product. For two positive numers nd, the geometric men of nd is the positive numer in the proportion. ross multipling gives, so. Emple Find the geometric men etween ech pir of numers. Find nd. 0. ;. ;. nd Let represent the geometric men. efinition of geometric men or Eercises ross multipl. Tke the squre root of ech side.. nd Let represent the geometric men..7 Find the geometric men etween ech pir of numers.. nd. nd.9. nd nd. nd.. nd 0 7. nd.. 0 nd nd nd. nd. nd 7. Lesson 7- ppliction. NES Khlih is mking nner for the dnce committee. The nner is to e s high s the wll of the gmnsium. To find the height of the wll, Khlih held ook up to her ees so tht the top nd ottom of the wll were in line with the top edge nd inding of the cover. If Khlih s ee level is feet off the ground nd she is stnding feet from the wll, how high is the wll?. ft The geometric men nd one etreme re given. Find the other etreme.. is the geometric men etween nd. Find if.. is the geometric men etween nd. Find if. etermine whether ech sttement is lws, sometimes, or never true.. The geometric men of two positive numers is greter thn the verge of the two numers. never. If the geometric men of two positive numers is less thn, then oth of the numers re less thn. sometimes 7- NME TE PERIO Prctice (verge) Geometric Men Skills Prctice, p. nd Prctice, p. (shown) Find the geometric men etween ech pir of numers to the nerest tenth.. nd. 7 nd 7. nd Find the mesure of ech ltitude. Stte ect nswers nd nswers to the nerest tenth.. U. J M 7 T V Find,, nd z.. 7. z.;.7; z.;.; ; 0.; 7. ; ; IVIL ENGINEERING n irport, fctor, nd shopping center re t the vertices of right tringle formed three highws. The irport nd fctor re.0 miles prt. Their distnces from the shopping center re. miles nd. miles, respectivel. service rod will e constructed from the shopping center to the highw tht connects the irport nd fctor. Wht is the shortest possile length for the service rod? Round to the nerest hundredth.. mi Reding to Lern Mthemtics, p. 7- L NME TE PERIO Reding to Lern Mthemtics Geometric Men Pre-ctivit How cn the geometric men e used to view pintings? Red the introduction to Lesson 7- t the top of pge in our tetook. Wht is disdvntge of stnding too close to pinting? Smple nswer: You don t get good overll view. Wht is disdvntge of stnding too fr from pinting? Smple nswer: You cn t see ll the detils in the pinting. Reding the Lesson. In the pst, when ou hve seen the word men in mthemtics, it referred to the verge or rithmetic men of the two numers.. omplete the following writing n lgeric epression in ech lnk. If nd re two positive numers, then the geometric men etween nd is nd their rithmetic men is.. Eplin in words, without using n mthemticl smols, the difference etween the geometric men nd the lgeric men. Smple nswer: The geometric men etween two numers is the squre root of their product. The rithmetic men of two numers is hlf their sum.. Let r nd s e two positive numers. In which of the following equtions is z equl to the geometric men etween r nd s?,,, F. z s z r. r s r z z z. s: z z: r. z s E. z r z s F. z s r z. Suppl the missing words or phrses to complete the sttement of ech theorem.. The mesure of the ltitude drwn from the verte of the right ngle of right tringle to its hpotenuse is the geometric men etween the mesures of the two segments of the hpotenuse.. If the ltitude is drwn from the verte of the right ngle of right tringle to its hpotenuse, then the mesure of leg of the tringle is the geometric men etween the mesure of the hpotenuse nd the segment of the hpotenuse djcent to tht leg. c. If the ltitude is drwn from the verte of the right ngle of right tringle to its hpotenuse, then the two tringles formed re similr to the given tringle nd to ech other. z z 0 0 K ELL Prctice nd ppl For Eercises 0 7 See Emples, Etr Prctice See pge hpter 7 Right Tringles nd Trigonometr NME TE PERIO 7- Enrichment, p. Mthemtics nd Music Pthgors, Greek philosopher who lived during the sith centur.., elieved tht ll nture, eut, nd hrmon could e epressed wholenumer reltionships. Most people rememer Pthgors for his techings out right tringles. (The sum of the squres of the legs equls the squre of the hpotenuse.) ut Pthgors lso discovered reltionships etween the musicl notes of scle. These reltionships cn e epressed s rtios. E F G 9 Find the geometric men etween ech pir of numers.. nd. nd. nd 0. nd 7 7. nd. nd 9. nd 0. nd 7 7 Find the mesure of the ltitude drwn to the hpotenuse...7. F. J H. Q. V W. Z P R U T S Find,, nd z. 7. See mrgin z 0... z 0 z 9 E z G N. Y z z M L 0 K. X Helping You Rememer. good w to rememer new mthemticl concept is to relte it to something ou lred know. How cn the mening of men in proportion help ou to rememer how to find the geometric men etween two numers? Smple nswer: Write proportion in which the two mens re equl. This common men is the geometric men etween the two etremes. When ou pl stringed instrument, ou produce different notes plcing our finger on different plces on string. This is the result of chnging the length of the virting prt of the string. The string cn e used to produce F plcing finger of the w long the string. of string hpter 7 Right Tringles nd Trigonometr g hs len

14 The geometric men nd one etreme re given. Find the other etreme.. 7 is the geometric men etween nd. Find if is the geometric men etween nd. Find if..9 etermine whether ech sttement is lws, sometimes, or never true.. The geometric men for consecutive positive integers is the verge of the two numers. never. The geometric men for two perfect squres is positive integer. lws 7. The geometric men for two positive integers is nother integer. sometimes. The mesure of the ltitude of tringle is the geometric men etween the mesures of the segments of the side it intersects. sometimes nswers 9. FGH is right tringle. OG is the ltitude from the verte of the right ngle to the hpotenuse of tht tringle. So, Theorem 7., OG is the geometric men etween OF nd OH, nd so on. 0. Smple nswer: The golden rtio occurs when the geometric men is pproimtel.. 9. IOLOGY The shpe of the shell of chmered nutilus cn e modeled geometric men. onsider the sequence of segments O, O, O, O, OE, OF, OG, OH, OI, nd OJ. The length of ech of these segments is the geometric men etween the lengths of the preceding segment nd the succeeding segment. Eplin this reltionship. (Hint: onsider FGH.) See mrgin. J F G O E I H You cn use geometric men nd the Qudrtic Formul to discover the golden men. Visit online.com/wequest to continue work on our WeQuest project. 0. RESERH Refer to the informtion t the left. Use the Internet or other resource to write rief description of the golden rtio.. ONSTRUTION In the United Sttes, most uilding codes limit the steepness of the slope of roof to, s shown t the right. uilder wnts to put support rce from point perpendiculr to P. Find the length of the rce.. d d P d d SOER For Eercises nd, refer to the grphic.. Find the geometric men etween the numer of plers from Indin nd North rolin re there two schools whose geometric men is the sme s the geometric men etween UL nd lemson? If so, which schools? es; Indin nd Virgini. RITIL THINKING Find the ect vlue of E, given nd. US TOY Snpshots ruins ring skills to MLS Universities producing the most plers in Mjor Legue Soccer this seson: Source: MLS UL Indin Virgini North rolin Wshington lemson E Ellen J. Horrow nd drienne Lewis, US TOY Lesson 7- Geometric Men 7 Kz hi/photoisc nswers (p. ) 7. 9.;.7; z.9. 0 ; 0; z ; ; z ; ; z..7;.9; z ;.7; z.9 Lesson 7- Geometric Men 7

15 ssess Open-Ended ssessment Writing rw nd lel severl figures on the ord, nd sk students to write down the correct geometric men proportions without working out the prolem. Then hve volunteers come to the ord nd write pproprite proportions tht correspond with the figures, nd llow seted students to check their work. Getting Red for Lesson 7- Prerequisite Skill Students will review the Pthgoren Theorem nd lern out its converse in Lesson 7-. Use Eercises to determine our students fmilirit with using the Pthgoren Theorem. nswers. Smple nswer: The geometric men cn e used to help determine the optimum viewing distnce. nswers should include the following. If ou re too fr from pinting, ou m not e le to see fine detils. If ou re too close, ou m not e le to see the entire pinting. curtor cn use the geometric men to help determine how fr from the pinting the roping should e. Stndrdized Test Prctice Mintin Your Skills Mied Review.,,.,, Getting Red for the Net Lesson PROOF Write the specified tpe proof for ech theorem. 7. See p two-column proof. prgrph proof of 7. two-column proof of Theorem 7. of Theorem 7. of Theorem 7.. WRITING IN MTH nswer the question tht ws posed t the eginning of the lesson. See mrgin. How cn the geometric men e used to view pintings? Include the following in our nswer: n eplntion of wht hppens when ou re too fr or too close to pinting, nd n eplntion of how the curtor of museum would determine where to plce roping in front of pintings on displ. 9. Find nd. nd. nd 7.. nd. nd 7 0. LGER Solve 0. Find the first three itertions of ech epression. (Lesson -)., where initill equls., where initill equls., where initill equls. ( ), where initill equls 7, 7, 07,,. The mesures of the sides of tringle re 0,, nd 0. Find the mesures of the segments formed where the isector of the smllest ngle meets the opposite side. (Lesson -) 9, 9 Use the Eterior ngle Inequlit Theorem to list ll ngles tht stisf the stted condition. (Lesson -). ll ngles with mesure less thn m,,, 7. ll ngles with mesure greter thn m, 7. ll ngles with mesure less thn m7, 9. ll ngles with mesure greter thn m, 7, Write n eqution in slope-intercept form for the line tht stisfies the given conditions. (Lesson -) 0. m, -intercept. -intercept is, -intercept. psses through (, ) nd (, 0). m, psses through (, ) PREREQUISITE SKILL Use the Pthgoren Theorem to find the length of the hpotenuse of ech right tringle. (To review using the Pthgoren Theorem, see Lesson -.). cm. ft ft.. in. cm ft cm in. cm 0 cm cm cm 7 cm in. hpter 7 Right Tringles nd Trigonometr hpter 7 Right Tringles nd Trigonometr

16 The Pthgoren Theorem Preview of Lesson 7- In hpter, ou lerned tht the Pthgoren Theorem reltes the mesures of the legs nd the hpotenuse of right tringle. ncient cultures used the Pthgoren Theorem efore it ws officill nmed in 909. Use squre pieces of ptt pper nd lger. Then ou too cn discover this reltionship mong the mesures of the sides of right tringle. ctivit Use pper folding to develop the Pthgoren Theorem. On piece of ptt pper, mke mrk long one side so tht the two resulting segments re not congruent. Lel one s nd the other s. op these mesures on the other sides in the order shown t the right. Fold the pper to divide the squre into four sections. Lel the re of ech section. On nother sheet of ptt pper, mrk the sme lengths nd on the sides in the different pttern shown t the right. Use our strightedge nd pencil to connect the mrks s shown t the right. Let c represent the length of ech hpotenuse. Step Lel the re of ech section, which is for ech tringle nd c for the squre. Step Plce the squres side side nd color the corresponding regions tht hve the sme re. For emple,. The prts tht re not shded tell us tht c. Model. Use ruler to find ctul mesures for,, nd c. o these mesures mesures confirm tht c? es. Repet the ctivit with different nd vlues. Wht do ou notice? c nlze the model. Eplin wh the drwing t the right is n illustrtion of the Pthgoren Theorem. Smple nswer: The sum of the res of the two smller squres is equl to the re of the lrgest squre. = c Geometr ctivit The Pthgoren Theorem 9 c c c c c c c c c Resource Mnger Geometr ctivit Preview of Lesson 7- Getting Strted Ojective Phsicll eplore side-length reltionships in the Pthgoren Theorem. Mterils ptt pper, ruler, pencil Tech For Step, hve students use ruler nd pencil to drw the lines tht the creses mke. dvise students tht the ke to this ctivit is mking sure mesures nd re ectl the sme on oth sheets of pper. Without using ruler, students m use one mrked edge of the first pper to mrk ccurte lengths on the second pper. fter Step, some students m still e skepticl tht the shded res of the two pieces of ptt pper re equl. Point out tht the two pieces of pper re the sme size, so the hve the sme re. Then sk students to cut out ll the shded tringles from the second piece of pper nd rrnge them so tht the fit over the shded res on the first piece of pper. ssess Eercises nd provide students with ctul mesurements the cn use to confirm the Pthgoren Theorem. In Eercise, students prove tht three lengths meet the criteri for the Pthgoren Theorem. Teching Geometr with Mnipultives p. (student recording sheet) p. 7 (ruler) Glencoe Mthemtics lssroom Mnipultive Kit rulers Stud Noteook sk students to summrize wht the hve lerned out the Pthgoren Theorem. Geometr ctivit The Pthgoren Theorem 9

17 Lesson Notes The Pthgoren Theorem nd Its onverse Focus -Minute heck Trnsprenc 7- Use s quiz or review of Lesson 7-. Mthemticl ckground notes re ville for this lesson on p. 0. re right tringles used to uild suspension ridges? sk students: How cn ou reword the Pthgoren Theorem sustituting the nmes of the prts of the Tlmdge Memoril ridge for the legs nd hpotenuse of the tringle? The sum of the squres of the mesure of the height of the tower from the rodw nd length of the rodw from the tower to the end of the ridge equls the squre of the mesure of the cle. You m notice tht most suspension cles hve some slck in them. In this cse, does the ctul length of the cle stisf the criteri of the Pthgoren Theorem? Eplin. No; suspension cle is cut to llow for epnsion nd contrction due to temperture chnges, nd when it hs slck, its length ctull eceeds the length required to stisf the Pthgoren Theorem. Voculr Pthgoren triple Stud Tip Look ck To review finding the hpotenuse of right tringle, see Lesson -. Use the Pthgoren Theorem. Use the converse of the Pthgoren Theorem. re right tringles used to uild suspension ridges? The Tlmdge Memoril ridge over the Svnnh River hs two soring towers of suspension cles. Note the right tringles eing formed the rodw, the perpendiculr tower, nd the suspension cles. The Pthgoren Theorem cn e used to find mesures in n right tringle. THE PYTHGOREN THEOREM In Lesson -, ou used the Pthgoren Theorem to find the distnce etween two points finding the length of the hpotenuse when given the lengths of the two legs of right tringle. You cn lso find the mesure of n side of right tringle given the other two mesures. Theorem 7. Pthgoren Theorem In right tringle, the sum of the squres of the mesures of the legs equls the squre of the mesure of the hpotenuse. Smols: c Proof The geometric men cn e used to prove the Pthgoren Theorem. Pthgoren Theorem Given: with right ngle t Prove: c Proof: rw right tringle so is the right ngle. Then drw the ltitude from to. Let c,,,,, nd h. Two geometric mens now eist. c nd c c nd c ross products dd the equtions. c c c( ) Fctor. c Since c, sustitute c for ( ). h c c 0 hpter 7 Right Tringles nd Trigonometr Resource Mnger Workook nd Reproducile Msters hpter 7 Resource Msters Stud Guide nd Intervention, pp. 7 Skills Prctice, p. 9 Prctice, p. 0 Reding to Lern Mthemtics, p. Enrichment, p. ssessment, p. 07 Grphing lcultor nd omputer Msters, p. 9 School-to-reer Msters, p. Prerequisite Skills Workook, pp., Teching Geometr With Mnipultives Msters, pp., Trnsprencies -Minute heck Trnsprenc 7- nswer Ke Trnsprencies Technolog GeomPSS: Tutoril Plus, Lesson Interctive hlkord

18 Mps ue to the curvture of Erth, the distnce etween two points is often epressed s degree distnce using ltitude nd longitude. This mesurement closel pproimtes the distnce on plne. Source: NS Emple Find. (XY) (YZ) (XZ) Theorem 7. Find the Length of the Hpotenuse LONGITUE N LTITUE NS rden is locted t out 7 degrees longitude nd degrees ltitude. NS mes is locted t out degrees longitude nd 7 degrees ltitude. Use the lines of longitude nd ltitude to find the degree distnce to the nerest tenth etween NS rden nd NS mes. The chnge in longitude etween the two loctions is 7 or degrees. Let this distnce e. The chnge in ltitude is 7 or degrees ltitude. Let this distnce e. Use the Pthgoren Theorem to find the distnce in degrees from NS rden to NS mes, represented c. c Pthgoren Theorem c, 9 c Simplif. c dd. c Tke the squre root of ech side.. c Use clcultor. The degree distnce etween NS rden nd NS mes is out. degrees. Emple Find the Length of Leg Pthgoren Theorem 7 XY 7, XZ 9 9 Simplif. 7 Sutrct 9 from ech side. 7 Tke the squre root of ech side. 7 Simplif.. Use clcultor. onverse of the Pthgoren Theorem If the sum of the squres of the mesures of two sides of tringle equls the squre of the mesure of the longest side, then the tringle is right tringle. Smols: If c, then is right tringle. NS mes 7 in. X Y c in. NS rden 0 ONVERSE OF THE PYTHGOREN THEOREM The converse of the Pthgoren Theorem cn help ou determine whether three mesures of the sides of tringle re those of right tringle. in. Z Tech THE PYTHGOREN THEOREM In-lss Emples Power Point LONGITUE N LTITUE rson it, Nevd, is locted t out 0 degrees longitude nd 9 degrees ltitude. Use the lines of longitude nd ltitude to find the degree distnce to the nerest tenth degree if ou were to trvel directl from NS mes to rson it, Nevd. out. degrees Teching Tip Students cn lso rewrite the Pthgoren Theorem s c or c when the re using it to find the mesure of either leg of right tringle. Find d. P cm cm Q d cm R d. uilding on Prior Knowledge Students used the istnce Formul in hpter when the were lerning how to clculte segment lengths. In this lesson, the use this formul to verif the side lengths of right tringles. You will prove this theorem in Eercise. Lesson 7- The Pthgoren Theorem nd Its onverse StockTrek/Photoisc Lesson 7- The Pthgoren Theorem nd Its onverse

19 ONVERSE OF THE PYTHGOREN THEOREM In-lss Emples Power Point OORINTE GEOMETRY Verif tht is right tringle. ( 9, ) O (, ) Stud Tip istnce Formul When using the istnce Formul, e sure to follow the order of opertions crefull. Perform the opertion inside the prentheses first, squre ech term, nd then dd. Emple Verif Tringle is Right Tringle OORINTE GEOMETRY Verif tht PQR is right tringle. Use the istnce Formul to determine the lengths of the sides. PQ ( ) ) (,,, ( ) Sutrct. Simplif. QR )] [ ( ) (,,, ) ( Sutrct. Simplif. Q(, ) O R(, ) P(, ) (, 7) 0 ; ; ; is right tringle ecuse. etermine whether ech set of mesures re the sides of right tringle. Then stte whether the form Pthgoren triple.. 9,, nd The segments form the sides of right tringle, nd the mesures form Pthgoren triple..,, nd The segments do not form the sides of right tringle, nd the mesures do not form Pthgoren triple. c.,, nd The segments form the sides of right tringle, ut the mesures do not form Pthgoren triple. PR ( ) ( ) =, =, =, = Sutrct. Simplif. the converse of the Pthgoren Theorem, if the sum of the squres of the mesures of two sides of tringle equls the squre of the mesure of the longest side, then the tringle is right tringle. PQ PR QR onverse of the Pthgoren Theorem PQ, PR, QR Simplif. dd. Since the sum of the squres of two sides equls the squre of the longest side, PQR is right tringle. Pthgoren triple is three whole numers tht stisf the eqution c, where c is the gretest numer. One common Pthgoren triple is --, in which the sides of right tringle re in the rtio ::. If the mesures of the sides of n right tringle re whole numers, the mesures form Pthgoren triple. Emple Pthgoren Triples etermine whether ech set of mesures cn e the sides of right tringle. Then stte whether the form Pthgoren triple..,, Since the mesure of the longest side is, must e c, nd or re nd. c Pthgoren Theorem,, c Simplif. 9 dd. hpter 7 Right Tringles nd Trigonometr Since 9, segments with these mesures cnnot form right tringle. Therefore, the do not form Pthgoren triple. hpter 7 Right Tringles nd Trigonometr ifferentited Instruction Visul/Sptil Eplin tht mn rtists use right tringles in their works ecuse the re so ppeling to the ee. Right tringles cn serve s guidelines to drw mountins in the ckground or to crete vnishing points nd perspective. sk students to construct one or more right tringles on lnk sheet of pper, find the lengths of the sides nd then tr to compose picture using the tringle in n imge or s guide for n imge. Emples could e picture of rod vnishing in the distnce or house with right tringles s prt of its roof.

