connected.mcgraw-hill.com Your Digital Math Portal In Chapter 8, you will: In Chapter 7, you learned to solve proportions.

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1 Right Tringles nd Trigonometr Then Now Wh? In Chpter 7, ou lerned to solve proportions. In Chpter 8, ou will: Properties of tringles cn be used in plnning nd preprtion for specil events including the height of decortions. Use the Pthgoren Theorem. Use properties of specil right tringles. Use trigonometr to find missing mesures of tringles. connected.mcgrw-hill.com Animtion Vocbulr eglossr Your Digitl Mth Portl Personl Tutor Virtul Mnipultives Grphing Clcultor Audio Foldbles Self-Check Prctice Worksheets

2 Tennessee Curriculum Stndrds CLE Get Red for the Chpter Dignose Rediness You hve two options for checking prerequisite skills. Tetbook Option Tke the Quick Check below. Refer to the Quick Review for help. Simplif. (Lesson 0-9) _ 4 4. _ 6 _ 18 Quick Check _ _ Emple 1 Simplif 6_ = 6_ 6_ _ = 6. _ Quick Review _ Multipl b. or Simplif. Find. (Lesson 1-) Emple Find BANNERS Ann is mking bnner out of 4 congruent tringles s shown below. How much blue trim will she need for ech side? 10 in. 14 in. + b = c 15 Pthgoren Theorem = = 8 nd b = = Simplif. 89 = Tke the positive squre root of ech side. 17 = Simplif. Grph the line segment with the given endpoints. (Lesson -4) 10. G (, -4) nd H (, 4) 11. E (-, 5) nd F (4, -) 1. COLLEGES Quinn is visiting college cmpus. He notices from his mp tht severl importnt buildings re locted round grss re the students cll the Qud. If the librr is represented on the mp b L (6, 8) nd the cfeteri is represented b C (0, 0), grph the line segment tht represents the shortest pth between the two buildings. Emple Grph the line segment with endpoints A (-4, ) nd B (, -). Plot points A nd B. Connect the points. Online Option Tke n online self-check Chpter Rediness Quiz t connected.mcgrw-hill.com. 59

3 Get Strted on the Chpter You will lern severl new concepts, skills, nd vocbulr terms s ou stud Chpter 8. To get red, identif importnt terms nd orgnize our resources. You m wish to refer to Chpter 0 to review prerequisite skills. Stud Orgnizer Right Angles nd Trigonometr Mke this Foldble to help ou orgnize our Chpter 8 notes bout right ngles nd trigonometr. Begin with three sheets of notebook pper nd one sheet of construction pper. 1 Stck the notebook pper on the construction pper. Fold the pper digonll to form tringle nd cut off the ecess. Open the pper nd stple the inside fold to form booklet. New Vocbulr English Espñol geometric men p. 51 medi geométric Pthgoren triple p. 54 triplete de Pitágors cosine p. 56 coseno sine p. 56 seno tngent p. 56 tngente trigonometric rtio p. 56 rzón trigonométric trigonometr p. 56 trigonométri ngle of depression p. 574 ángulo de depresión ngle of elevtion p. 574 ángulo de elevción Lw of Sines p. 58 Le de los senos Lw of Cosines p. 58 Le do los cosenos component form p. 59 componente mgnitude p. 59 mgnitud vector p. 59 vector resultnt p. 595 resultnte 4 Lbel ech pge with lesson number nd title. Right Tringles Review Vocbulr ltitude p. 5 ltur segment drwn from verte of tringle perpendiculr to the line contining the other side Pthgoren Theorem p. 6 Teorem de Pitágors If nd b re the mesures of the legs of right tringle nd c is the mesure of the hpotenuse, then + b = c. is n ltitude from to 50 Chpter 8 Right Tringles nd Trigonometr