20 Stud Tip ompring Numers If ou cnnot quickl identif the gretest numer, use clcultor to find deciml vlues for ech numer nd compre. oncept heck. Mri; olin does not hve the longest side s the vlue of c.. Since the numers in Pthgoren triple stisf the eqution c, the represent the sides of right tringle the converse of the Pthgoren Theorem. Guided Prctice GUIE PRTIE KEY Eercises Emples, 7 0 nswers. F. 0,, nd c Pthgoren Theorem 0 0,, c Simplif dd. These segments form the sides of right tringle since the stisf the Pthgoren Theorem. The mesures re whole numers nd form Pthgoren triple. c.,, nd c 9 Simplif. 9 9 dd. Pthgoren Theorem,, c 9 Since 9, segments with these mesures form right tringle. However, the three numers re not whole numers. Therefore, the do not form Pthgoren triple.. FIN THE ERROR Mri nd olin re determining whether -- is Pthgoren triple. olin + = c + =? 9 + =? 9 no Mri + = c + =? + =? 9 9 = 9 es Who is correct? Eplin our resoning.. Eplin wh Pthgoren triple cn represent the mesures of the sides of right tringle.. OPEN ENE rw pir of similr right tringles. List the corresponding sides, the corresponding ngles, nd the scle fctor. re the mesures of the sides of ech tringle Pthgoren triple? See mrgin. Find E 7 Lesson 7- The Pthgoren Theorem nd Its onverse Smple nswer : EF,, E, nd F, corresponds to E, corresponds to EF, nd corresponds to F. The scle fctor is. No; the mesures of the sides do not form Pthgoren triple since nd re not whole numers. 0 Prctice/ppl Stud Noteook Hve students dd the definitions/emples of the voculr terms to their Voculr uilder worksheets for hpter 7. include n other item(s) tht the find helpful in mstering the skills in this lesson. FIN THE ERROR Point out tht in ll right tringles, the hpotenuse is lws the longest side ecuse it is opposite the 90º ngle. Students hve to mke sure tht the vlue the use to check for c is the lrgest vlue given. lso, tell students tht the order of nd does not mtter. Mri could hve used for nd for, nd she still would hve een correct. out the Eercises Orgniztion Ojective The Pthgoren Theorem: 7 onverse of the Pthgoren Theorem: 9 Odd/Even ssignments Eercises 9 re structured so tht students prctice the sme concepts whether the re ssigned odd or even prolems. lert! Eercises nd 9 require grphing clcultor. ssignment Guide sic: 7 odd,, 70 verge: odd, 70 dvnced: even, 0 (optionl: 70) Lesson 7- The Pthgoren Theorem nd Its onverse

21 Stud Stud Guide nd nd Intervention Intervention, p. 7 (shown) nd p. 7- NME TE PERIO The Pthgoren Theorem nd Its onverse The Pthgoren Theorem In right tringle, the sum of the squres of the mesures of the legs equls the squre of the mesure of the hpotenuse. is right tringle, so c. Emple Prove the Pthgoren Theorem. With ltitude, ech leg nd is geometric men etween hpotenuse c nd the segment of the hpotenuse djcent to tht leg. c nd c, so c nd c. dd the two equtions nd sustitute c to get c c c( ) c. Emple. Find. c Pthgoren Theorem, c 9 Eercises Simplif. Sutrct. Tke the squre root of ech side.. Find c. c c h NME 7 TE PERIO Prctice (verge) 0 0 c Pthgoren Theorem 0 0 c 0, c Simplif. 00 c dd. 00 c c Tke the squre root of ech side.. c Use clcultor. Find Skills Prctice, p. 9 nd Prctice, p. 0 (shown) Find The Pthgoren Theorem nd Its onverse NME 0 TE PERIO 9.. etermine whether GHI is right tringle for the given vertices. Eplin. 7. G(, 7), H(, ), I(, ). G(, ), H(, ), I(, ) es; GH, HI 9, IG 00, () (9 ) (00 ) no; GH 9, HI 0, IG 7, (0 ) (7 ) (9 ) 9. G(, ), H(, ), I(, ) 0. G(, ), H(, ), I(, 9) es; GH 9, HI, IG 9, (9 ) (9 ) ( ) es; GH, HI, IG 70, ( ) ( ) (70 ) etermine whether ech set of mesures cn e the mesures of the sides of right tringle. Then stte whether the form Pthgoren triple.. 9, 0,. 7,, 9.,, 0 es, es no, no es, es. 9,,, 7 7.,,., 7 es, no es, no es, no 7. ONSTRUTION The ottom end of rmp t wrehouse is 0 feet from the se of the min dock nd is feet long. How high is the dock? out. ft high Reding to Lern Mthemtics, p. 7- Reding to Lern Mthemtics The Pthgoren Theorem nd Its onverse ft rmp 0 ft Pre-ctivit How re right tringles used to uild suspension ridges? Red the introduction to Lesson 7- t the top of pge 0 in our tetook. o the two right tringles shown in the drwing pper to e similr? Eplin our resoning. Smple nswer: No; their sides re not proportionl. In the smller tringle, the longer leg is more thn twice the length of the shorter leg, while in the lrger tringle, the longer leg is less thn twice the length of the shorter leg. Reding the Lesson. Eplin in our own words the difference etween how the Pthgoren Theorem is used nd how the onverse of the Pthgoren Theorem is used. Smple nswer: The Pthgoren Theorem is used to find the third side of right tringle if ou know the lengths of n two of the sides. The converse is used to tell whether tringle with three given side lengths is right tringle.. Refer to the figure. For this figure, which sttements re true?. m n p. n m p m, E, F, G. m n p. m p n E. p n m F. n p m G. n mp H. p mn. Is the following sttement true or flse? Pthgoren triple is n group of three numers for which the sum of the squres of the smller two numers is equl to the squre of the lrgest numer. Eplin our resoning. Smple nswer: The sttement is flse ecuse in Pthgoren triple, ll three numers must e whole numers.. If,, nd z form Pthgoren triple nd k is positive integer, which of the following groups of numers re lso Pthgoren triples?,.,, z.,, z.,, z. k, k, kz ELL n p? dock Lesson 7- indictes incresed difficult Prctice nd ppl For Eercises,,,, 7 9 ppliction See Emples Etr Prctice See pge 77.. es, es. es, es. no, no. no, no. --. Smple nswer: The triples re ll multiples of the triple --.. Smple nswer: The consist of n numer of similr tringles.. Yes; the mesures of the sides re lws multiples of,, nd.. -0-; ; 7-0- c. --0; -7-7 hpter 7 Right Tringles nd Trigonometr NME TE PERIO 7- Enrichment, p. onverse of Right Tringle Theorem You hve lerned tht the mesure of the ltitude from the verte of the right ngle of right tringle to its hpotenuse is the geometric men etween the mesures of the two segments of the hpotenuse. Is the converse of this theorem true? In order to find out, it will help to rewrite the originl theorem in if-then form s follows. 7. OORINTE GEOMETRY etermine whether JKL with vertices J(, ), K(, ), nd L(, ) is right tringle. Eplin. es; JK 7, KL 7, JL ; 7 7 etermine whether ech set of numers cn e the mesures of the sides of right tringle. Then stte whether the form Pthgoren triple..,, 9 es, es 9. 0, 0, no, no 0.,, 0 es, no. OMPUTERS omputer monitors re usull mesured long the digonl of the screen. 9-inch monitor hs digonl tht mesures 9 inches. If the height of the screen is. inches, how wide is the screen? out. in. Find in in.. 9. OORINTE GEOMETRY etermine whether QRS is right tringle for the given vertices. Eplin.. See mrgin for eplntions.. Q(, 0), R(, ), S(9, 0) es 9. Q(, ), R(0, ), S(, ) no 0. Q(, ), R(, ), S(, ) no. Q( 9, ), R(, ), S(, 9) es etermine whether ech set of numers cn e the mesures of the sides of right tringle. Then stte whether the form Pthgoren triple.. primitive Pthgoren triple is Pthgoren triple with no common fctors ecept. Nme n primitive Pthgoren triples contined in the tle.. escrie the pttern tht reltes these sets of Pthgoren triples.. These Pthgoren triples re clled fmil. Wh do ou think this is?. re the tringles descried fmil of Pthgoren triples similr? Eplin.. For ech Pthgoren triple, find two triples in the sme fmil..,, 7. 9, 0, c. 7,, 0 7..,, 7. 7,,. 0,,. 7,,., 7, 7 7.,,,, 9. 7, 7, 0 7 es, no no, no es, no es, no For Eercises 0, use the tle of Pthgoren triples. 0. op nd complete the tle. c 0? 0? 9? 0 Helping You Rememer. Mn students who studied geometr long go rememer the Pthgoren Theorem s the eqution c, ut cnnot tell ou wht this eqution mens. formul is useless if ou don t know wht it mens nd how to use it. How could ou help someone who hs forgotten the Pthgoren Theorem rememer the mening of the eqution c? Smple nswer: rw right tringle. Lel the lengths of the two legs s nd nd the length of the hpotenuse s c. hpter 7 Right Tringles nd Trigonometr If Q is right tringle with right ngle t Q, then QP is the geometric men etween P nd P, where P is etween nd nd QP is perpendiculr to.. Write the converse of the if-then form of the theorem. f QP is geometric men etween P nd t e Q P

22 Militr ll rnches of the militr use nvigtion. Some of the jos using nvigtion include rdr/sonr opertors, ot opertors, irplne nvigtors, nd spce opertions officers. Online Reserch For informtion out creer in the militr, visit: com/creers GEOGRPHY For Eercises nd 7, use the following informtion. enver is locted t out 0 degrees longitude nd 0 degrees ltitude. Sn Frncisco is locted t out degrees longitude nd degrees ltitude. Ls Vegs is locted t out degrees longitude nd degrees ltitude. Using the lines of longitude nd ltitude, find ech degree distnce.. Sn Frncisco to enver 7. degrees 7. Ls Vegs to enver 0. degrees. PROOF Write prgrph proof of Theorem 7.. See mrgin. 9. PROOF Use the Pthgoren Theorem nd the figure t the right to prove the istnce Formul. See mrgin. 0. PINTING pinter sets ldder up to rech the ottom of second-stor window feet ove the ground. The se of the ldder is feet from the house. While the pinter mies the pint, neighor s dog umps the ldder, which moves the se feet frther w from the house. How fr up the side of the house does the ldder rech? ft. ft. SILING The mst of silot is supported wires clled shrouds. Wht is the totl length of wire needed to form these shrouds? ft ft ft shrouds 9 ft O (, ) (, ) ft ft. LNSPING Si congruent squre stones re rrnged in n L-shped wlkw through grden. If inches, then find the re of the L-shped wlkw. (, ). NVIGTION fishing trwler off the cost of lsk ws ordered the U.S. ost Gurd to chnge course. The were to trvel miles west nd then sil miles south to miss lrge iceerg efore continuing on the originl course. How mn miles out of the w did the trwler trvel? out. mi. RITIL THINKING The figure t the right is rectngulr prism with,, nd F, nd M is the midpoint of. Find nd HM. How re EM, FM, nd GM relted to HM? 0, HM 9 9.; HM EM FM GM nswers (p. ) out 7. ft. QR, RS 0, QS ; 0 9. QR, RS, QS ; Sn Frncisco Ls Vegs Lesson 7- The Pthgoren Theorem nd Its onverse 0. QR, RS, QS ; ( ) ( ) ( ). QR 9, RS 9, QS ; ( 9 ) ( 9 ) ( ) 0 E H d ft 70 in M enver G F Phil Mislinski/Gett Imges nswers. Given: with sides of mesure,, nd c, where c Prove: is right tringle. m F Proof: rw E on line with mesure equl to. t, drw line m E. Locte point F on m so tht F. rw FE nd cll its mesure. ecuse FE is right tringle,. ut c, so c or c. Thus, FE SSS. This mens. Therefore, must e right ngle, mking right tringle. 9. Given: with right ngle t, d Prove: d ( ) ( ) O c (, ) d (, ) (, ) Proof: Sttements (Resons). with right ngle t, d (Given). () () () (Pthgoren Theorem). ; (istnce on numer line). d (Sustitution). ( ) ( ) d (Sustitution). ( ) ( ) d (Tke the squre root of ech side.) 7. d ( ) ( ) (Refleive Propert) E Lesson 7- The Pthgoren Theorem nd Its onverse

23 ssess Open-Ended ssessment Modeling sk students to recll how right tringles re modeled in suspension ridges, nd hve them demonstrte how right tringle could model rel world ojects such s silots, flgs, houses, ookshelves, nd so on. Getting Red for Lesson 7- Prerequisite Skill In Lesson 7-, students will e working more etensivel with rdicl epressions to find side lengths of specil right tringles. Use Eercises 70 to determine our students fmilirit with simplifing rdicl epressions. ssessment Options Quiz (Lessons 7- nd 7-) is ville on p. 07 of the hpter 7 Resource Msters. nswers. Smple nswer: The rod, the tower tht is perpendiculr to the rod, nd the cles form the right tringles. nswers should include the following. Right tringles re formed the ridge, the towers, nd the cles. The cle is the hpotenuse in ech tringle. Stndrdized Test Prctice Grphing lcultor. --, --0, --0, --0, 7-- Mintin Your Skills Mied Review. 0; it converges to. 7. ; it pproches positive infinit.. ; it converges to ; it lterntes etween 0. nd. Getting Red for the Net Lesson. WRITING IN MTH nswer the question tht ws posed t the eginning of the lesson. See mrgin. How re right tringles used to uild suspension ridges? Include the following in our nswer: the loctions of the right tringles, nd n eplntion of which prts of the right tringle re formed the cles.. In the figure, if E 0, wht is the vlue of h? 0 7. LGER If (9 ), then find. no solution. 0 PROGRMMING For Eercises nd 9, use the following informtion. The TI- Plus progrm uses procedure for finding Pthgoren triples PROGRM: PYTHTRIP tht ws developed Euclid round 0.. Run the progrm to generte :For (X,, ) list of Pthgoren triples. :For (Y,, ) :If X > Y :Then :int (X Y + 0.) :XY :int (X +Y +0.) :If > :Then. List ll the memers of the -- fmil tht re generted the progrm. 9. geometr student mde the conjecture tht if three whole numers re Pthgoren triple, then their product is divisile 0. oes this conjecture hold true for ech triple tht is produced the progrm? es O E(, h) :isp,, :Else :isp,, :End :End :Puse :isp :End :End :Stop Find the geometric men etween ech pir of numers. (Lesson 7-) 0. nd. 9 nd 0.. nd nd nd 7.7. nd 0. Find the vlue of ech epression. Then use tht vlue s the net in the epression. Repet the process nd descrie our oservtions. (Lesson -)., where initill equls 7., where initill equls., where initill equls 9., where initill equls 0. etermine whether the sides of tringle could hve the lengths,, nd. Eplin. (Lesson -) no; PREREQUISITE SKILL Simplif ech epression rtionlizing the denomintor. (To review simplifing rdicl epressions, see pges 7 nd 7.) hpter 7 Right Tringles nd Trigonometr hpter 7 Right Tringles nd Trigonometr