4 Geometric Men Then You used proportionl reltionships of corresponding ngle bisectors, ltitudes, nd medins of similr tringles. (Lesson 7-5) Now 1Find the geometric men between two numbers. Solve problems involving reltionships between prts of right tringle nd the ltitude to its hpotenuse. Wh? Photogrphing ver tll or ver wide objects cn be chllenging. It cn be difficult to include the entire object in our shot without distorting the imge. If our cmer is set for verticl viewing ngle of 90 nd ou know the height of the object ou wish to photogrph, ou cn use the geometric men of the distnce from the top of the object to our cmer level nd the distnce from the bottom of the object to cmer level. New Vocbulr geometric men Tennessee Curriculum Stndrds Use geometric men to solve problems involving reltionships tht eist when the ltitude is drwn to the hpotenuse of right tringle. SPI Use properties of right tringles to solve problems. Geometric Men When the mens of proportion re the sme number, tht 1 number is clled the geometric men of the etremes. The geometric men between two numbers is the positive squre root of their product. Words etreme men Ke Concept Geometric Men Emple men etreme The geometric men of two positive numbers nd b is the number such tht _ = _ b. So, = b nd = b. The geometric men of = 9 nd b = 4 is 6, becuse 6 = 9 4. Emple 1 Geometric Men = b Find the geometric men between 8 nd 10. = b Definition of geometric men = 8 10 = 8 nd b = 10 = (4 ) ( 5) Fctor. = 16 5 Associtive Propert = 4 5 Simplif. The geometric men between 8 nd 10 is 4 5 or bout 8.9. GuidedPrctice Find the geometric men between ech pir of numbers. 1A. 5 nd 45 1B. 1 nd 15 Geometric Mens in Right Tringles In right tringle, n ltitude drwn from the verte of the right ngle to the hpotenuse forms two dditionl right tringles. These three right tringles shre specil reltionship. connected.mcgrw-hill.com 51

5 Review Vocbulr ltitude (of tringle) segment from verte to the line contining the opposite side nd perpendiculr to the line contining tht side (Lesson 5-) Theorem 8.1 If the ltitude is drwn to the hpotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd to ech other. Emple If CD is the ltitude to hpotenuse AB of right ABC, then ACD ABC, CBD ABC, nd ACD CBD. You will prove Theorem 8.1 in Eercise 9. Emple Identif Similr Right Tringles Stud Tip Reorienting Tringles To reorient the right tringles in Emple, first mtch up the right ngles. Then mtch up the shorter sides. Write similrit sttement identifing the three similr right tringles in the figure. Seprte the tringle into two tringles long the ltitude. Then sketch the three tringles, reorienting the smller ones so tht their corresponding ngles nd sides re in the sme positions s the originl tringle. So b Theorem 8.1, FJG GJH FGH. GuidedPrctice A. B. From Theorem 8.1, ou know tht ltitude CD drwn to the hpotenuse of right tringle ABC forms three similr tringles: ACB ADC CDB. B the definition of similr polgons, ou cn write the following proportions compring the side lengths of these tringles. b h c b h h c b = = = = = = = = = 5 Lesson 8-1 Geometric Men Notice tht the circled reltionships involve geometric mens. This leds to the theorems t the top of the net pge.

6 Theorems Right Tringle Geometric Men Theorems 8. Geometric Men (Altitude) Theorem The ltitude drwn to the hpotenuse of right tringle seprtes the hpotenuse into two segments. The length of this ltitude is the geometric men between the lengths of these two segments. Emple If CD is the ltitude to hpotenuse AB of right ABC, then _ h = h_ or h =. h 8. Geometric Men (Leg) Theorem The ltitude drwn to the hpotenuse of right tringle seprtes the hpotenuse into two segments. The length of leg of this tringle is the geometric men between the length of the hpotenuse nd the segment of the hpotenuse djcent to tht leg. Emple If CD is the ltitude to hpotenuse AB of right ABC, then _ c b = _ b or b = c nd _ c = _ or = c. b c You will prove Theorems 8. nd 8. in Eercises 40 nd 41, respectivel. Emple Use Geometric Men with Right Tringles Stud Tip Use Proportion In Emple, the vlue of could lso be found b solving the proportion 5_ = _ 0. Find,, nd z. Since is the mesure of the ltitude drwn to the hpotenuse of right JKL, is the geometric men of the lengths of the two segments tht mke up the hpotenuse, JM nd MK. = JM MK = 5 0 Substitution = 100 or 10 Simplif. Geometric Men (Altitude) Theorem 5 0 z Since is the mesure of leg JL, is the geometric men of JM, the mesure of the segment djcent to this leg, nd the mesure of the hpotenuse JK. = JM JK Geometric Men (Leg) Theorem = 5 (0 + 5) Substitution = 15 or bout 11. Use clcultor to simplif. Since z is the mesure of leg KL, z is the geometric men of MK, the mesure of the segment djcent to KL, nd the mesure of the hpotenuse JK. z = MK JK Geometric Men (Leg) Theorem = 0 (0 + 5) Substitution = 500 or bout.4 Use clcultor to simplif. connected.mcgrw-hill.com 5