24 Lesson Notes Specil Right Tringles Use properties of tringles. Use properties of tringles. Focus is tringle tiling used in wllpper design? -Minute heck Trnsprenc 7- Use s quiz or review of Lesson 7-. Tringle tiling is the process of ling copies of single tringle net to ech other to fill n re. One tpe of tringle tiling is wllpper tiling. There re ectl 7 tpes of tringle tiles tht cn e used for wllpper tiling. 0 7 Mthemticl ckground notes re ville for this lesson on p Tile is mde up of two tringles tht form squre. This tile is rotted to mke the wllpper design shown t the right. PROPERTIES OF TRINGLES Fcts out tringles re used to solve mn geometr prolems. The Pthgoren Theorem llows us to discover specil reltionships tht eist mong the sides of tringle. rw digonl of squre. The two tringles formed re isosceles right tringles. Let represent the mesure of ech side nd let d represent the mesure of the hpotenuse. d Pthgoren Theorem d dd. d d is tringle tiling used in wllpper design? sk students: How mn of the wllpper tiles re mde up of right tringles? Which ones re the? tiles; tiles,,,, 7, 0,,,,,, nd 7 Wht percent of wllpper tiling depends on right tringles? out of 7 or out 7% If the length of the hpotenuse of one of the tringles in tile centimeters, then how is 7 long is ech leg of the tringle? 7 cm Tke the positive squre root of ech side. d Fctor. d Simplif. This lgeric proof verifies tht the length of the hpotenuse of n tringle is times the length of its leg. The rtio of the sides is : :. Theorem 7. In tringle, the length of the hpotenuse is times the length of leg. n n n Lesson 7- Specil Right Tringles 7 Resource Mnger Workook nd Reproducile Msters hpter 7 Resource Msters Stud Guide nd Intervention, pp. Skills Prctice, p. Prctice, p. Reding to Lern Mthemtics, p. 7 Enrichment, p. Prerequisite Skills Workook, pp. 7 Teching Geometr With Mnipultives Msters, p. Trnsprencies -Minute heck Trnsprenc 7- nswer Ke Trnsprencies Technolog GeomPSS: Tutoril Plus, Lesson Interctive hlkord Lesson - Lesson Title 7

25 Tech PROPERTIES OF --90 TRINGLES In-lss Emples Power Point WLLPPER TILING Note tht the figure in Emple cn e divided into four equl squre qudrnts so tht ech squre contins tringles. Wht is the re of one of these squres if the hpotenuse of ech º-º-90º tringle mesures 7 millimeters? 9 mm Find. R in. cm S cm T Teching Tip Point out to students tht one corner of the wllpper squre consists of four º-º-90º tringles, the comintion of which forms new lrger º-º-90º tringle. Eplin how it is interesting tht tiled groups of,, nd of the smller tringles form lrger º-º-90º tringles. Stud Tip Rtionlizing enomintors To rtionlize denomintor, multipl the frction in the form of rdicl over itself so tht the product in the denomintor is rtionl numer. Emple Find the Mesure of the Hpotenuse WLLPPER TILING ssume tht the length of one of the legs of the tringles in the wllpper in the figure is inches. Wht is the length of the digonl of the entire wllpper squre? The length of ech leg of the tringle is inches. The length of the hpotenuse is times s long s leg. The length of the hpotenuse of one of the tringles is. There re four tringles long the digonl of the squre. So, the length of the digonl of the squre is or inches. Emple Find the Mesure of the Legs Find. The length of the hpotenuse of tringle is times the length of leg of the tringle. (), ivide ech side. Rtionlize the denomintor. Multipl. ivide. PROPERTIES OF TRINGLES There is lso specil reltionship mong the mesures of the sides of tringle. When n ltitude is drwn from n verte of n equilterl tringle, two congruent tringles re formed. LM nd KM re congruent segments, so let LM nd KM. the Segment ddition Postulte, LM KM KL. Thus, KL. Since JKL is n equilterl tringle, KL JL JK. Therefore, JL nd JK. Let represent the mesure of the ltitude. Use the Pthgoren Theorem to find. m 0 L m m J 0 0 M 0 K Intervention Some students m hve difficult keeping the properties of º-º-90º nd 0º-0º-90º tringles stright. Suggest tht the periodicll reproduce the figures in Theorem 7. nd Theorem 7.7 to keep these reltionships fresh. dvise them, however, tht the hve the intuitive knowledge to derive either of these two figures using the Pthgoren Theorem with squre or n equilterl tringle. New hpter 7 Right Tringles nd Trigonometr hpter 7 Right Tringles nd Trigonometr (JM) (LM) (JL) () Pthgoren Theorem JM, LM, JL Simplif. Sutrct from ech side. Tke the positive squre root of ech side. Fctor. Simplif. So, in tringle, the mesures of the sides re,, nd. The rtio of the sides is ::. ifferentited Instruction Logicl/Mthemticl Suggest tht students close their ooks nd divide piece of pper into two columns. t the top of one column, sk them to drw squre with digonl d nd side. t the top of the other column, sk students to drw n equilterl tringle with one ltitude nd tell them to lel the segment tht the ltitude divides with two s. Then hve students sstemticll use the Pthgoren Theorem to figure out the side reltionships of the º-º-90º nd 0º-0º-90º tringles in these two figures.

26 Stud Tip Tringle The shorter leg is opposite the 0 ngle, nd the longer leg is opposite the 0 ngle. The reltionship of the side mesures leds to Theorem 7.7. Theorem 7.7 In tringle, the length of the hpotenuse is twice the length of the shorter leg, nd the length of the longer leg is times the length of the shorter leg. n 0 n 0 n PROPERTIES OF TRINGLES In-lss Emples Find QR. P cm Power Point Q 0 R Emple Find Tringles is the longer leg, is the shorter leg, nd is the hpotenuse. () () or 7 () (7) or in. cm WXY is 0º-0º-90º tringle with right ngle X nd WX s the longer leg. Grph points X(, 7) nd Y(7, 7), nd locte point W in Qudrnt III. Y X Emple Specil Tringles in oordinte Plne OORINTE GEOMETRY Tringle P is tringle with right ngle. is the longer leg with endpoints (, ) nd (9, ). Locte point P in Qudrnt I. Grph nd. lies on horizontl gridline of the coordinte plne. Since P will e perpendiculr to, it lies on verticl gridline. Find the length of. 9 O W W(, 7 ) is the longer leg. P is the shorter leg. So, (P). Use to find P. (P) (P) P ivide ech side. P Rtionlize the denomintor. (, ) (9, ) O P 0 0 P P Multipl. Simplif. Point P hs the sme -coordinte s. P is locted units ove. So, the coordintes of P re, or out (,.). Lesson 7- Specil Right Tringles 9 Unlocking Misconceptions Tringles Point out tht common mistke is for students to quickl ssume tht the longer leg of 0º-0º-90º tringle is twice the length of the shorter leg. emonstrte for students tht this cnnot e so drwing 0º-0º-90º tringle on the ord nd etending the figure into n equilterl tringle. Eplin tht the shorter leg of the originl right tringle is ectl hlf the distnce of one side of the equilterl tringle. Tell students tht the hpotenuse is lso side of the equilterl tringle, so this hs to e the side tht is twice the length of the shorter leg. Lesson 7- Specil Right Tringles 9

27 Prctice/ppl Stud Noteook Hve students dd the definitions/emples of the voculr terms to their Voculr uilder worksheets for hpter 7. include n other item(s) tht the find helpful in mstering the skills in this lesson. out the Eercises Orgniztion Ojective Properties of --90 Tringles:,, 7,, Properties of Tringles:,,, 7 Odd/Even ssignments Eercises re structured so tht students prctice the sme concepts whether the re ssigned odd or even prolems. ssignment Guide sic: 7 odd, verge: odd, dvnced: 0 even, (optionl: 9 ) ll: Quiz ( ) oncept heck Guided Prctice GUIE PRTIE KEY Eercises Emples,,, 9, 0 ppliction indictes incresed difficult Prctice nd ppl For Eercises See Emples,, 7,,,,,, 7 Etr Prctice See pge 77.. OPEN ENE rw tringle. e sure to lel the ngles nd the sides nd to eplin how ou mde the drwing.. See mrgin.. rw tringle with the shorter leg centimeters long. Lel the ngles nd the remining sides.. Write n eqution to find the length of rectngle tht hs digonl twice s long s its width. The length of the rectngle is times the width; w. Find nd.... ; ; ; Find the missing mesures. 7. If c, find nd. ; c 0. If, find nd c. ; c 0 Tringle is tringle with right ngle nd with s the shorter leg. Grph nd, nd locte point in Qudrnt I See p (, 0), (, ) 0. (, ), (, ). SOFTLL Find the distnce from home plte to second se if the ses re 90 feet prt. 90 or 7. ft nd se 90 ft 90 ft rd st se se 90 ft 90 ft Find nd. 7 ; ; 9 ; ;.;. ; For Eercises nd 9, use the figure t the right.. If 0, find E nd. E ; E c 0 9. If 7, find, E,, nd. ; E ; ; 0 0 d Home Plte 0 hpter 7 Right Tringles nd Trigonometr nswers. Smple nswer: onstruct two perpendiculr lines. Use ruler to mesure cm from the point of intersection on the one r. Use the compss to cop the cm segment. onnect the two endpoints to form tringle with sides of cm nd hpotenuse of cm.. Smple nswer: rw line using ruler. Then use protrctor to mesure 90 ngle. On one r mrk cm length, nd t tht endpoint use the protrctor to mesure 0 ngle towrd the other r. Where this r intersects the other r should form 0 ngle, completing the tringle with sides cm, cm, nd hpotenuse of cm. 0 hpter 7 Right Tringles nd Trigonometr

28 0. The length of n ltitude of n equilterl tringle is feet. Find the length of side of the tringle. ft. ft. The perimeter of n equilterl tringle is centimeters. Find the length of n ltitude of the tringle. 7. cm.99 cm. The length of digonl of squre is millimeters. Find the perimeter of the squre. mm. The ltitude of n equilterl tringle is 7. meters long. Find the perimeter of the tringle.. m. m. The digonls of rectngle re inches long nd intersect t n ngle of 0. Find the perimeter of the rectngle. or out.7 in.. The sum of the squres of the mesures of the sides of squre is. Find the mesure of digonl of the squre... Find,, z, nd the perimeter of. ; ; z ; or.9 units 7. P is tringle with right ngle. Find the coordintes of P in Qudrnt I for (, ) nd (, ). (, ). PGH is tringle with mp 90. Find the coordintes of P in Qudrnt I for G(,) nd H(, ). (, ), (7, ) 9. P is tringle with right ngle nd the longer leg. Find 9., the coordintes of P in Qudrnt III for (, ) nd (, 7). or out (0., ) 0. P is tringle with m 0 nd hpotenuse. Find the coordintes of P for (, ) nd (0, ) if P lies ove.,. If PQ SR, use the figure to find,, c, nd d. P (, ) Q (, d), 9, c, d 9 Tringle Tiling uildings in Federtion Squre in Melourne, ustrli, feture tiling pttern clled pinwheel tiling. The sides of ech right tringle re in the rtio ::. Source: There re no gps ecuse when 0 ngle is rotted times, it rottes 0. nswer. Smple nswer: TRINGLE TILING For Eercises, use the following informtion. Tringle tiling refers to the process of tking mn copies of single tringle nd ling them net to ech other to fill n re. For emple, the pttern shown is composed of tiles like the one outlined.. How mn tringles re used to crete the sic circulr pttern?. Which ngle of the tringle is eing rotted to mke the sic shpe? 0 ngle. Eplin wh there re no gps in the sic pttern.. Use grid pper to cut out tringles. olor the sme pttern on ech tringle. rete one sic figure tht would e prt of wllpper tiling. See mrgin. S (0, 0) 0 7- Enrichment, p. onstructing Vlues of Squre Roots The digrm t the right shows right isosceles tringle with two legs of length inch. the Pthgoren Theorem, the length of the hpotenuse is inches. constructing n djcent right tringle with legs of inches nd inch, ou cn crete segment of length. continuing this process s shown elow, ou cn construct wheel of squre roots. This wheel is clled the Wheel of Theodorus fter Greek philosopher who lived out T (c, 0) U (e, 0) R (f, 0) NME TE PERIO z Lesson 7- Specil Right Tringles John Gollings, courtes Federtion Squre Stud Stud Guide nd nd Intervention Intervention, p. (shown) nd p. 7- NME TE PERIO Specil Right Tringles Properties of Tringles The sides of right tringle hve specil reltionship. Emple If the leg of Emple In right right tringle is units, show tringle the hpotenuse is times tht the hpotenuse is units. the leg. If the hpotenuse is units, find the length of ech leg. The hpotenuse is times the leg, so divide the length of the hpotenuse. Using the Pthgoren Theorem with, then c c units Eercises Find Find the perimeter of squre with digonl centimeters..9 cm. Find the digonl of squre with perimeter 0 inches. 7. in. 9. Find the digonl of squre with perimeter meters m 7- NME TE PERIO Prctice (verge) Skills Prctice, p. nd Prctice, p. (shown) Find nd , 9, 0,... 0.,., 7 ; For Eercises 7 9, use the figure t the right. 7. If, find nd c., c. If, find nd., 9 9. If, find,, nd.,, 0. The perimeter of n equilterl tringle is 9 centimeters. Find the length of n ltitude of the tringle.. in. or out. in.. MIP is tringle with right ngle t I, nd IP the longer leg. Find the coordintes of M in Qudrnt I for I(, ) nd P(, ). (, ) or out (,.9). TJK is tringle with right ngle t J. Find the coordintes of T in Qudrnt II for J(, ) nd K(, ). (, ) Specil Right Tringles. OTNIL GRENS One of the displs t otnicl grden is n her grden plnted in the shpe of squre. The squre mesures rds on ech side. Visitors cn view the hers from digonl pthw through the grden. How long is the pthw? d or out. d Reding to Lern Mthemtics, p NME TE PERIO Reding to Lern Mthemtics Specil Tringles 0 c 0 d d d ELL Pre-ctivit How is tringle tiling used in wllpper design? Red the introduction to Lesson 7- t the top of pge 7 in our tetook. How cn ou most completel descrie the lrger tringle nd the two smller tringles in tile? Smple nswer: The lrger tringle is n isosceles otuse tringle. The two smller tringles re congruent sclene right tringles. How cn ou most completel descrie the lrger tringle nd the two smller tringles in tile? (Include ngle mesures in descriing ll the tringles.) Smple nswer: The lrger tringle is equilterl, so ech of its ngle mesures is 0. The two smller tringles re congruent right tringles in which the ngle mesures re 0, 0, nd 90. Reding the Lesson. Suppl the correct numer or numers to complete ech sttement.. In tringle, to find the length of the hpotenuse, multipl the length of leg.. In tringle, to find the length of the hpotenuse, multipl the length of the shorter leg. c. In tringle, the longer leg is opposite the ngle with mesure of 0. d. In tringle, to find the length of the longer leg, multipl the length of the shorter leg. e. In n isosceles right tringle, ech leg is opposite n ngle with mesure of. f. In tringle, to find the length of the shorter leg, divide the length of the longer leg. g. In tringle, to find the length of the longer leg, divide the length of the hpotenuse nd multipl the result. h. To find the length of side of squre, divide the length of the digonl.. Indicte whether ech sttement is lws, sometimes, or never true.. The lengths of the three sides of n isosceles tringle stisf the Pthgoren Theorem. sometimes. The lengths of the sides of tringle form Pthgoren triple. never c. The lengths of ll three sides of tringle re positive integers. never d Lesson 7- Lesson 7- ontinue constructing the wheel until ou mke segment of length. Helping You Rememer. Some students find it esier to rememer mthemticl concepts in terms of specific numers rther thn vriles. How cn ou use specific numers to help ou rememer the reltionship etween the lengths of the three sides in tringle? Smple nswer: rw tringle. Lel the length of the shorter leg s. Then the length of the hpotenuse is, nd the length of the longer leg is. Just rememer:,,. Lesson 7- Specil Right Tringles

29 nswers. Smple nswer: ongruent tringles of different color cn e rrnged to crete ptterns. nswers should include the following., 9,, nd 7;,,, 7, 0,, nd Plcing ngles net to one nother forms 90 ngles, which cn e plced net to ech other, leving no holes.. Find,, nd z. ; ; 7. If nd mh 0, z find H. z H G. Ech tringle in the figure is 9. In regulr hegon UVWXYZ, ech tringle. Find.. side is centimeters long. Find WY. W X 0.7 cm E V U F Z Y 0. SELL The digrm t the right shows some dimensions of omiske Prk in hicgo, Illinois. is segment from home plte to ded center field, nd E is segment from the left field foul-ll pole to the right field foul-ll pole. If the center fielder is stnding t, how fr is he from home plte? 7. ft 7 ft 7 ft E. RITIL THINKING Given figure, with, m 0, m,, nd, find the perimeter. units 0. WRITING IN MTH nswer the question tht ws posed t the eginning of the lesson. See mrgin. How is tringle tiling used in wllpper design? Include the following in our nswer: which of the numered designs contin tringles nd which contin tringles, nd reson wh rottions of the sic design left no holes in the completed design. Stndrdized Test Prctice. In the right tringle, wht is if? units units units units. SHORT RESPONSE For rel numers nd, where 0, if, then ( )( ).? hpter 7 Right Tringles nd Trigonometr hpter 7 Right Tringles nd Trigonometr

30 Mintin Your Skills P rctice Quiz Find the mesure of the ltitude drwn to the hpotenuse. (Lesson 7-).. Mied Review. 9.;.7;.9. 0 ; ; 0.. JK, KL, JL, RS, ST, RT ; JKL RST SSS Getting Red for the Net Lesson 7 etermine whether ech set of mesures cn e the sides of right tringle. Then stte whether the form Pthgoren triple. (Lesson 7-).,, es, es. 9, 0, es, es 7. 0,, no, no. 0,, es, es 9. 7,, es, es 0.,, 7 no, no Find,, nd z. (Lesson 7-)... 0 z Write n inequlit or eqution relting ech pir of ngles. (Lesson -). mlk, mln mlk mln 9.; ;. mlk, mnlo mlk mnlo. molk, mnlo molk mnlo 7. mklo, mln mklo mln. etermine whether JKL with vertices J(, ), K(, ), nd L(, ) is congruent to RST with vertices R(, ), S(, ), nd T(, ). Eplin. (Lesson -) PREREQUISITE SKILL Solve ech eqution. (To review solving equtions, see pges 77 nd 7.) k g n m X Z 9 K Y z. L. O z. N Lessons 7- through 7- ssess Open-Ended ssessment Speking Hve students nme the side reltionships for º-º- 90º tringles nd 0º-0º-90º tringles. For some emples nd prolems in the ook, sk students to cll out these side reltionships using the numers nd vriles in the figures. Getting Red for Lesson 7- Prerequisite Skill Students will lern out trigonometric rtios in Lesson 7-. The will use frctions to solve equtions nd epress side lengths. Use Eercises 9 to determine our students fmilirit with solving equtions tht involve frctions. ssessment Options Prctice Quiz The quiz provides students with rief review of the concepts nd skills in Lessons 7- through 7-. Lesson numers re given to the right of the eercises or instruction lines so students cn review concepts not et mstered.. etermine whether with vertices (, ), (, 0), nd (, 7) is right tringle. Eplin. (Lesson 7-) es;, 0, ; 0 Find nd. (Lesson 7-). ;. ; 0 Lesson 7- Specil Right Tringles Lesson 7- Specil Right Tringles

31 Lesson Notes Trigonometr Focus -Minute heck Trnsprenc 7- Use s quiz or review of Lesson 7-. Mthemticl ckground notes re ville for this lesson on p. 0. cn surveors determine ngle mesures? sk students: Wht side of right tringle is represented the theodolite s line of sight? the hpotenuse The Sttue of Liert is pproimtel 9. meters tll from the se of the pedestl to the tip of the torch. If theodolite were plced 00 meters from the se of the pedestl, out how fr would the theodolite e from the top of the torch? out. m Voculr trigonometr trigonometric rtio sine cosine tngent TEHING TIP In the epression sin, represents the mesure of in degrees. Stud Tip Reding Mth SOH-H-TO is mnemonic device for lerning the rtios for sine, cosine, nd tngent using the first letter of ech word in the rtios. sin o pp hp dj cos h p tn o p p dj Find trigonometric rtios using right tringles. Solve prolems using trigonometric rtios. cn surveors determine ngle mesures? The old surveor s telescope shown t right is clled theodolite (thee H duh lite). It is n opticl instrument used to mesure ngles in surveing, nvigtion, nd meteorolog. It consists of telescope fitted with level nd mounted on tripod so tht it is free to rotte out its verticl nd horizontl es. fter mesuring ngles, surveors ppl trigonometr to clculte distnce or height. TRIGONOMETRI RTIOS The word trigonometr comes from two Greek terms, trigon, mening tringle, nd metron, mening mesure. The stud of trigonometr involves tringle mesurement. rtio of the lengths of sides of right tringle is clled trigonometric rtio. The three most common trigonometric rtios re sine, cosine, nd tngent. Words Smols Models mesure of leg opposite sine of mesure of hpotenuse sine of cosine of cosine of tngent of tngent of mesure of leg opposite mesure of hpotenuse mesure of leg djcent to mesure of hpotenuse mesure of leg djcent to mesure of hpotenuse mesure of leg opposite mesure of leg djcent to mesure of leg opposite mesure of leg djcent to sin sin cos cos tn tn Trigonometric Rtios hpotenuse leg opposite hpotenuse leg djcent to hpotenuse leg djcent to nd opposite leg opposite leg djcent to leg opposite nd djcent to Resource Mnger hpter 7 Right Tringles nd Trigonometr rthur Thevenrt/ORIS Workook nd Reproducile Msters hpter 7 Resource Msters Stud Guide nd Intervention, pp Skills Prctice, p. 7 Prctice, p. 7 Reding to Lern Mthemtics, p. 7 Enrichment, p. 7 ssessment, pp. 07, 09 School-to-reer Msters, p. Prerequisite Skills Workook, pp. Teching Geometr With Mnipultives Msters, pp., 7,, 7, Trnsprencies -Minute heck Trnsprenc 7- nswer Ke Trnsprencies Technolog Interctive hlkord

32 Stud Tip Equivlent Rtios Notice tht the rtio leg opposite is the hpotenuse sme s leg djcent. hpotenuse Thus, sin cos. Likewise, cos sin c. Trigonometric rtios re relted to the cute ngles of right tringle, not the right ngle. Trigonometric Rtios Fold rectngulr piece of pper long digonl from to. Then cut long the fold to form right tringle. Write the nme of ech ngle on the inside of the tringle. Fold the tringle so tht there re two segments perpendiculr to. Lel points, E, F, nd G s shown. Use ruler to mesure,,, F, G, FG,, E, nd E to the nerest millimeter. nlze. Wht is true of E, GF, nd? The re similr tringles.. op the tle. Write the rtio of the side lengths for ech trigonometric rtio. Then clculte vlue for ech rtio to the nerest ten-thousndth. E G F Tech TRIGONOMETRI RTIOS uilding on Prior Knowledge In Lesson 7-, students lerned tht the could use the istnce Formul nd the Pthgoren Theorem to find lengths in coordinte plne. The revisit these methods in this lesson nd comine them with trigonometr to find the ngles of tringle in coordinte plne.. Smple nswer: Regrdless of the side lengths, the trigonometric rtio is the sme when compring ngles in similr tringles. sin cos tn. Stud the tle. Write sentence out the ptterns ou oserve with the trigonometric rtios.. Wht is true out m in ech tringle? m is the sme in ll tringles. Emple Find Sine, osine, nd Tngent Rtios Find sin R, cos R, tn R, sin S, cos S, nd tn S. Epress ech rtio s frction nd s deciml. sin R o pposite leg hpotenuse ST R S or 0. sin S o pposite leg hpotenuse R T RS or 0. In E in GF In E 0. F G F E 0.79 G F E FG E G s the Geometr ctivit suggests, the vlue of trigonometric rtio depends onl on the mesure of the ngle. It does not depend on the size of the tringle. cos R djcent leg hpotenuse R T RS or 0. cos S djcent leg hpotenuse ST R S or 0. T R S tn R o pposite leg djcentleg ST R T or. tn S o pposite leg djcentleg R T ST or 0.7 Lesson 7- Trigonometr In-lss Emple Power Point Teching Tip ssure the clss tht sine, cosine, nd tngent re ver useful nd not s imposing s the first seem. Eplin tht if students regulrl prctice using these rtios to solve prolems, the will ecome s fmilir s squring nd tking the squre root of numers nd re just s es to ke into clcultor. Find sin L, cos L, tn L, sin N, cos N, nd tn N. Epress ech rtio s frction nd s deciml. N 7 M sin L or cos L or 0. 7 tn L or 0. sin N or 0. 7 cos N or tn N or. L Geometr ctivit Mterils: rectngulr pper, scissors, metric ruler, pencil When students re folding the tringle to form two segments tht re prllel to, tell them tht it does not mtter how close or fr prt these segments re, nd encourge the clss to come up with vriet of different lengths to demonstrte tht this ctivit would work for n lengths. Students cn lso egin the ctivit with different sizes of rectngulr pper. Teching Tip Encourge students to fmilirize themselves with using their clcultors to find cosines, sines nd tngents nd their inverses. llow students to prctice with es vlues, such s cos 0, sin 0 0., cos 0. 0, etc., so the cn e sure the re using their clcultors correctl. Lesson 7- Trigonometr

33 In-lss Emple Power Point Emple Use lcultor to Evlute Epressions Stud Tip Grphing lcultor Find ech vlue to the nerest ten thousndth.. tn º.. cos 9º 0.07 e sure our clcultor is in degree mode rther thn rdin mode. Your clcultor m require ou to input the ngle efore using the trigonometric ke. USE TRIGONOMETRI RTIOS In-lss Emples. sin 7 SIN 7 ENTER KEYSTROKES: sin USE TRIGONOMETRI RTIOS You cn use trigonometric rtios to find the missing mesures of right tringle if ou know the mesures of two sides of tringle or the mesure of one side nd one cute ngle. Power Point Emple Use Trigonometric Rtios to Find Length EXERISING fitness triner sets the incline on tredmill to 7º. The wlking surfce is feet long. pproimtel how mn inches did the triner rise the end of the tredmill from the floor? ft Use clcultor to find ech vlue to the nerest ten thousndth.. cos 9 KEYSTROKES: OS 9 ENTER cos in. SURVEYING kot is stnding on the ground 97 rds from the se of cliff. Using theodolite, he noted tht the ngle formed the ground nd the line of sight to the top of the cliff ws. Find the height of the cliff to the nerest rd. Let e the height of the cliff in rds. Stud Tip 97 The tngent of ngle is the sme s the slope of the line through the origin tht forms n ngle of mesure with the positive -is. leg opposite tn tn leg djcent Tngent 97 tn d Multipl ech side 97. Use clcultor to find. KEYSTROKES: 97 TN ENTER d The cliff is out rds high. OORINTE GEOMETRY Find m X in right XYZ for X(, ), Y(, ), nd Z(, ). When solving equtions like 7, ou use the inverse of multipliction to find. In trigonometr, ou cn find the mesure of the ngle using the inverse of sine, cosine, or tngent. X(, ) Y (, ) Z (, ) O Given eqution To find the ngle Red s sin sin () equls the inverse sine of. cos cos () equls the inverse cosine of. tn z tn (z) equls the inverse tngent of z. Emple Use Trigonometric Rtios to Find n ngle Mesure OORINTE GEOMETRY Find m in right tringle for (, ), (, ), nd (, )..9 Eplore You know the coordintes of the vertices of right tringle nd tht is the right ngle. You need to find the mesure of one of the ngles. Pln (, ) (, ) Use the istnce Formul to find the mesure of ech side. Then use one of the trigonometric rtios to write n eqution. Use the inverse to find m. O hpter 7 Right Tringles nd Trigonometr ifferentited Instruction uditor/musicl The esiest w for uditor lerners to rememer the rtios for sine, cosine nd tngent is for them to chnt SOH-H-TO. When introducing this mnemonic device to students, hve them repet it s clss few times in rhthm. Point out tht SOH nd H ech hve one sllle ecuse the H is silent, so students cn rememer tht one silent hpotenuse is involved for the sine nd cosine rtios. TO hs two sllles nd involves the two legs for the tngent rtio. hpter 7 Right Tringles nd Trigonometr (, )

34 Stud Tip lcultors The second functions of the SIN, OS, nd TN kes re usull the inverses. Solve Emine () ( ) ( ) ( ) 0 or ( ) ( ) 0 or Use the cosine rtio. cos cos l eg djcent hpotenuse or cos nd cos Solve for. Use clcultor to find m. KEYSTROKES: nd [OS ] nd [0] ) ENTER m.0 The mesure of is out.. Use the sine rtio to check the nswer. sin sin l eg opposite hpotenuse sin nd Prctice/ppl Stud Noteook Hve students dd the definitions/emples of the voculr terms to their Voculr uilder worksheets for hpter 7. include the SOH-H-TO mnemonic device nd n emple similr to Emple on p. for quick reference. include n other item(s) tht the find helpful in mstering the skills in this lesson. oncept heck. The tringles re similr, so the rtios remin the sme. Guided Prctice GUIE PRTIE KEY Eercises Emples, 7 7 KEYSTROKES: nd [SIN ] nd [0] ) ENTER m.0 The nswer is correct.. Eplin wh trigonometric rtios do not depend on the size of the right tringle.. OPEN ENE rw right tringle nd lel the mesures of one cute ngle nd the mesure of the side opposite tht ngle. Then solve for the remining mesures.. See mrgin.. ompre nd contrst the sine, cosine, nd tngent rtios.. Eplin the difference etween tn nd tn. Use to find sin, cos, tn, sin, cos, nd tn. Epress ech rtio s frction nd s deciml to the nerest hundredth.. See mrgin..,, nd c 0.,, nd c 7 Use clcultor to find ech vlue. Round to the nerest ten-thousndth. 7. sin cos cos tn tn sin 0.99 c Lesson 7- Trigonometr 7 out the Eercises Orgniztion Ojective Trigonometric Rtios: Use Trigonometric Rtios: 7, Odd/Even ssignments Eercises re structured so tht students prctice the sme concepts whether the re ssigned odd or even prolems. lert! Eercises 7 require scientific clcultor. ssignment Guide sic: 9 9 odd, 9,, 9 (optionl: ) verge: 9 odd,, 9 (optionl: ) dvnced: 0 even, 7 (optionl: 77 ) nswers. Smple nswer: m 90, m,., c.. ll three rtios involve two sides of right tringle. The sine rtio is the mesure of the opposite leg divided the mesure of the hpotenuse. The cosine rtio is the mesure of the djcent leg divided the mesure of the hpotenuse. The tngent rtio is the mesure of the opposite leg divided the mesure of the djcent leg.. The tn is the rtio of the mesure of the opposite leg divided the mesure of the djcent leg for given ngle in right tringle. The tn is the mesure of the ngle with certin tngent rtio.. 0.; 0.9; 0.9; ; 0.; ; 0.; 0.; ; 0.7;. 7 7 Lesson 7- Trigonometr 7

35 Stud Stud Guide nd nd Intervention Intervention, p. 9 (shown) nd p NME TE PERIO Trigonometr Trigonometric Rtios The rtio of the lengths of two sides of right S tringle is clled trigonometric rtio. The three most common rtios r t re sine, cosine, nd tngent, which re revited sin, cos, nd tn, T respectivel. s sin R leg opposite R leg djcent to R leg opposite R cos R tn R hpotenuse hpotenuse leg djcent to R r t s t r s Emple Find sin, cos, nd tn. Epress ech rtio s deciml to the nerest thousndth. sin o pposite leg hpotenuse cos djcent leg hpotenuse opposite leg tn djcent leg Eercises Find the indicted trigonometric rtio s frction nd s deciml. If necessr, round to the nerest ten-thousndth.. sin. tn 7 ; 0. ; 0.. cos. cos 7 ; 0.70 ; sin. tn E ; 0. ; cos E. cos ; 0. ; NME 9 TE PERIO Prctice (verge) Skills Prctice, p. 7 nd Prctice, p. 7 (shown) Trigonometr Use LMN to find sin L, cos L, tn L, sin M, cos M, nd tn M. Epress ech rtio s frction nd s deciml to the nerest hundredth. L., m, n 9., m, n sin L 9 0.; sin L 0.0; cos L 0.9; cos L 0.7; 9 tn L 0.; tn L 0.; sin M 0.9; sin M 0.7; 9 cos M 9 0.; cos M 0.0; tn M. tn M.7 Use clcultor to find ech vlue. Round to the nerest ten-thousndth.. sin tn cos. 0. Use the figure to find ech trigonometric rtio. Epress nswers s frction nd s deciml rounded to the nerest ten-thousndth.. cos 7. tn. sin E F Find the mesure of ech cute ngle to the nerest tenth of degree. 9. sin tn 0... cos R Find. Round to the nerest tenth GEOGRPHY iego used theodolite to mp region of lnd for his clss in geomorpholog. To determine the elevtion of verticl rock formtion, he mesured the distnce from the se of the formtion to his position nd the ngle etween the ground nd the line of sight to the top of the formtion. The distnce ws meters nd the ngle ws degrees. Wht is the height of the formtion to the nerest meter? m Reding to Lern Mthemtics, p m NME 7 TE PERIO Reding to Lern Mthemtics Trigonometr Pre-ctivit How cn surveors determine ngle mesures? Red the introduction to Lesson 7- t the top of pge in our tetook. Wh is it importnt to determine the reltive positions ccurtel in nvigtion? (Give two possile resons.) Smple nswers: () To void collisions etween ships, nd () to prevent ships from losing their erings nd getting lost t se. Wht does clirted men? Smple nswer: mrked precisel to permit ccurte mesurements to e mde Reding the Lesson. Refer to the figure. Write rtio using the side lengths in the M N figure to represent ech of the following trigonometric rtios. MP NP. sin N. cos N M N M N P. tn N M P NP. tn M NP M P NP MP E. sin M F. cos M M N M N. ssume tht ou enter ech of the epressions in the list on the left into our clcultor. Mtch ech of these epressions with description from the list on the right to tell wht ou re finding when ou enter this epression.. sin 0 v. cos 0 ii c. sin 0. vi d. tn 0. iii e. tn 0 iv f. cos 0. i 0 i. the degree mesure of n cute ngle whose cosine is 0. ii. the rtio of the length of the leg djcent to the 0 ngle to the length of hpotenuse in tringle iii.the degree mesure of n cute ngle in right tringle for which the rtio of the length of the opposite leg to the length of the djcent leg is 0. iv. the rtio of the length of the leg opposite the 0 ngle to the length of the leg djcent to it in tringle v. the rtio of the length of the leg opposite the 0 ngle to the length of hpotenuse in tringle vi. the degree mesure of n cute ngle in right tringle for which the rtio of the length of the opposite leg to the length of the hpotenuse is 0. Helping You Rememer. How cn the co in cosine help ou to rememer the reltionship etween the sines nd cosines of the two cute ngles of right tringle? Smple nswer: The co in cosine comes from complement, s in complementr ngles. The cosine of n cute ngle is equl to the sine of its complement. ELL N R M Lesson 7- indictes incresed difficult Prctice nd ppl For Eercises 7- Enrichment, p. 7 Sine nd osine of ngles The following digrm cn e used to otin pproimte vlues for the sine nd cosine of ngles from 0 to 90. The rdius of the circle is. So, the sine nd cosine vlues cn e red directl from the verticl nd horizontl es , 7 7, 90 NME TE PERIO 0 ppliction See Emples Find the mesure of ech ngle to the nerest tenth of degree.. tn.7 m.. sin 0.07 m 9. OORINTE GEOMETRY Find the mesure of the ngle to the nerest tenth in ech right tringle.. in, for (, 0), (, ), nd (0, ) m.7. in, for (, ), (7, ), nd (7, ) m. 7. SURVEYING Mureen is stnding on horizontl ground level with the se of the N Tower in Toronto, Ontrio. The ngle formed the ground nd the line segment from her position to the top of the tower is.. She knows tht the height of the tower to the top of the ntenne is out feet. Find her distnce from the N Tower to the nerest foot. 997 ft Use PQR with right ngle R to find sin P, cos P, tn P, Q sin Q, cos Q, nd tn Q. Epress ech rtio s frction, nd s deciml to the nerest hundredth.. See mrgin.. p, q, nd r 7 9. p, q, nd r 0. p, q, nd r 0 0. p, q, nd r Use clcultor to find ech vlue. Round to the nerest ten-thousndth.. sin 0.0. tn cos sin tn cos Use the figure to find ech trigonometric rtio. Epress nswers s frction nd s deciml rounded to the nerest ten-thousndth.. sin 9. tn 0. cos. sin. cos. tn. cos. sin. tn Find the mesure of ech ngle to the nerest tenth of degree. 7. sin cos tn E sin cos tn F Find. Round to the nerest tenth hpter 7 Right Tringles nd Trigonometr vid R. Frzier/Photo Reserchers Etr Prctice See pge R P hpter 7 Right Tringles nd Trigonometr