7 GuidedPrctice Find,, nd z. A. 8 B. 9 z 5 1 z You cn use geometric men to mesure height indirectl. Rel-World Emple 4 Indirect Mesurement ADVERTISING Zch wnts to order bnner tht will hng over the side of his high school bsebll stdium grndstnd nd rech the ground. Rel-World Creer Event Plnner Event plnners orgnize events including choosing loction, rrnging for food, nd scheduling entertinment. The lso coordinte services like trnsporttion nd photogrph. Most of the skills required for event plnning re cquired through on-the-job eperience. To find this height, he uses crdbord squre to line up the top nd bottom of the grndstnd. He mesures his distnce from the grndstnd nd from the ground to his ee level. Find the height of the grndstnd to the nerest foot. The distnce from Zch to the grndstnd is the ltitude to the hpotenuse of right tringle. The length of this ltitude is the geometric men of the two segments tht mke up the hpotenuse. The shorter segment hs the mesure of 5.75 feet. Let the unknown mesure be feet ft ft 5.75 ft Note: Not drwn to scle 10.5 = 5.75 Geometric Men (Altitude) Theorem = 5.75 Squre ech side Divide ech side b The height of the grndstnd is the totl length of the hpotenuse, , or bout 5 feet. GuidedPrctice 4. SPORTS A communit center needs to estimte the cost of instlling rock climbing wll b estimting the height of the wll. Sue holds book up to her ees so tht the top nd bottom of the wll re in line with the bottom edge nd binding of the cover. If her ee level is 5 feet bove the ground nd she stnds 11 feet from the wll, how high is the wll? Drw digrm nd eplin our resoning. 54 Lesson 8-1 Geometric Men

8 Check Your Understnding = Step-b-Step Solutions begin on pge R0. Emple 1 Emple Find the geometric men between ech pir of numbers nd 0. 6 nd nd Write similrit sttement identifing the three similr tringles in the figure. Emple Find,, nd z z 16 z Emple 4 7. MONUMENTS Core is visiting the Jefferson Memoril with his fmil. He wnts to estimte the height of the sttue of Thoms Jefferson. Core stnds so tht his line of vision to the top nd bse of the sttue form right ngle s shown in the digrm. About how tll is the sttue? 5 ft 8 in. 8 ft 8 in. Note: Not drwn to scle. Prctice nd Problem Solving Etr Prctice begins on pge 947. Emple 1 Find the geometric men between ech pir of numbers nd nd nd nd nd nd 1.5 Emple Write similrit sttement identifing the three similr tringles in the figure connected.mcgrw-hill.com 55

9 Emple Find,, nd z. 18. z z z z z z EVENT PLANNING Evelin is hnging silver strs from the gm ceiling using string for the homecoming dnce. She wnts the ends of the strings where the strs will be ttched to be 7 feet from the floor. Use the digrm to determine how long she should mke the strings. 5 ft 10 ft Note: Not drwn to scle. 5. WATERFALLS Mkl is using book to sight the top of wterfll. Her ee level is 5 feet from the ground nd she is horizontl distnce of 8 feet from the wterfll. Find the height of the wterfll to the nerest tenth of foot. 5 ft 8 ft Note: Not drwn to scle B Find the geometric men between ech pir of numbers _ nd _ 5 7. _ nd _ 1 6. _ nd Find,, nd z z z 1. ALGEBRA The geometric men of number nd four times the number is. Wht is the number? 56 Lesson 8-1 Geometric Men 1 4 Emple 4

10 Use similr tringles to find the vlue of ft 15.6 ft 4.8 ft 9. ft 5 ft 14. ft 1 ft. ft 7.6 ft ALGEBRA Find the vlue(s) of the vrible t+1 m w m CONSTRUCTION A room-in-ttic truss is truss design tht provides support while leving re tht cn be enclosed s living spce. In the digrm, BCA nd EGB re right ngles, BEF is isosceles, CD is n ltitude of ABC, nd EG is n ltitude of BEF. If DB = 5 feet, CD = 6 feet 4 inches, BF = 10 feet 10 inches, nd EG = 4 feet 6 inches, wht is AE? & # % " ' ( $ PROOF Write proof for ech theorem. 9. Theorem Theorem TRUCKS In photogrph, the ngle formed b the top of the subject, the cmer, nd the bottom of the subject is clled the viewing ngle, s shown t the right. Ntlie is tking picture of Bigfoot #5, which is 5 feet 6 inches tll. She sets her cmer on tripod tht is 5 feet bove ground level. The verticl viewing ngle of her cmer is set for Theorem 8. top of subject viewing ngle bottom of subject. Sketch digrm of this sitution. b. How fr w from the truck should Ntlie stnd so tht she perfectl frmes the entire height of the truck in her shot? 4 FINANCE The verge rte of return on n investment over two ers is the geometric men of the two nnul returns. If n investment returns 1% one er nd 7% the net er, wht is the verge rte of return on this investment over the two-er period? connected.mcgrw-hill.com 57

11 C 44. PROOF Derive the Pthgoren Theorem using the figure t the right nd the Geometric Men (Leg) Theorem. b c Determine whether ech sttement is lws, sometimes, or never true. Eplin our resoning. 45 The geometric men for consecutive positive integers is the men of the two numbers. 46. The geometric men for two perfect squres is positive integer. 47. The geometric men for two positive integers is nother integer. 48. MULTIPLE REPRESENTATIONS In this problem, ou will investigte geometric men.. Tbulr Cop nd complete the tble of five ordered pirs (, ) such 8 tht = 8. 8 b. Grphicl Grph the ordered pirs from our tble in sctter plot. 8 8 c. Verbl Mke conjecture s to the tpe of grph tht would be formed if 8 ou connected the points from our sctter plot. Do ou think the grph of n set of ordered pirs tht results in the sme geometric men would hve the sme generl shpe? Eplin our resoning. H.O.T. Problems Use Higher-Order Thinking Skills 49. ERROR ANALYSIS Aiden nd Ti re finding the vlue in the tringle shown. Is either of them correct? Eplin our resoning. Aiden 4_ = _ 7 5. Ti 4_ = _ CHALLENGE Refer to the figure t the right. Find,, nd z OPEN ENDED Find two pirs of whole numbers with geometric men tht is lso whole number. Wht condition must be met in order for pir of numbers to produce whole-number geometric men? 8.5 z 5. REASONING Refer to the figure t the right. The orthocenter of ABC is locted 6.4 units from point D. Find BC WRITING IN MATH Compre nd contrst the rithmetic nd geometric mens of two numbers. When will the two mens be equl? Justif our resoning. 58 Lesson 8-1 Geometric Men

12 Stndrdized Test Prctice 54. Wht is the geometric men of 8 nd in simplest form? A 4 11 C B 15 D SHORT RESPONSE If MN PQ, use proportion to find the vlue of. Show our work ALGEBRA Wht re the solutions of the qudrtic eqution 0 = 8? F, 10 H -1, 0 G 0, 1 J -, SAT/ACT In the figure, AD is perpendiculr to BC, nd AB is perpendiculr to AC. Wht is BC? A 5 B 5 C 0 D 5 E 75 SPI , SPI , SPI Spirl Review 58. MAPS Use the mp to estimte how long it would tke to drive from Chicgo to Springfield if ou verged 65 miles per hour. (Lesson 7-7) Grph the originl figure nd its dilted imge. Then verif tht the diltion is similrit trnsformtion. (Lesson 7-6) 59. A(-, 1), B(9, 7), C(, -); D(-1, 1), E(, ), F(1, 0) 60. G(-4, -4), H(-1, ), J(, -1); K(-, -), L(1, 0) 61. M(7, -4), N(5, -4), P(7, -1); Q(, -8), R(6, -8), S(, -) The interior ngle mesure of regulr polgon is given. Identif the polgon. (Lesson 6-1) Find nd in ech figure. (Lesson -) 64. ( + 1) (4-5) ( + 11) 65. ( - 15) 68 ( ) MISSOURI 55 0 IOWA WISCONSIN 88 Peori Springfield 0.5 in.: 100 mi mi ( - 11) 90 Dectur ILLINOIS 64 4 Lke Michign Chicgo MICH. INDIANA KENTUCKY Identif ech solid. Nme the bses, fces, edges, nd vertices. (Lesson 1-7) 67. G P A B C D F E Q Skills Review Simplif ech epression b rtionlizing the denomintor. (Lesson 0-9) 70. _ _ 7. _ 6 4 _ _ connected.mcgrw-hill.com 59