36 ..; ;...;. 9. out 7,7 stronomicl units 0. The stellr prll would e too smll.. See mrgin. 9. VITION plne is one mile ove se level when it egins to clim t constnt ngle of for the net 0 ground miles. out how fr ove se level is the plne fter its clim?. mi SFETY For Eercises 0 nd, use the following informtion. To gurd ginst fll, ldder should mke n ngle of 7 or less with the ground. 0. Wht is the mimum height tht 0-foot ldder cn rech sfel? 9. ft. How fr from the uilding is the se of the ldder t the 7 mimum height?. ft OORINTE GEOMETRY Find the mesure of ech ngle to the nerest tenth in ech right tringle.. J in JL for J(, ), (, ), nd L(7, ) out.. in for (, ), (, ), nd (, ) out.. X in XYZ for X(, 0), Y(7, 0), nd Z(0, ) 9.. Find the perimeter of if m, m 90, nd 0 inches. out 7.9 in. Find nd. Round to the nerest tenth STRONOMY For Eercises 9 nd 0, use the following informtion. One w to find the distnce etween the sun nd reltivel close str is to determine the ngles of sight for the str ectl si months prt. Hlf the mesure formed these two ngles of sight is clled the stellr prll. istnces in spce re sometimes mesured in stronomicl units. n stronomicl unit is equl to the verge distnce etween Erth nd the sun. 0 mi nswer. Smple nswer: Surveors use theodolite to mesure ngles to determine distnces nd heights. nswers should include the following. Theodolites re used in surveing, nvigtion, nd meteorolog. The re used to mesure ngles. The ngle mesures from two points, which re fied distnce prt, to third point. stronom The stellr prll is one of severl methods of tringultion used to determine the distnce of strs from the sun. nother method is trigonometric prll, which mesures the displcement of ner str reltive to more distnt str. Source: lph (stellr prll) enturi (stellr prll) 9. Find the distnce etween lph enturi nd the sun. 0. Mke conjecture s to wh this method is used onl for close strs.. RITIL THINKING Use the figure t the right to find sin. 0. WRITING IN MTH nswer the question tht ws posed t the eginning of the lesson. How do surveors determine ngle mesures? 0 Include the following in our nswer: where theodolites re used, nd Eercise the kind of informtion one otins from theodolite. Erth Sun Erth ( months lter) Lesson 7- Trigonometr 9 StockTrek/ORIS nswers (p. ). 0.; 0.9; 0.; 0.9; 0.; ; 0.; 0.7; 0.; 0.; ; 0.7; 0.; 0.7; 0.; ; 0.7; 0.9; 0.7; 0.7;. Lesson 7- Trigonometr 9

37 ssess Open-Ended ssessment Speking rw nd lel the sides nd ngles of severl right tringles on the ord, nd cll on volunteers to nme the sides the would use to find the sine, cosine, nd tngent of the ngles. Encourge students to use the terms opposite, djcent, nd hpotenuse when nming the sides. Getting Red for Lesson 7- Prerequisite Skill Students will lern out ngles of elevtion nd depression in Lesson 7-. The will use ngle reltionships nd trigonometr to find distnces nd ngles. Use Eercises 77 to determine our students fmilirit with determining ngle reltionships mong ngles formed prllel lines nd trnsversl. ssessment Options Quiz (Lessons 7- nd 7-) is ville on p. 07 of the hpter 7 Resource Msters. Mid-hpter Test (Lessons 7- through 7-) is ville on p. 09 of the hpter 7 Resource Msters. Stndrdized Test Prctice Etending the Lesson. Find cos.. LGER If (), find Ech of the sic trigonometric rtios hs reciprocl rtio. The reciprocls of the sine, cosine, nd tngent re clled the cosecnt, secnt, nd the cotngent, respectivel. Mintin Your Skills Mied Review Find ech mesure. (Lesson 7-) Reciprocl Trigonometric Rtio revition efinition mesure of the hpotenuse c cosecnt of csc sin mesure of the leg opposite co s secnt of sec mesure of the hpotenuse c mesure of the leg djcent mesure of the leg djcent cotngent of cot tn mesure of the leg opposite Use to find csc, sec, cot, csc, sec, nd cot. Epress ech rtio s frction or s rdicl in simplest form.. See mrgin..,, nd c.,, nd c 7.,, nd c.,, nd c 9. If, find nd c., c 70. If, find nd c., c 7. If c, find nd..,. etermine whether ech set of mesures cn e the sides of right tringle. Then stte whether the form Pthgoren triple. (Lesson 7-) 7.,, no, no 7.,, es, es 7. 9,, es, es 7.,, no, no 7. TELEVISION uring 0-minute television progrm, the rtio of minutes of commercils to minutes of the ctul show is :. How mn minutes re spent on commercils? (Lesson -) min 0 c 0 c nswers. csc ; sec ; cot ; Getting Red for the Net Lesson PREREQUISITE SKILL Find ech ngle mesure if h k. (To review ngles formed prllel lines nd trnsversl, see Lesson -.) s 77. m 7 7. m7 0 t csc ; sec ; cot. csc ; sec ; cot ; csc ; sec ; cot 7. csc ; sec ; 70 hpter 7 Right Tringles nd Trigonometr 79. m 0 0. m. m. m h k cot ; csc ; sec ; cot. csc ; sec ; cot ; csc ; sec ; cot 70 hpter 7 Right Tringles nd Trigonometr

38 ngles of Elevtion nd epression Lesson Notes Voculr ngle of elevtion ngle of depression Solve prolems involving ngles of elevtion. Solve prolems involving ngles of depression. do irline pilots use ngles of elevtion nd depression? pilot is getting red to tke off from Mountin Vlle irport. She looks up t the pek of mountin immeditel in front of her. The pilot must estimte the speed needed nd the ngle formed line long the runw nd line from the plne to the pek of the mountin to cler the mountin. NGLES OF ELEVTION n ngle of elevtion is the ngle etween the line of sight nd the horizontl when n oserver looks upwrd. Emple ngle of Elevtion VITION The pek of Goose Mountin is 00 meters higher thn the end of locl irstrip. The pek rises ove point 0 meters from the end of the irstrip. plne tkes off from the end of the runw in the direction of the mountin t n ngle tht is kept constnt until the pek hs een clered. If the pilot wnts to cler the mountin 0 meters, wht should the ngle of elevtion e for the tkeoff to the nerest tenth of degree? Mke drwing. 0 m 00 m Since is 00 meters nd is 0 meters, is 0 meters. Let represent m. tn tn o pposite djcent 0 tn 0, 0 0 tn 0 0 Solve for.. Use clcultor. 0 m line of sight ngle of elevtion Focus -Minute heck Trnsprenc 7- Use s quiz or review of Lesson 7-. Mthemticl ckground notes re ville for this lesson on p. 0. do irline pilots use ngles of elevtion nd depression? sk students: In the right tringle formed the pilot s scenrio, wht tringle side does the height of the mountin represent? Wht does the line long the runw represent? the side opposite the unknown ngle, the side djcent to the unknown ngle If the pilot is going to clculte the ngle tht she needs to cler the mountin given the height of the mountin nd the horizontl distnce from her tkeoff point to the se of the mountin, wht trigonometric rtio would she use? tngent The ngle of elevtion for the tkeoff should e more thn.. Lesson 7- ngles of Elevtion nd epression 7 Roert Holmes/ORIS Resource Mnger Workook nd Reproducile Msters hpter 7 Resource Msters Stud Guide nd Intervention, pp. 7 7 Skills Prctice, p. 77 Prctice, p. 7 Reding to Lern Mthemtics, p. 79 Enrichment, p. 0 Teching Geometr With Mnipultives Msters, p. Trnsprencies -Minute heck Trnsprenc 7- Rel-World Trnsprenc 7 nswer Ke Trnsprencies Technolog GeomPSS: Tutoril Plus, Lesson Interctive hlkord Lesson - Lesson Title 7

39 Tech NGLE OF ELEVTION In-lss Emple In-lss Emples Power Point IRUS TS t the circus, person in the udience t ground level wtches the high-wire routine. -foot- -inch tll crot is stnding on pltform tht is feet off the ground. How fr is the udience memer from the se of the pltform, if the ngle of elevtion from the udience memer s line of sight to the top of the crot is 7º? out 0 ft NGLE OF EPRESSION Power Point Teching Tip Remind students to drw upon their previous knowledge of ngle reltionships formed two prllel lines nd trnsversl. Students m wnt to review these reltionships. wheelchir rmp is meters long nd inclines t º. Find the height of the rmp to the nerest tenth centimeter. out. cm Vernon is on the top deck of cruise ship nd oserves two dolphins following ech other directl w from the ship in stright line. Vernon s position is meters ove se level, nd the ngles of depression to the two dolphins re º nd º. Find the distnce etween the two dolphins to the nerest meter. out m Stud Tip Stndrdized Test Prctice ommon Misconception The ngle of depression is often not n ngle of the tringle, ut the complement to n ngle of the tringle. In, the ngle of depression is E, not. 7 hpter 7 Right Tringles nd Trigonometr NGLES OF EPRESSION n ngle of depression is the ngle etween the line of sight when n oserver looks downwrd, nd the horizontl. Emple Short-Response Test Item ngle of epression The tilgte of moving vn is. feet ove the ground. loding rmp is ttched to the rer of the vn t n incline of 0. Find the length of the rmp to the nerest tenth foot. Solve the Test Item Method The ground nd the horizontl level with the ck of the vn re prllel. Therefore,mm since the re lternte interior ngles.. sin 0 sin 0.. sin 0 0. The rmp is out 0. feet long. Method line of sight. ft ngle of depression Red the Test Item The ngle of depression etween the rmp nd the horizontl is 0. Use trigonometr to find the length of the rmp.. ft 0 0 Indirect Mesurement Olivi is in lighthouse on cliff. She oserves two silots due est of the lighthouse. The ngles of depression to the two ots re nd 7. Find the distnce etween the two silots to the nerest foot. nd re right tringles, nd nd 0 or 9. The distnce etween the ots is or. Use the right tringles to find these two lengths.. ft The horizontl line from the ck of the vn nd the segment from the ground to the ck of the vn re perpendiculr. So, nd re complementr ngles. Therefore, m 900 or 0.. cos 0 cos 0.. co s 0 0. ngles of elevtion or depression to two different ojects cn e used to find the distnce etween those ojects. Emple ft 0 ft 7 E 7 hpter 7 Right Tringles nd Trigonometr ifferentited Instruction Kinesthetic Using meterstick nd clcultor, groups of students cn find ngles of elevtion nd depression for different ojects in the clssroom. Groups cn mesure one person s ee level from the floor, nd the topmost height of wll clock from the floor. The person stnds feet w from the clock, nd the group clcultes the ngle of elevtion from the person s line of sight to the top of the oject. Repet for items plced on the floor, nd include vritions like hving the person stnd on chir, or plcing two ojects on the floor certin distnce from ech other.

40 ecuse E nd re horizontl lines, the re prllel. Thus, E nd E ecuse the re lternte interior ngles. This mens tht m nd m 7. tn 9 tn 9 9 t n 00.7 tn 7 9 tn t n 7. tn o pposite ; m djcent Multipl ech side. ivide ech side tn. Use clcultor. tn o pposite ; m 7 djcent Multipl ech side. ivide ech side tn 7. Use clcultor. The distnce etween the ots is or out 7 feet. Prctice/ppl Stud Noteook Hve students dd the definitions/emples of the voculr terms to their Voculr uilder worksheets for hpter 7. include n other item(s) tht the find helpful in mstering the skills in this lesson. oncept heck Guided Prctice GUIE PRTIE KEY Eercises Emples, 7 Stndrdized Test Prctice. OPEN ENE Find rel-life emple of. See mrgin. n ngle of depression. rw digrm nd identif the ngle of depression. P. Eplin wh n ngle of elevtion is given tht nme.. Nme the ngles of depression nd elevtion in the figure. The ngle of depression is FP nd the ngle of elevtion is TP.. VITION pilot is fling t 0,000 feet nd wnts to tke the plne up to 0,000 feet over the net 0 miles. Wht should e his ngle of elevtion to the nerest tenth? (Hint: There re 0 feet in mile.) out.. SHOWS Find the ngle of elevtion of the sun when 7.-meter flgpole csts.-meter shdow. Round to the nerest tenth of degree..7. SLVGE slvge ship uses sonr to determine tht the ngle of depression to wreck on the ocen floor is.. The depth chrt shows tht the ocen floor is 0 meters elow the surfce. How fr must diver lowered from the slvge ship wlk long the ocen floor to rech the wreck? out 9.9 m 7. SHORT RESPONSE From the top of 0-foot high tower, n ir trffic controller oserves n irplne on the runw. To the nerest foot, how fr from the se of the tower is the irplne? 70 ft T. Lesson 7- ngles of Elevtion nd epression 7 F 0 ft 0 m out the Eercises Orgniztion Ojective ngles of Elevtion:, ngles of epression:,, 9 0 Odd/Even ssignments Eercises 0 re structured so tht students prctice the sme concepts whether the re ssigned odd or even prolems. lert! Eercise requires the Internet or other reserch mterils. ssignment Guide sic: 9 odd, verge: 9 odd, dvnced: even, 7 0 (optionl: ) nswers. Smple nswer:. Smple nswer: n ngle of elevtion is clled tht ecuse the ngle formed horizontl line nd segment joining two endpoints rises ove the horizontl line. Lesson 7- ngles of Elevtion nd epression 7