13 Geometr Lb The Pthgoren Theorem In Chpter 1, ou lerned tht the Pthgoren Theorem reltes the mesures of the legs nd the hpotenuse of right tringle. You cn use squre pieces of ptt pper nd lgebr to discover this reltionship. Activit Develop the Pthgoren Theorem b using pper nd lgebr. Tennessee Curriculum Stndrds CLE Develop the tools of right tringle trigonometr in the contetul pplictions, including the Pthgoren Theorem, Lw of Sines nd Lw of Cosines. Step 1 Step Step b b b b b b b b b b b b On piece of ptt pper, mrk one side nd b s shown bove. Step 4 Cop these mesures on ech of the other sides. Step 5 Fold the pper into four sections nd lbel the re of ech section. Step 6 b b b b c c c c 1 b c 1 b c c c 1 b c 1 b On nother piece of ptt pper, mrk ech side nd b s shown bove. Connect the mrks using strightedge. Let c represent the length of ech hpotenuse. Lbel the re of ech tringle 1_ b nd the re of ech squre c. Step 7 Plce the squres side b side nd color the corresponding regions tht hve the sme re. For emple, b = 1_ b + 1_ b. The prts tht re not shded tell us tht + b = c. b b b 1 b 1 b c 1 b 1 b Anlze the Results 1. Use ruler to mesure, b, nd c. Do these mesures confirm tht + b = c?. Repet the ctivit with different nd b vlues. Wht do ou notice?. WRITING IN MATH Eplin wh the drwing t the right is n illustrtion of the Pthgoren Theorem. 4. CHALLENGE Drw geometric digrm to show tht for n positive numbers nd b, + b > + b. Eplin. 540 Eplore 8- Geometr Lb: The Pthgoren Theorem

14 The Pthgoren Theorem nd Its Converse Then Now Wh? You used the Pthgoren Theorem to develop the Distnce Formul. 1 Tether lines re used to sted n infltble snowmn. Suppose ou know the height t which the tether lines re ttched to the snowmn nd how fr w ou wnt to nchor the tether in the ground. You cn use the converse of the Pthgoren Theorem to djust the lengths of the tethers to keep the snowmn perpendiculr to the ground. (Lesson 1-) NewVocbulr Pthgoren triple Use the Pthgoren Theorem. Use the Converse of the Pthgoren Theorem. 1 The Pthgoren Theorem The Pthgoren hpotenuse Theorem is perhps one of the most fmous theorems in mthemtics. It reltes the lengths of the hpotenuse (side opposite the right ngle) nd legs (sides djcent to the right ngle) of right tringle. Tennessee Curriculum Stndrds Theorem 8.4 Pthgoren Theorem CLE Develop the tools of right tringle trigonometr in the contetul pplictions, including the Pthgoren Theorem, Lw of Sines nd Lw of Cosines Identif nd use Pthgoren triples in right tringles to find lengths of n unknown side in two- or three-dimensionl situtions Use the converse of the Pthgoren Theorem to clssif tringle b its ngles (right, cute, or obtuse). Also ddresses nd SPI Words leg leg In right tringle, the sum of the squres of the lengths of the legs is equl to the squre of the length of the hpotenuse. Smbols # c If ABC is right tringle with right ngle C, then + b = c. b $ " The geometric men cn be used to prove the Pthgoren Theorem. Proof Pthgoren Theorem Given: ABC with right ngle t C $ Prove: + b = c b h Proof: Drw right tringle ABC so C is the right ngle. Then drw the ltitude from C to AB. Let AB = c, AC = b, BC =, AD =, DB =, nd CD = h. Two geometric mens now eist. _c = _ nd = c + b = c + c # % _c = _b Geometric Men (Leg) Theorem b = c Cross products b " c Add the equtions. + b = c( + ) Fctor. + b = c c Since c = +, substitute c for ( + ). + b = c Simplif. connected.mcgrw-hill.com 541