41 Stud Stud Guide nd nd Intervention Intervention, p. 7 (shown) nd p NME TE PERIO ngles of Elevtion nd epression ngles of Elevtion Mn rel-world prolems tht involve looking up to n oject cn e descried in terms of n ngle of elevtion, which is the ngle etween n oserver s line of sight nd horizontl line. Emple The ngle of elevtion from point to the top of cliff is. If point is 000 feet from the se of the cliff, how high is the cliff? Let the height of the cliff. tn 0 00 tn o pposite djcent 000(tn ) Multipl ech side Use clcultor. The height of the cliff is out 7. feet. Eercises Solve ech prolem. Round mesures of segments to the nerest whole numer nd ngles to the nerest degree.. The ngle of elevtion from point to the top of hill is 9. If point is 00 feet from the se of the hill, how high is the hill?? 0 ft 9 00 ft. Find the ngle of elevtion of the sun when.-meter-tll telephone pole csts -meter-long shdow.. ldder lening ginst uilding mkes n ngle of 7 with the ground. The foot of the ldder is feet from the uilding. How long is the ldder? ft. person whose ees re feet ove the ground is stnding on the runw of n irport 00 feet from the control tower. Tht person oserves n ir trffic controller t the window of the -foot tower. Wht is the ngle of elevtion? 7- NME 7 TE PERIO Prctice (verge) ngles of Elevtion nd epression Skills Prctice, p. 77 nd Prctice, p. 7 (shown) Nme the ngle of depression or ngle of elevtion in ech figure.. T R. R P Z Y TRZ; YZR PRM; LMR ft? m? 7 ft? 000 ft 00 ft line of sight ngle of elevtion. WTER TOWERS student cn see wter tower from the closest point of the soccer field t Sn Loos High School. The edge of the soccer field is out 0 feet from the wter tower nd the wter tower stnds t height of. feet. Wht is the ngle of elevtion if the ee level of the student viewing the tower from the edge of the soccer field is feet ove the ground? Round to the nerest tenth degree. out.. ONSTRUTION roofer props ldder ginst wll so tht the top of the ldder reches 0-foot roof tht needs repir. If the ngle of elevtion from the ottom of the ldder to the roof is, how fr is the ldder from the se of the wll? Round our nswer to the nerest foot. out ft. TOWN ORINNES The town of elmont restricts the height of flgpoles to feet on n propert. Linds wnts to determine whether her school is in complince with the regultion. Her ee level is. feet from the ground nd she stnds feet from the flgpole. If the ngle of elevtion is out, wht is the height of the flgpole to the nerest tenth foot? out. ft. GEOGRPHY Stephn is stnding on mes t the Pinted esert. The elevtion of the mes is out 0 meters nd Stephn s ee level is. meters ove ground. If Stephn cn see nd of multicolored shle t the ottom nd the ngle of depression is 9, out how fr is the nd of shle from his ees? Round to the nerest meter. out 0 m 7. INIRET MESUREMENT Mr. ominguez is stnding on 0-foot ocen luff ner his home. He cn see his two dogs on the ech elow. If his line of sight is feet ove the ground nd the ngles of depression to his dogs re nd, how fr prt re the dogs to the nerest foot? out 7 ft Reding to Lern Mthemtics, p Mr. ominguez sun. m NME 7 TE PERIO Reding to Lern Mthemtics ngles of Elevtion nd epression Pre-ctivit How do irline pilots use ngles of elevtion nd depression? Red the introduction to Lesson 7- t the top of pge 7 in our tetook. Wht does the ngle mesure tell the pilot? Smple nswer: how steep her scent must e to cler the pek Reding the Lesson. Refer to the figure. The two oservers re looking t U one nother. Select the correct choice for ech question.. Wht is the line of sight? iii oserver (i) line RS (ii) line ST (iii) line RT (iv) line TU on ground R. Wht is the ngle of elevtion? ii (i) RST (ii) SRT (iii) RTS (iv) UTR c. Wht is the ngle of depression? iv (i) RST (ii) SRT (iii) RTS (iv) UTR d. How re the ngle of elevtion nd the ngle of depression relted? ii (i) The re complementr. (ii) The re congruent. (iii) The re supplementr. (iv) The ngle of elevtion is lrger thn the ngle of depression. e. Which postulte or theorem tht ou lerned in hpter supports our nswer for prt c? iv (i) orresponding ngles Postulte (ii) lternte Eterior ngles Theorem (iii) onsecutive Interior ngles Theorem (iv) lternte Interior ngles Theorem. student ss tht the ngle of elevtion from his ee to the top of flgpole is. Wht is wrong with the student s sttement? n ngle of elevtion cnnot e otuse. Helping You Rememer. good w to rememer something is to eplin it to someone else. Suppose clssmte finds it difficult to distinguish etween ngles of elevtion nd ngles of depression. Wht re some hints ou cn give her to help her get it right ever time? Smple nswers: () The ngle of depression nd the ngle of elevtion re oth mesured etween the horizontl nd the line of sight. () The ngle of depression is lws congruent to the ngle of elevtion in the sme digrm. () ssocite the word elevtion with the word up nd the word depression with the word down. L M ft 0 ft luff. ft ELL ft S ft oserver t T top of uilding Lesson 7- indictes incresed difficult Prctice nd ppl For Eercises, 9,, 9, 0 7 hpter 7 Right Tringles nd Trigonometr NME TE PERIO 7- Enrichment, p. 0 Reding Mthemtics See Emples Etr Prctice See pge 7. Rilrods The Mononghel Incline is feet long with verticl rise of 9.9 feet. It ws uilt t cost of $0,000 nd opened on M, 70. It is still used commuters to nd from Pittsurgh. Source: The three most common trigonometric rtios re sine, cosine, nd tngent. Three other rtios re the cosecnt, secnt, nd cotngent. The chrt elow shows revitions nd definitions for ll si rtios. Refer to the tringle t the right. revition Red s: Rtio leg opposite sin the sine of c hpotenuse leg djcent to cos the cosine of c hpotenuse leg opposite tn the tngent of leg djcent to csc the cosecnt of hpotenuse c leg opposite hpotenuse c sec the secnt of leg djcent to. OTING Two ots re oserved prsiler 7 meters ove lke. The ngles of depression re. nd 7. How fr prt re the ots? out 7 m 9. GOLF golfer is stnding t the tee, looking up to the green on hill. If the tee is rds lower thn the green nd the ngle of elevtion from the tee to the hole is, find the distnce from the tee to the hole. out 7. d 0. VITION fter fling t n ltitude of 00 meters, helicopter strts to descend when its ground distnce from the lnding pd is kilometers. Wht is the ngle of depression for this prt of the flight? out.. SLEING sledding run is 00 rds long with verticl drop of 7. rds. Find the ngle of depression of the run. out.. RILROS The Mononghel Incline overlooks the cit of Pittsurgh, Pennslvni. Refer to the informtion t the left to determine the incline of the rilw. out.. MUSEMENT PRKS From the top of roller coster, 0 rds ove the ground, rider looks down nd sees the merr-go-round nd the Ferris wheel. If the ngles of depression re nd respectivel, how fr prt re the merr-go-round nd the Ferris wheel? out. d IVIL ENGINEERING For Eercises nd, use the following informtion. The percent grde of highw is the rtio of the verticl rise or fll over given horizontl distnce. The rtio is epressed s percent to the nerest whole numer. Suppose highw hs verticl rise of 0 feet for ever 000 feet of horizontl distnce.. lculte the percent grde of the highw. 7%. Find the ngle of elevtion tht the highw mkes with the horizontl. out. SKIING ski run hs n ngle of elevtion of. nd verticl drop of 00 feet. To the nerest foot, how long is the ski run? ft GEYSERS For Eercises 7 nd, use the following informtion. Kirk visits Yellowstone Prk nd Old Fithful on perfect d. His ees re feet from the ground, nd the geser cn rech heights rnging from 90 feet to feet. 7. If Kirk stnds 00 feet from the geser nd the eruption rises 7 feet in the ir, wht is the ngle of elevtion to the top of the spr to the nerest tenth? out 0.. In the fternoon, Kirk returns nd oserves the geser s spr rech height of feet when the ngle of elevtion is 7. How fr from the geser is Kirk stnding to the nerest tenth of foot? out. ft 9. IRWTHING Two oservers re 00 feet prt, in line with tree contining ird s nest. The ngles of elevtion to the ird s nest re 0 nd 0. How fr is ech oserver from the se of the tree? 00 ft, 00 ft c 0 d d otngent of le o 7 hpter 7 Right Tringles nd Trigonometr

42 0. METEOROLOGY The ltitude of the se of cloud formtion is clled the ceiling. To find the ceiling one night, meteorologist directed spotlight verticll t the clouds. Using theodolite plced meters from the spotlight nd. meters ove the ground, he found the ngle of elevtion to e.7. How high ws the ceiling? out. m.7 m MEIINE For Eercises, use the following informtion. doctor is using tredmill to ssess the strength of ptient s hert. t the eginning of the em, the -inch long tredmill is set t n incline of 0.. How fr off the horizontl is the rised end of the tredmill t the eginning of the em? out. in.. uring one stge of the em, the end of the tredmill is 0 inches ove the horizontl. Wht is the incline of the tredmill to the nerest degree?. Suppose the em is divided into five stges nd the incline of the tredmill is incresed for ech stge. oes the end of the tredmill rise the sme distnce etween ech stge? no. m nswer 7. nswers should include the following. Pilots use ngles of elevtion when the re scending nd ngles of depression when descending. ngles of elevtion re formed when person looks upwrd nd ngles of depression re formed when person looks downwrd. Trvel ers Rock is the lrgest monolith, tpe of rock formtion, in the world. It is pproimtel. kilometers long nd kilometers wide. The rock is elieved to e the tip of mountin, two thirds of which is underground. Source: TRVEL Kwn-Yong uses theodolite to mesure the ngle of elevtion from the ground to the top of ers Rock to e.. He wlks hlf kilometer closer nd mesures the ngle of elevtion to e.. How high is ers Rock to the nerest meter? out m. EROSPE On Jul 0, 99, Neil rmstrong ecme the first humn to wlk on the moon. uring this mission, pollo orited the moon three miles ove the surfce. t one point in the orit, the onord guidnce sstem mesured the ngles of depression to the fr nd ner edges of lrge crter. The ngles mesured nd 9, respectivel. Find the distnce cross the crter. out. mi orit f 9 mi n Online Reserch t Updte Use the Internet to determine the ngle of depression formed someone ord the interntionl spce sttion looking down to our communit. Visit to lern more.. RITIL THINKING Two wether oservtion sttions re 7 miles prt. wether lloon is locted etween the sttions. From Sttion, the ngle of elevtion to the wether lloon is. From Sttion, the ngle of elevtion to the lloon is. Find the ltitude of the lloon to the nerest tenth of mile. out.0 mi 7. WRITING IN MTH nswer the question tht ws posed t the eginning of the lesson. See mrgin. How do irline pilots use ngles of elevtion nd depression? Include the following in our nswer: when pilots use ngles of elevtion or depression, nd the difference etween ngles of elevtion nd depression. Lesson 7- ngles of Elevtion nd epression 7 John Med/Science Photo Lirr/Photo Reserchers Techer to Techer Susn M. Prece Plmouth South High School, Plmouth, M I like to mke clinometer (lso clled hpsometer) nd hve m students tr to mesure ojects in the schoolrd, such s height of the uilding, height of the flgpole, nd height of the lights. (irections cn e found t Lesson 7- ngles of Elevtion nd epression 7

43 ssess Open-Ended ssessment Modeling Using ruler, hve students model different ngles of elevtion nd depression nd nme rel-world situtions tht the ngle depicted could e modeling. Stndrdized Test Prctice. The top of signl tower is 0 meters ove se level. The ngle of depression from the top of tower to pssing ship is. How mn meters from the foot of the tower is the ship?.9 m 7. m. m m 0 m 9. LGER If, then find when Getting Red for Lesson 7- Prerequisite Skill Students will lern out the Lw of Sines in Lesson 7-. The will use proportions to solve for side lengths nd ngle mesures. Use Eercises to determine our students fmilirit with solving proportions. Mintin Your Skills Mied Review Find the mesure of ech ngle to the nerest tenth of degree. (Lesson 7-) 0. cos sin tn.7.. cos sin E 0... tn F Find nd. (Lesson 7-) ; ; 0 ; 0 9. HOIES twin-engine irplne used for medium-rnge flights hs length of 7 meters nd wingspn of 90 meters. If scle model is mde with wingspn of centimeters, find its length. (Lesson -). cm 0. op nd complete the flow proof. (Lesson -) Given: FR GS Prove: Proof: R F G X S Given FX GX FR GS Given.? Vert. s re. FXR GXS?. S c. d.?? PT? e. f.? Isos. Th. Getting Red for the Net Lesson PREREQUISITE SKILL Solve ech proportion. (To review solving proportions, see Lesson -.) hpter 7 Right Tringles nd Trigonometr 7 hpter 7 Right Tringles nd Trigonometr

44 The Lw of Sines Lesson Notes Voculr Lw of Sines solving tringle Stud Tip Otuse ngles There re lso vlues for sin, cos, nd tn, when 90. Vlues of the rtios for these ngles will e found using the trigonometric functions on our clcultor. Use the Lw of Sines to solve tringles. Solve prolems using the Lw of Sines. re tringles used in rdio stronom? The Ver Lrge rr (VL), one of the world s premier stronomicl rdio oservtories, consists of 7 rdio ntenns in Y-shped configurtion on the Plins of Sn gustin in New Meico. stronomers use the VL to mke pictures from the rdio wves emitted stronomicl ojects. onstruction of the ntenns is supported vriet of tringles, mn of which re not right tringles. THE LW OF SINES In trigonometr, the Lw of Sines cn e used to find missing prts of tringles tht re not right tringles. Let e n tringle with,, nd c representing the mesures of the sides opposite the ngles with mesures,, nd, respectivel. Then sin sin sin. c Lw of Sines c Focus -Minute heck Trnsprenc 7- Use s quiz or review of Lesson 7-. Mthemticl ckground notes re ville for this lesson on p. 0. re tringles used in rdio stronom? sk students: Wh is it clled rdio stronom? ecuse the re nlzing rdio signls Wht re some emples of stronomicl ojects? Smple nswers: strs, plnets, qusrs, rown dwrfs, glies Proof Lw of Sines is tringle with n ltitude from tht intersects t. Let h represent the mesure of. Since nd re right tringles, we cn find sin nd sin. h sin h sin h efinition of sine sin h sin h ross products sin sin sin sin Sustitution ivide ech side. The proof cn e completed using similr technique with the other ltitudes to show tht sin sin nd sin sin. c c Workook nd Reproducile Msters hpter 7 Resource Msters Stud Guide nd Intervention, pp. Skills Prctice, p. Prctice, p. Reding to Lern Mthemtics, p. Enrichment, p. ssessment, p. 0 Grphing lcultor nd omputer Msters, p. 0 Lesson 7- The Lw of Sines 77 Roger Ressmeer/ORIS Resource Mnger Trnsprencies -Minute heck Trnsprenc 7- nswer Ke Trnsprencies Technolog Interctive hlkord Multimedi pplictions: Virtul ctivities Lesson - Lesson Title 77

45 Tech THE LW OF SINES In-lss Emples Power Point. Find p. Round to the nerest tenth. Q 7 9 P R.. Find ml to the nerest degree in LMN if n 7, 9, nd mn. Teching Tip To sve step, students cn lso write sin sin nd then sustitute vlues to solve.. Solve EF if m, mf, nd f. d.; e.; me 0. Solve IJK if mj, i 0, nd j. k 7.; mi.; mk. Stud Tip Rounding If ou round efore the finl nswer, our results m differ from results in which rounding ws not done until the finl nswer. Stud Tip Look ck To review the ngle Sum Theorem, see Lesson -. Emple Use the Lw of Sines. Find. Round to the nerest tenth. Use the Lw of Sines to write proportion. sin sin Lw of Sines sin 7 sin m 7,, m sin 7 sin ross products sin sin 7 ivide ech side sin 7.. Use clcultor.. Find mz to the nerest degree in XYZ if 7, z, nd my 9. Write proportion relting Y, Z,, nd z. sin Y sin Z Lw of Sines z sin 9 si n Z 7 my 9, 7, z sin 9 7 sin Z ross products s in 9 sin Z 7 ivide ech side 7. sin s in 9 7 Z Solve for Z. Z Use clcultor. So, mz. The Lw of Sines cn e used to solve tringle. Solving tringle mens finding the mesures of ll of the ngles nd sides of tringle. Emple Solve Tringles Therefore, m 00, 0., nd c Solve if m, m 7, nd. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth. We know the mesures of two ngles of the tringle. Use the ngle Sum Theorem to find m. c m m m 0 ngle Sum Theorem 7 m 0 m, m 7 0 m 0 dd. m 00 Sutrct 0 from ech side. Since we know m nd, use proportions involving sin. To find : To find c: sin sin Lw of Sines sin sin c sin 7 sin Sustitute. sin 7 sin 00 c sin 7 sin ross products c sin 7 sin 00 sin sin 7 sin 00 s in 7 0. Use clcultor. c.9 7 hpter 7 Right Tringles nd Trigonometr ifferentited Instruction Interpersonl Seprte the clss into groups of three students to work through the eercises. One group memer cn select the eercise. The second memer sets up the eqution for the Lw of Sines nd fills in the vlues. The third person uses clcultor to solve the prolem. Group memers cn rotte tsks so ech memer prticiptes in ech responsiilit. 7 hpter 7 Right Tringles nd Trigonometr

46 Stud Tip n Equivlent Proportion The Lw of Sines m lso e written s c. sin sin sin You m wish to use this form when finding the length of side.. Solve if m 9,, nd c 0. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth. We know the mesures of two sides nd n ngle opposite one of the sides. sin sin c Lw of Sines si n 9 0 m 9,, nd c 0 0 sin sin 9 ross products sin s in 9 0 ivide ech side 0. sin s in 9 0 Solve for. Use clcultor. m m m 0 ngle Sum Theorem m 9 0 m nd m 9 m 0 dd. m Sutrct from ech side. sin sin c Lw of Sines sin sin 9 0 m, m 9, nd c 0 0 sin sin 9 ross products 0 sin sin 9 ivide ech side sin 9.. Use clcultor. Therefore,,, nd.. USE THE LW OF SINES TO SOLVE PROLEMS In-lss Emples Power Point In Emple, wht hppens to the length of the shdow lter in the d when the ngle of elevtion decreses to º? ssume tht the length nd position of the telephone pole re still the sme. The length of the shdow increses to out 7. feet. USE THE LW OF SINES TO SOLVE PROLEMS The Lw of Sines is ver useful in solving direct nd indirect mesurement pplictions. Emple Indirect Mesurement When the ngle of elevtion to the sun is, telephone pole tilted t n ngle of 7 from the verticl csts shdow of 0 feet long on the ground. Find the length of the telephone pole to the nerest tenth of foot. rw digrm. rw S G. Then find the mgp nd mgp. mgp 90 7 or mgp 0 or mgp Since ou know the mesures of two ngles of the tringle, mgp nd mgp, nd the length of side opposite one of the ngles (G is opposite GP) ou cn use the Lw of Sines to find the length of the pole. S P 7 pole G 0-foot shdow (continued on the net pge) Lesson 7- The Lw of Sines 79 Lesson 7- The Lw of Sines 79

47 Prctice/ppl Stud Noteook Hve students dd the definitions/emples of the voculr terms to their Voculr uilder worksheets for hpter 7. include one completel worked emple similr to Emple on pp include n other item(s) tht the find helpful in mstering the skills in this lesson. FIN THE ERROR Eplin tht if F in the figure hd een 90º, then Mkl nd Felipe would hve found the sme vlue for d. Students should recognize tht the could use the Lw of Sines to solve right tringles, ut trigonometric rtios re more efficient. out the Eercises Orgniztion Ojective The Lw of Sines: 9 Use the Lw of Sines to Solve Prolems: 0 Odd/Even ssignments Eercises re structured so tht students prctice the sme concepts whether the re ssigned odd or even prolems. ssignment Guide sic: 7 9 odd, 0 verge: 7 9 odd, 0 dvnced: 0 even, (optionl: ) ll: Quiz ( ) oncept heck. Felipe; Mkl is using the definition of the sine rtio for right tringle, ut this is not right tringle. Guided Prctice GUIE PRTIE KEY Eercises Emples 7. mp, q 7., r mr 9, mq, q mq 9, p.0, r.7. mq, mr 7, r 9. 0 hpter 7 Right Tringles nd Trigonometr nswer P G Lw of Sines sin GP sin GP P 0 sin sin mgp, mgp, nd G 0 P sin 0 sin ross products P 0 sin ivide ech side sin. sin P. Use clcultor. The telephone pole is out. feet long. The Lw of Sines cn e used to solve tringle in the following cses. se se You know the mesures of two ngles nd n side of tringle. (S or S) You know the mesures of two sides nd n ngle opposite one of these sides of the tringle. (SS). FIN THE ERROR Mkl nd Felipe re tring to find d in EF. Mkl sin 9 d Felipe sin 9 sin d Who is correct? Eplin our resoning.. OPEN ENE rw n cute tringle nd lel the mesures of two ngles nd the length of one side. Eplin how to solve the tringle.. ompre the two cses for the Lw of Sines.. See mrgin. Find ech mesure using the given mesures of XYZ. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth.. Find if, mx 7, nd my... Find if., mx 7, nd mz 7... Find my if 7, z, nd mz Find mz if 7, z, nd my 9. Solve ech PQR descried elow. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth.. mr, mq 9, p 7 9. p, r, mp 0 0. mp, mr, q. p, q, mp 0. mp 0, mq, p. q 7., r 9., mq 0.7 mr, q., r. mp 7, p.; mr. Find the perimeter of prllelogrm to the nerest tenth.. units F 7 d f. Smple nswer: Let m, me 7, nd d. Then F sin is the fied rtio or scle fctor for the Lw of Sines etended proportion. The length of e is found using sin sin 7. The e d e mf is found evluting 0 (m me). In this prolem mf. The length of f is found using sin sin. E f f Lw of Sines E 0 hpter 7 Right Tringles nd Trigonometr