15 You cn use the Pthgoren Theorem to find the mesure of n side of right tringle given the lengths of the other two sides. Emple 1 Find Missing Mesures Using the Pthgoren Theorem Find The side opposite the right ngle is the hpotenuse, so c =. + b = c Pthgoren Theorem = = 6 nd b = = Simplif. 61 = Tke the positive squre root of ech side. Stud Tip 9 = Simplif. Positive Squre Root When finding the length of side using the Pthgoren Theorem, use onl the positive nd not the negtive squre root, since length cnnot be negtive. b The hpotenuse is 11, so c = b = c Pthgoren Theorem + 9 = 11 = nd b = = 11 Simplif. = 40 Subtrct 81 from ech side. = 40 or 10 Tke the positive squre root of ech side nd simplif. GuidedPrctice 1A B. 5 Stud Tip Pthgoren Triples If the mesures of the sides of n right tringle re not whole numbers, the mesures do not form Pthgoren triple. A Pthgoren triple is set of three nonzero whole numbers, b, nd c, such tht + b = c. One common Pthgoren triple is, 4, 5; tht is, the sides of right tringle re in the rtio :4:5. The most common Pthgoren triples re shown below in blue. The triples below these re found b multipling ech number in the triple b the sme fctor. Ke Concept Common Pthgoren Triples, 4, 5 5, 1, 1 8, 15, 17 7, 4, 5 6, 8, 10 10, 4, 6 16, 0, 4 14, 48, 50 9, 1, 15 15, 6, 9 4, 45, 51 1, 7, , 4, 5 5, 1, 1 8, 15, 17 7, 4, 5 The lrgest number in ech triple is the length of the hpotenuse. 54 Lesson 8- The Pthgoren Theorem nd Its Converse

16 Emple Use Pthgoren Triple Use Pthgoren triple to find. Eplin our resoning. RedingMth -4-5 A right tringle with side lengths, 4, nd 5 is clled -4-5 right tringle. Notice tht 15 nd 1 re both multiples of : 15 = 5 nd 1 = 4. Since, 4, 5 is Pthgoren triple, the missing leg length is or 9. CHECK Pthgoren Theorem 5 = 5 1 Simplif. GuidedPrctice 48 A. 50 B The Pthgoren Theorem cn be used to solve mn rel-world problems. SPI Test Emple Dmon is locked out of his house, with the onl open window on the second floor, 1 feet bove the ground. He needs to borrow ldder from his neighbor. If he must plce the ldder 5 feet from the house to void some bushes, wht length of ldder does Dmon need? A 7 feet C 1 feet B 11 feet D 17 feet 1 ft 5 ft Note: Not drwn to scle Red the Test Item Test-TkingTip Eliminte Unresonble Answers Since the hpotenuse of right tringle is lws the longest side, the length of the ldder in Emple must be greter thn 5 or 1 feet. Since 7 nd 11 feet re both less thn 1 feet, choices A nd B cn be eliminted. The distnce the ldder is from the house, the height the ldder reches, nd the length of the ldder itself mke up the lengths of the sides of right tringle. You need to find the length of the ldder, which is the hpotenuse. Solve the Test Item Method 1 Use Pthgoren triple. The lengths of the legs re 5 nd 1. 5, 1, 1 is Pthgoren triple, so the length of the ldder is 1 feet. Method Use the Pthgoren Theorem. Let the length of the ldder be = 169 = 169 = 1 = Pthgoren Theorem Simplif. Tke the positive squre root of ech side. Simplif. So the nswer is choice C. connected.mcgrw-hill.com 54