48 ppliction. SURVEYING To find the distnce etween two points nd tht re on opposite sides of river, surveor mesures the distnce to point on the sme side of the river s point. The distnce from to is 0 feet. He then mesures the ngle from to s nd mesures the ngle from to s. Find the distnce from to. out 7. ft 7- NME TE PERIO The Lw of Sines The Lw of Sines In n tringle, there is specil reltionship etween the ngles of the tringle nd the lengths of the sides opposite the ngles. Lw of Sines sin sin sin c Emple In, find. Emple Stud Stud Guide nd nd Intervention Intervention, p. (shown) nd p. 7 0 sin sin c Lw of Sines E F sin sin E d e In EF, find m. Lw of Sines Lesson 7- Prctice nd ppl sin sin 7 m, c 0, m 7 0 sin 0 sin 7 ross multipl. 0 sin 7 ivide ech side sin. sin 0. Use clcultor. si n sin d, me, e sin sin ross multipl. sin s in ivide ech side. sin s in Use the inverse sine.. Use clcultor. For Eercises 9 0 See Emples Etr Prctice See pge 7.. mw, w 7.,.. mx., mw., w 0.. my 0, w.,.. mx 9., mw.7, w.. mx 7,.,. 7. mx,.,.7. my 7., mx.,. 9. mx 9., my.,. Find ech mesure using the given mesures of KLM. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth.. If ml, mk, nd, find k If k., ml, nd mk 70, find..7. If m 0., k., nd mk 7, find mm. 9. If k 0, m., nd mk 9, find mm If ml, mk, nd m., find... If mm 9,., nd m., find ml. 9 Solve ech WXY descried elow. Round mesures to the nerest tenth.. my 7, 7., mx. 0.,.7, my 9. mx, mw,.. my, 0,. mw, my, w. 7. mw, my, w.. w 0, 9., mw 07 9., w, mw 0. n isosceles tringle hs se of centimeters nd verte ngle of. Find the perimeter. out. cm. Find the perimeter of qudrilterl to the nerest tenth..9 units. GRENING Elen is plnning tringulr grden. She wnts to uild fence round the grden to keep out the deer. The length of one side of the grden is feet. If the ngles t the end of this side re 7 nd, find the length of fence needed to enclose the grden. out 77. ft. VITION Two rdr sttions tht re 0 miles prt locted plne t the sme time. The first sttion indicted tht the position of the plne mde n ngle of with the line etween the sttions. The second sttion indicted tht it mde n ngle of with the sme line. How fr is ech sttion from the plne? out.9 mi, out. mi. SURVEYING Mri Lopez is surveor who must determine the distnce cross section of the Rio Grnde Gorge in New Meico. Stnding on one side of the ridge, she mesures the ngle formed the edge of the ridge nd the line of sight to tree on the other side of the ridge. She then wlks long the ridge feet, pssing the tree nd mesures the ngle formed the edge of the ridge nd the new line of sight to the sme tree. If the first ngle is 0 nd the second ngle is, find the distnce cross the gorge. out 9 ft 0 Eercises Find ech mesure using the given mesures of. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth.. If c, m 0, nd m 0, find... If 0, c, nd m, find m. 7. If, c, nd m, find m.. If, m 7, nd m 7, find If, m 9, nd m 0, find... If 0, c 0, nd m 0, find m. 7- NME TE PERIO Prctice (verge) Skills Prctice, p. nd Prctice, p. (shown) The Lw of Sines Find ech mesure using the given mesures from EFG. Round ngle mesures to the nerest tenth degree nd side mesures to the nerest tenth.. If mg, me 7, nd e, find g..7. If e.7, me, nd mf, find f... If g, f., nd mg, find mf... If e 9., mg, nd me, find g..9. If f 9., g 7., nd mg, find mf.. Solve ech STU descried elow. Round mesures to the nerest tenth.. mt, s., t. ms., mu., u s 0, u, ms 7 mt., mu 0., t.9. mu 7, t., mt 7 ms, s., u.7 9. ms, mu 9, s 7. mt 9, t 7., u t., u.7, mt ms 9.7, mu., s 9.. ms 9, s., t mt., mu., u.. mt 9, mu 7, u 9. ms, s., t 9.9. t., ms, mt 7 mu 9, s.0, u.. INIRET MESUREMENT To find the distnce from the edge of the lke to the tree on the islnd in the lke, Hnnh set up tringulr configurtion s shown in the digrm. The distnce from loction to loction is meters. The mesures of the ngles t nd re nd, respectivel. Wht is the distnce from the edge of the lke t to the tree on the islnd t? out 9. m Reding to Lern Mthemtics, p. 7- NME TE PERIO Reding to Lern Mthemtics The Lw of Sines Pre-ctivit How re tringles used in rdio stronom? Red the introduction to Lesson 7- t the top of pge 77 in our tetook. Wh might severl ntenns e etter thn one single ntenn when studing distnt ojects? Smple nswer: Oserving n oject from onl one position often does not provide enough informtion to clculte things such s the distnce from the oserver to the oject. Reding the Lesson. Refer to the figure. ccording to the Lw of Sines, which of the following re correct sttements?, F m n p.. sin m si n n sin p sin M sin N sin P M N P ELL M n P p m N Lesson 7- Lesson 7- The Lw of Sines. co s M cos N co s P m n p sin M. sin N sin P m n p E. (sin M) (sin N) (sin P) F. sin P sin M sin N p m n nswers (pge 0). In one cse ou need the mesures of two sides nd the mesure of n ngle opposite one of the sides. In the other cse ou need the mesures of two ngles nd the mesure of side. 7- Enrichment, p. Identities NME TE PERIO n identit is n eqution tht is true for ll vlues of the vrile for which oth sides re defined. One w to verif n identit is to use right tringle nd the definitions for trigonometric functions. Emple Verif tht (sin ) (cos ) is n identit. (sin ) (cos ) c c c c c To check whether n eqution m e n identit, ou cn test severl vlues. However, since ou cnnot test ll vlues, ou cnnot e certin tht the eqution is n identit. Emple Test sin sin cos to see if it could e n identit. clcultor to ev epr c. Stte whether ech of the following sttements is true or flse. If the sttement is flse, eplin wh.. The Lw of Sines pplies to ll tringles. true. The Pthgoren Theorem pplies to ll tringles. Flse; smple nswer: It onl pplies to right tringles. c. If ou re given the length of one side of tringle nd the mesures of n two ngles, ou cn use the Lw of Sines to find the lengths of the other two sides. true d. If ou know the mesures of two ngles of tringle, ou should use the Lw of Sines to find the mesure of the third ngle. Flse; smple nswer: You should use the ngle Sum Theorem. e. friend tells ou tht in tringle RST, mr, r centimeters, nd s centimeters. n ou use the Lw of Sines to solve the tringle? Eplin. No; smple nswer: In n tringle, the longest side is opposite the lrgest ngle. ecuse tringle cn hve onl one otuse ngle, R must e the lrgest ngle, ut s r, so it is impossile to hve tringle with the given mesures. Helping You Rememer. Mn students rememer mthemticl equtions nd formuls etter if the cn stte them in words. Stte the Lw of Sines in our own words without using vriles or mthemticl smols. Smple nswer: In n tringle, the rtio of the sine of n ngle to the length of the opposite side is the sme for ll three ngles. Lesson 7- The Lw of Sines

49 nswers 0. Yes; in right sin sin c where is the right ngle, sin s in. Since m 90, c then sin sin 90. Since the c sin 90, then sin, which c is the definition of the sine rtio.. Smple nswer: Tringles re used to determine distnces in spce. nswers should include the following. The VL is one of the world s premier stronomicl rdio oservtories. It is used to mke pictures from the rdio wves emitted stronomicl ojects. Tringles re used in the construction of the ntenns. vition From Jnur 00 to M 00, Poll Vcher flew over 9,000 miles to ecome the first womn to fl the smllest single-engined ircrft round the world vi ustrli nd the Pcific. Source: REL ESTTE house is uilt on tringulr plot of lnd. Two sides of the plot re 0 feet long, nd the meet t n ngle of. If fence is to e plced long the perimeter of the propert, how much fencing mteril is needed? out ft MIRRORS For Eercises nd 7, use the following informtion. Kl, Jenn, nd Pige live in town nicknmed Perpendiculr it ecuse the plnners nd uilders took gret cre to hve ll the streets oriented north-south or est-west. The three of them pl gme where the signl ech other using mirrors. Kl nd Jenn signl ech other from distnce of meters. Jenn turns 7 to signl Pige. Kl rottes 0 to signl Pige.. To the nerest tenth of meter, how fr prt re Kl nd Pige? out 70. m 7. To the nerest tenth of meter, how fr prt re Jenn nd Pige? out m VITION For Eercises nd 9, use the following informtion. Keish Jefferson is fling smll plne due west. 7 To void the jet strem, she must chnge her course. She turns the plne 7 to the south nd flies 0 miles. Then she mkes turn of heds ck to her originl course.. out. mi. How fr must she fl fter the second turn to return to the originl course? 9. How mn miles did she dd to the flight chnging course? out. mi 0 ft 0. RITIL THINKING oes the Lw of Sines ppl to the cute ngles of right tringle? Eplin our nswer. See mrgin. 0 ft Stndrdized Test Prctice. WRITING IN MTH nswer the question tht ws posed t the eginning of the lesson. See mrgin. How re tringles used in rdio stronom? Include the following in our nswer: description of wht the VL is nd the purpose of the VL, nd the purpose of the tringles in the VL.. SHORT RESPONSE In XYZ, if, mx, nd my, solve the tringle to the nerest tenth. mz 0,.0, nd z.. LGER The tle elow shows the customer rtings of five resturnts in the Metro it Guide to Resturnts. The rting scle is from, the worst, to 0, the est. Which of the five resturnts hs the est verge rting? Resturnt Food ecor Service Vlue el lnco s quvent 9 Le ircus 0 Sushi Mmo 7 7 Metropolis Grill Metropolis Grill quvent Le ircus el lnco s hpter 7 Right Tringles nd Trigonometr Re US Ltd. hpter 7 Right Tringles nd Trigonometr

50 Mintin Your Skills Mied Review Getting Red for the Net Lesson P rctice Quiz Find to the nerest tenth. (Lesson 7-) RHITETURE For Eercises nd, use the following informtion. Mr. Mrtinez is n rchitect who designs houses so tht the windows receive minimum sun in the summer nd mimum sun in the winter. For olumus, Ohio, the ngle of elevtion of the sun t noon on the longest d is 7. nd on the shortest d is.. Suppose house is designed with south-fcing window tht is feet tll. The top of the window is to e instlled foot elow the overhng. (Lesson 7-). How long should the rchitect mke the overhng so tht the window gets no direct sunlight t noon on the longest d? out.07 ft. Using the overhng from Eercise, how much of the window will get direct sunlight t noon on the shortest d? out.97 ft Use JKL to find sin J, cos J, tn J, sin L, cos L, nd tn L. J Epress ech rtio s frction nd s deciml to the nerest hundredth. (Lesson 7-) 9. See mrgin.. j, k 7, 7. j 0, k 9,. j, k, 9. j 7, k, 7 If KH is prllel to JI, find the mesure of ech ngle. (Lesson -) PREREQUISITE SKILL Evlute c for the given vlues of,, nd c. (To review evluting epressions, see pge 7.) ,, c 0., 9, c.,, c 0.,, c 7., 0, c 9., 7, c OMMUNITIONS To secure 00-foot. Solve EF. rdio tower ginst high winds, gu wires re (Lesson 7-) ttched to the tower feet from the top. m The wires form ngle with the tower. me 7 Find the distnce from the centerline of the e 0. tower to the nchor point of the wires. (Lesson 7-). ft H Lessons 7- through 7- K ngle of elevtion of sun I F J window ft high E ft K L ssess Open-Ended ssessment Writing sk students to drw three tringulr figures on single sheet of pper nd lel nd ssign vlues to them in correspondence with the three tpes of tringles tht cn e solved using the Lw of Sines (S, S nd SS). For emple, one tringle would hve two ngle mesures, one djcent side mesure nd one djcent side vrile. Students cn use the emples in the ook if necessr. Then tell them to set up the pproprite proportions to find the unknown vlues. Getting Red for Lesson 7-7 Prerequisite Skill Students will lern out the Lw of osines in Lesson 7-7. The will solve prolems using multivrile equtions. Use Eercises to determine our students fmilirit with evluting multivrile epressions. ssessment Options Prctice Quiz This quiz provides students with rief review of the concepts nd skills in Lessons 7- through 7-. Lesson numers re given to the right of the eercises or instruction lines so students cn review concepts not et mstered. Quiz (Lessons 7- nd 7-) is ville on p. 0 of the hpter 7 Resource Msters. Lesson 7- The Lw of Sines nswers. 0.7; 0.; 0.; 0.; 0.7; ; 0.7; 0 0.9; 0.7; 0 0.9; ; 0.7; 0.; 0.7; 0.0; ; 0.7;.00; 0.7; 0.7;.00 Lesson 7- The Lw of Sines

51 Geometr Softwre Investigtion Follow-Up of Lesson 7- Getting Strted The miguous se of the Lw of Sines s vrition, students cn lel the points nd where the circle intersects nd construct two rdii nd. Then the cn find nd displ ll of the mesures on the screen t one time. Tech When students re compring results, point out tht the two possile mesures for ngle re supplementr, nd discuss wh this is the cse. Eplin to students tht if circle intersects t just one point, then is tngent to circle, nd is right tringle with right ngle. If students ecome concerned out the possiilit of finding two solutions for SS prolem, ssure them tht the prolems will either e designed to ield one nswer or set up in w so students will know which solution is pproprite. Sometimes it might e pproprite for students to find oth solutions. ssess Eercises fter working through the eercises, students will hve demonstrted numerous emples of the miguous cse of the Lw of Sines nd discussed the possiilities of finding one solution or no solution. Model onstruct nd. onstruct circle whose center is so tht it intersects t two points. Then, construct n rdius. Find the mesures of,, nd. Use the rotte tool to move so tht it lies on one of the intersection points of circle nd. In, find the mesures of,, nd. nlze. Which mesures re the sme in oth tringles?,, m hpter 7 Right Tringles nd Trigonometr nswers Follow-Up of Lesson 7- The miguous se of the Lw of Sines In Lesson 7-, ou lerned tht ou could solve tringle using the Lw of Sines if ou know the mesures of two ngles nd n side of the tringle (S or S). You cn lso solve tringle the Lw of Sines if ou know the mesures of two sides nd n ngle opposite one of the sides (SS). When ou use SS to solve tringle, nd the given ngle is cute, sometimes it is possile to find two different tringles. You cn use The Geometer s Sketchpd to eplore this cse, clled the miguous cse, of the Lw of Sines. miguous se =.0 cm m = 97. =.79 cm m =. m = 9. = 7. cm Step Using the rotte tool, move to the other intersection point of circle nd. Note the mesures of,, nd in. miguous se =.0 cm m =. =.79 cm m =. m = 9.. Repet the ctivit using different mesures for,, nd. How do the results compre to the erlier results? Smple nswer: There re two different tringles. Mke onjecture. See mrgin.. ompre our results with those of our clssmtes. How do the results compre?. Wht would hve to e true out circle in order for there to e one unique solution? Test our conjecture repeting the ctivit.. Is it possile, given the mesures of,, nd, to hve no solution? Test our conjecture nd eplin.. Smple nswer: The results re the sme. In ech cse, two tringles re possile.. Smple nswer: ircle intersects t onl one point. See students work.. Yes; smple nswer: There is no solution if circle does not intersect. =.00 cm hpter 7 Right Tringles nd Trigonometr

52 The Lw of osines Lesson Notes Voculr Lw of osines Stud Tip Side nd ngle Note tht the letter of the side length on the left-hnd side of ech eqution corresponds to the ngle mesure used with the cosine. Use the Lw of osines to solve tringles. Solve prolems using the Lw of osines. re tringles used in uilding design? The hicgo Metropolitn orrectionl enter is 7-stor tringulr federl detention center. The cells re rrnged round lounge-like common re. The rchitect found tht tringulr floor pln llowed for the mimum numer of cells to e most efficientl centered round the lounge. THE LW OF OSINES Suppose ou know the lengths of the sides of the tringulr uilding nd wnt to solve the tringle. The Lw of osines llows us to solve tringle when the Lw of Sines cnnot e used. Let e n tringle with,, nd c representing the mesures of sides opposite ngles,, nd, respectivel. Then the following equtions re true. c c cos c c cos c cos Lw of osines The Lw of osines cn e used to find missing mesures in tringle if ou know the mesures of two sides nd their included ngle. c Focus -Minute heck Trnsprenc 7-7 Use s quiz or review of Lesson 7-. Mthemticl ckground notes re ville for this lesson on p. 0. re tringles used in uilding design? sk students: Wht dvntges might the tringulr design offer the correction center emploees? The residents would e more confined nd esier to monitor. How might the tringulr design of the correction fcilit e more cost efficient? tringulr uilding m utilize less spce thn rectngulr one, so fewer mterils would e needed, nd the uilding could hve more compct electricl nd pluming sstems. Emple Two Sides nd the Included ngle Find if c, 0, nd m 0. Use the Lw of osines since the mesures of two sides nd the included re known. 0 0 c c cos Lw of osines 0 (0)() cos 0 0, c, nd m 0 0 cos 0 s 0 co0 9. Simplif. Tke the squre root of ech side. Use clcultor. Workook nd Reproducile Msters hpter 7 Resource Msters Stud Guide nd Intervention, pp. 7 Skills Prctice, p. 9 Prctice, p. 90 Reding to Lern Mthemtics, p. 9 Enrichment, p. 9 ssessment, p. 0 Lesson 7-7 The Lw of osines Phil Mrtin/PhotoEdit Prerequisite Skills Workook, pp., Teching Geometr With Mnipultives Msters, p. Resource Mnger Trnsprencies -Minute heck Trnsprenc 7-7 nswer Ke Trnsprencies Technolog GeomPSS: Tutoril Plus, Lesson Interctive hlkord Lesson - Lesson Title

53 Tech THE LW OF OSINES In-lss Emples In-lss Emple Power Point Find if, z, nd mx..9 X Y Teching Tip In Emple, show students how the Lw of osines cn e rewritten to provide n lternte form: cos R q s r. qs Find ml. M L. Z 7 USE THE LW OF OSINES TO SOLVE PROLEMS N Power Point Teching Tip In Emple, students cn store the ctul vlue of k in their clcultors. Then if the use the stored vlue to find L, the will get dvise students tht nswers will vr when using rounded vlues insted of more ect vlues. etermine whether the Lw of Sines or the Lw of osines should e used first to solve EF. Then solve EF. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth. 7 F E osines, f.9; m ; me Stud Tip Lw of osines If ou use the Lw of osines to find nother mesure, our nswer m differ slightl from one found using the Lw of Sines. This is due to rounding. TEHING TIP Help students lern tht when using clcultor, the cn skip some steps. For emple, sin L sin. 0.. Emple hpter 7 Right Tringles nd Trigonometr You cn lso use the Lw of osines to find the mesures of ngles of tringle when ou know the mesures of the three sides. Three Sides Find mr. r q s qs cos R Lw of osines 7 (7)() cos R r, q 7, s 9 9 cos R Simplif. cos R Sutrct 9 from ech side. cos R ivide ech side. R cos Solve for R. R 9. Use clcultor. USE THE LW OF OSINES TO SOLVE PROLEMS Most prolems cn e solved using more thn one method. hoosing the most efficient w to solve prolem is sometimes not ovious. When solving right tringles, ou cn use sine, cosine, or tngent rtios. When solving other tringles, ou cn use the Lw of Sines or the Lw of osines. You must decide how to solve ech prolem depending on the given informtion. Emple Select Strteg Solve KLM. Round ngle mesure to the nerest degree nd side mesure to the nerest tenth. We do not know whether KLM is right tringle, so we must use the Lw of osines or the Lw of Sines. We know the mesures of two sides nd the included ngle. This is SS, so use the Lw of osines. k m m cos K Lw of osines k ()() cos, m, nd mk k ()( ) cos Tke the squre root of ech side. k. Use clcultor. Net, we cn find ml or mm. If we decide to find ml, we cn use either the Lw of Sines or the Lw of osines to find this vlue. In this cse, we will use the Lw of Sines. sin L sin K k Lw of Sines si n L n., k., nd mk. sin L sin ross products sin L sin. ivide ech side.. L sin sin Tke the inverse sine of ech side.. L 0 Use clcultor. ifferentited Instruction K R ELL Verl/Linguistic Hve students rewrite the equtions for the Lw of osines in their own words without using vriles. Then the cn descrie scenrios for which the Lw of osines is the more useful. Finll, the cn close their ooks, drw nd lel tringulr figure, nd tr to reproduce the equtions for the Lw of osines using onl their written eplntions nd descriptions. L Q 7 k M S hpter 7 Right Tringles nd Trigonometr

54 oncept heck. If two ngles nd one side re given, then the Lw of osines cnnot e used. Guided Prctice GUIE PRTIE KEY Eercises Emples 7, 0 9 Use the ngle Sum Theorem to find mm. mk ml mm 0 ngle Sum Theorem 0 mm 0 mk nd ml 0 mm 9 Sutrct from ech side. Therefore, k., mk 0, nd mm 9. Emple Use Lw of osines to Solve Prolems REL ESTTE Ms. Jenkins is uing some propert tht is shped like qudrilterl. Find the perimeter of the propert. Use the Pthgoren Theorem to find in. () () () Pthgoren Theorem 0 0 () 0, 0 90,000 () Simplif. 00 Tke the squre root of ech side. Net, use the Lw of osines to find in. () () () ()() cos Lw of osines () (00)(00) cos 0 00, 00, m 0 () 0,000 0,000 cos 0 Simplif. 0,00 0 0,000 cos 0 Tke the squre root of ech side.. Use clcultor. 0 ft 00 ft 0 ft The perimeter of the propert is or out. feet.. OPEN ENE rw nd lel one cute nd one otuse tringle, illustrting when ou cn use the Lw of osines to find the missing mesures.. Eplin when ou should use the Lw of Sines or the Lw of osines to solve tringle.. See mrgin.. Find counteremple for the following sttement. The Lw of osines cn e used to find the length of missing side in n tringle. In, given the following mesures, find the mesure of the missing side.. c, d, m.. 07, c 9, m In RST, given the lengths of the sides, find the mesure of the stted ngle to the nerest degree.. r, s, t ; ms r., s., t.; mr 9 0 In-lss Emple 7. m Power Point IRRFT From the digrm of the plne shown, determine the pproimte eterior perimeter of ech wing. Round to the nerest tenth meter. 9 m I H 0 0 m K oncept heck Hve students compose chrt in which the include ll of the tpes of tringles the could solve using the Pthgoren Theorem, trigonometric rtios, Lw of Sines, nd Lw of osines. The cn lso include column listing the formuls nd/or rtios nd column for the limittions of ech method. Students should recognize tht the Pthgoren Theorem is the most limited ut the most efficient, nd the Lw of osines is the most diverse ut ver involved nd time-consuming. The m wnt to form hit of checking for the most efficient w to solve prolem, keeping ll these methods in mind. J. mx 0; my ; mz 7 Solve ech tringle using the given informtion. Round ngle mesure to the nerest degree nd side mesure to the nerest tenth.. XYZ:, 0, z 9. KLM: k 0, m, ml 7 7.9; mk ; mm 7 Lesson 7-7 The Lw of osines 7 nswers. Smple nswer: Use the Lw of osines when ou hve ll three sides given (SSS) or two sides nd the included ngle (SS). R Y 0 Q S X Z. If ou hve ll three sides (SSS) or two sides nd the included ngle (SS) given, then use the Lw of osines. If two ngles nd one side (S or S) or two sides with ngle opposite one of the sides (SS) re given, then use the Lw of Sines. Lesson 7-7 The Lw of osines 7

55 Prctice/ppl Stud Noteook Hve students dd the definitions/emples of the voculr terms to their Voculr uilder worksheets for hpter 7. include n emple ech for using the Lw of osines with SS nd with SSS. include n other item(s) tht the find helpful in mstering the skills in this lesson. ppliction indictes incresed difficult Prctice nd ppl For Eercises, 7, 9 See Emples Etr Prctice See pge RFTS Jmie, ge, is creting logo for herself nd two cousins, ges 0 nd. She is using qurter ( cents), Q dime (0 cents), nd nickel ( cents) to represent their ges. She will hold the coins together soldering tringulr piece of silver wire so tht the three vertices of the tringle lie t the centers of the three circulr coins. The dimeter of the qurter N is millimeters, the dimeter of the nickel is millimeters, nd the dimeter of dime is 0 millimeters. Find the mesures of the three ngles in the tringle. mq.; m.; mn 7. In TUV, given the following mesures, find the mesure of the missing side.. t 9., v., mu u.9. t, u 7, mv 7 v.. u, v 7, mt 0 t.. v, u 7, mt 9 t.7 In EFG, given the lengths of the sides, find the mesure of the stted ngle to the nerest degree.. e 9., f., g.7; mf. e, f 9, g ; me 7. e, f 9, g 0; mf. e.9, f.9, g 0; mg 7 out the Eercises Orgniztion Ojective Solve Tringles:, Use the Lw of osines to Solve Prolems: 7, 9 Odd/Even ssignments Eercises re structured so tht students prctice the sme concepts whether the re ssigned odd or even prolems. ssignment Guide sic: 7 odd,, 9 verge: odd, 9 dvnced: even, (optionl: 7 9) 7. See mrgin. Solve ech tringle using the given informtion. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth. 9. See mrgin. 9. G 0. Q. F 0 0 g H. : m, m 77, c. : 0., 9., m 7. :, 9, c. : m, m, c.9. KITES eth is uilding kite like the one t the right. If is feet long, is feet long, nd is 7 feet long, find the mesure of the ngle etween the short sides nd the ngle etween the long sides to the nerest degree. 00; 7 P 0 Solve ech LMN descried elow. Round mesures to the nerest tenth. 7. m,, ml. m,, n 0 9. m 9, n, ml 9 0. mm, ml, n. m,, n. mm,., n.7. mm 7,, n 0. n 7, m 0,., n, mm 0., m, n 7. ml, 0, n. 0, m, n p M nswers 9. mh ;mg 09; g.7 0. p.9; mm 79; mq. m ; m ; m. m ;.;.. c.; m 0; m. m 0; m 9; m hpter 7 Right Tringles nd Trigonometr. m 99;.;. 7. mm.; mn.; n 9.. ml.; mm.9; mn ; mm.; mn. 0. mn 79;.; m.7. ml 0.9; mm.; mn.. m.0; ml 9.; mn.7. m.0; ml.; mn 0.. ml.; mm 79.; mn 7.0. m.; ml 0.9; mn 79.. ml.; mm 9.; mn. 7. mn.; mm.; m.. ml.; mm 9.; mn 9. hpter 7 Right Tringles nd Trigonometr

56 9. In qudrilterl,,, mp 70, nd P is the midpoint of nd. Find the perimeter of qudrilterl.. units P Stud Stud Guide nd nd Intervention Intervention, p. 7 (shown) nd p. 7-7 NME TE PERIO The Lw of osines The Lw of osines nother reltionship etween the sides nd ngles of n tringle is clled the Lw of osines.you cn use the Lw of osines if ou know three sides of tringle or if ou know two sides nd the included ngle of tringle. 0. In qudrilterl PQRS, PQ 7, QR 7, RS 9, PS 7, nd mp. Find QS, mpqs, nd mr. QS 7.7; mpqs.; mr 77.. UILINGS Refer to the informtion t the left. Find the mesures of the ngles of the tringulr uilding to the nerest tenth. 9.,.,. P S Q R Lw of osines Let e n tringle with,, nd c representing the mesures of the sides opposite the ngles with mesures,, nd, respectivel. Then the following equtions re true. c c cos c c cos c cos Emple In, find c. c cos Lw of osines c 0 ()(0)cos, 0, m c 0 (0)co () s Tke the squre root of ech side. c 9. Use clcultor. Emple In, find m. c c cos Lw of osines 7 ()() cos 7,, c 9 0 cos Multipl. 0 0 cos Sutrct 9 from ech side. cos ivide ech side 0. cos Use the inverse cosine. 0 c 7 Lesson 7-7 uildings The Swissôtel in hicgo, Illinois, is uilt in the shpe of tringulr prism. The lengths of the sides of the tringle re 0 feet, feet, nd 7 feet. Source: Swissôtel. SOER rlos nd dm re pling soccer. rlos is stnding 0 feet from one post of the gol nd 0 feet from the other post. dm is stnding 0 feet from one post of the gol nd feet from the other post. If the gol is feet wide, which pler hs greter ngle to mke shot on gol? dm;.. PROOF Justif ech sttement for the derivtion of the Lw of osines. Given: is n ltitude of. c Prove: c cos Proof: Sttement Resons. c ( ) h.? Pthgoren Theorem. c h.? Sustitution c. h c.? Pthgoren Theorem d. c d.? Sustitution e. cos e.? ef. of cosine f. cos f.? ross products g. c ( cos ) g.? Sustitution h. c cos h.? ommuttive Propert. RITIL THINKING Grph (, ), (0, ), (, ), nd (, ) on coordinte plne. Find the mesure of interior ngle nd the mesure of eterior ngle. m.; m.7. WRITING IN MTH nswer the question tht ws posed t the eginning of the lesson. See mrgin. How re tringles used in uilding construction? Include the following in our nswer: wh the uilding ws tringulr insted of rectngulr, nd wh the Lw of Sines could not e used to solve the tringle. nswers. Smple nswer: Tringles re used to uild supports, wlls, nd foundtions. nswers should include the following. The tringulr uilding ws more efficient with the cells round the edge. The Lw of Sines requires two ngles nd side or two sides nd n ngle opposite one of those sides. NME TE PERIO 7-7 Enrichment, p. 9 Sphericl Tringles Sphericl trigonometr is n etension of plne trigonometr. Figures re drwn on the surfce of sphere. rcs of gret circles correspond to line segments in the plne. The rcs of three gret circles intersecting on sphere form sphericl tringle. ngles hve the sme mesure s the tngent lines drwn to ech gret circle t the verte. Since the sides re rcs, the too cn e mesured in degrees. The sum of the sides of sphericl tringle is less thn 0. The sum of the ngles is greter thn 0 nd less thn 0. The Lw of Sines for sphericl tringles is s follows. sin sin sin c s in s in s in rlos There is lso Lw of osines for sphericl tringles. cos cos cos c sin sin c cos cos cos cos c sin sin c cos cos c cos cos sin sin cos 0 ft dm 0 ft 0 ft ft Lesson 7-7 The Lw of osines 9 h ft Pierre urnugh/photoedit c 0 Use clcultor. Eercises Find ech mesure using the given mesures from. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth.. If, c, nd m, find... If, 0, nd c, find m.. If,, nd c, find m.. If 0, c, nd m, find. 9.. If, c, nd m 9, find... If, 9, nd c, find m. 7-7 NME 7 TE PERIO Prctice (verge) The Lw of osines Skills Prctice, p. 9 nd Prctice, p. 90 (shown) In JKL, given the following mesures, find the mesure of the missing side.. j., k 0, ml j 9.,.7, mk k.. j, k 7, ml 0.0. k.7,., mj j. In MNQ, given the lengths of the sides, find the mesure of the stted ngle to the nerest tenth.. m 7, n, q ; mq 7.7. m, n, q ; mm. 7. m.9, n, q 0.; mn 0.. m, n 0., q ; mq 0.7 etermine whether the Lw of Sines or the Lw of osines should e used first to solve. Then sole ech tringle. Round ngle mesures to the nerest degree nd side mesure to the nerest tenth. 9.,, c 9 0., 9, m osines; m ; osines; m ; m ; m 7 m 7; c.7. 7,, m 9..,, m 7 Sines; m ; osines; m ; m 9; c 9.0 m 0; c 9. Solve ech FGH descried elow. Round mesures to the nerest tenth.. mf, f., g mg., mh 0., h.. f 0, g, mh 7 mf 7., mg 7., h 7.. f., g, h mf 77., mg., mh 9.. f, h 0, mg mf 7., mh.9, g REL ESTTE The Esposito fmil purchsed tringulr plot of lnd on which the pln to uild rn nd corrl. The lengths of the sides of the plot re 0 feet, feet, nd 0 feet. Wht re the mesures of the ngles formed on ech side of the propert?.,., 0. Reding to Lern Mthemtics, p NME 90 TE PERIO Reding to Lern Mthemtics The Lw of osines Pre-ctivit How re tringles used in uilding design? Red the introduction to Lesson 7-7 t the top of pge in our tetook. Wht could e disdvntge of tringulr room? Smple nswer: Furniture will not fit in the corners. Reding the Lesson. Refer to the figure. ccording to the Lw of osines, which sttements re correct for EF?, E, H. d e f ef cos. e d f df cos E. d e f ef cos. f d e ef cos F E. f d e de cos F F. d e f G. sin sin E sin F d e f H. d e f f e cos ELL. Ech of the following descries three given prts of tringle. In ech cse, indicte whether ou would use the Lw of Sines or the Lw of osines first in solving tringle with those given prts. (In some cses, onl one of the two lws would e used in solving the tringle.). SSS Lw of osines. S Lw of Sines c. S Lw of Sines d. SS Lw of osines e. SS Lw of Sines. Indicte whether ech sttement is true or flse. If the sttement is flse, eplin wh.. The Lw of osines pplies to right tringles. true. The Pthgoren Theorem pplies to cute tringles. Flse; smple nswer: It onl pplies to right tringles. c. The Lw of osines is used to find the third side of tringle when ou re given the mesures of two sides nd the nonincluded ngle. Flse; smple nswer: It is used when ou re given the mesures of two sides nd the included ngle. d. The Lw of osines cn e used to solve tringle in which the mesures of the three sides re centimeters, centimeters, nd centimeters. Flse; smple nswer:, so, the Tringle Inequlit Theorem, no such tringle eists. Helping You Rememer. good w to rememer new mthemticl formul is to relte it to one ou lred know. The Lw of osines looks somewht like the Pthgoren Theorem. oth formuls must e true for right tringle. How cn tht e? cos 90 0, so in right tringle, where the included ngle is the right ngle, the Lw of osines ecomes the Pthgoren Theorem. Lesson 7-7 The Lw of osines 9 e F d f E Lesson 7-7

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