17 Emple Use Pthgoren Triple Use Pthgoren triple to find. Eplin our resoning. RedingMth -4-5 A right tringle with side lengths, 4, nd 5 is clled -4-5 right tringle. Notice tht 15 nd 1 re both multiples of : 15 = 5 nd 1 = 4. Since, 4, 5 is Pthgoren triple, the missing leg length is or 9. CHECK Pthgoren Theorem 5 = 5 1 Simplif. GuidedPrctice 48 A. 50 B The Pthgoren Theorem cn be used to solve mn rel-world problems. SPI Test Emple Dmon is locked out of his house, with the onl open window on the second floor, 1 feet bove the ground. He needs to borrow ldder from his neighbor. If he must plce the ldder 5 feet from the house to void some bushes, wht length of ldder does Dmon need? A 7 feet C 1 feet B 11 feet D 17 feet 1 ft 5 ft Note: Not drwn to scle Red the Test Item Test-TkingTip Eliminte Unresonble Answers Since the hpotenuse of right tringle is lws the longest side, the length of the ldder in Emple must be greter thn 5 or 1 feet. Since 7 nd 11 feet re both less thn 1 feet, choices A nd B cn be eliminted. The distnce the ldder is from the house, the height the ldder reches, nd the length of the ldder itself mke up the lengths of the sides of right tringle. You need to find the length of the ldder, which is the hpotenuse. Solve the Test Item Method 1 Use Pthgoren triple. The lengths of the legs re 5 nd 1. 5, 1, 1 is Pthgoren triple, so the length of the ldder is 1 feet. Method Use the Pthgoren Theorem. Let the length of the ldder be = 169 = 169 = 1 = Pthgoren Theorem Simplif. Tke the positive squre root of ech side. Simplif. So the nswer is choice C. connected.mcgrw-hill.com 54

18 GuidedPrctice. According to our compn s sfet regultions, the distnce from the bse of ldder to wll tht it lens ginst should be t lest one fourth of the ldder s totl length. You re given 0-foot ldder to plce ginst wll t job site. If ou follow the compn s sfet regultions, wht is the mimum distnce up the wll the ldder will rech, to the nerest tenth? 0 ft ft F 1 feet H 0.6 feet Note: Not drwn to scle. G 19.4 feet J 0.6 feet Converse of the Pthgoren Theorem The converse of the Pthgoren Theorem lso holds. You cn use this theorem to help ou determine whether tringle is right tringle given the mesures of ll three sides. Theorem 8.5 Converse of the Pthgoren Theorem Words If the sum of the squres of the lengths of the shortest sides of tringle is equl to the squre of the length of the longest side, then the tringle is right tringle. c SmbolsIf + b = c, then ABC is right tringle. b You will prove Theorem 8.5 in Eercise 5. Stud Tip Determining the Longest Side If the mesures of n of the sides of tringle re epressed s rdicls, ou m wish to use clcultor to determine which length is the longest. You cn lso use side lengths to clssif tringle s cute or obtuse. Theorems Pthgoren Inequlit Theorems 8.6 If the squre of the length of the longest side of tringle is less thn the sum of the squres of the lengths of the other two sides, then the tringle is n cute tringle. Smbols If c < + b, then ABC is cute. b c 8.7 If the squre of the length of the longest side of tringle is greter thn the sum of the squres of the lengths of the other two sides, then the tringle is n obtuse tringle. c Smbols If c < + b, then ABC is obtuse. b 544 Lesson 8- The Pthgoren Theorem nd Its Converse You will prove Theorems 8.6 nd 8.7 in Eercises 6 nd 7, respectivel.

19 Emple 4 Clssif Tringles Determine whether ech set of numbers cn be the mesures of the sides of tringle. If so, clssif the tringle s cute, right, or obtuse. Justif our nswer.. 7, 14, 16 Review Vocbulr Tringle Inequlit Theorem The sum of the lengths of n two sides of tringle must be greter thn the length of the third side. (Lesson 5-5) Step 1 Determine whether the mesures cn form tringle using the Tringle Inequlit Theorem > > > 14 The side lengths 7, 14, nd 16 cn form tringle. Step Clssif the tringle b compring the squre of the longest side to the sum of the squres of the other two sides. c + b Compre c nd + b Substitution 56 > 45 Simplif nd compre. Since c > + b, the tringle is obtuse. b. 9, 40, 41 Step 1 Determine whether the mesures cn form tringle > > > 40 The side lengths 9, 40, nd 41 cn form tringle. Step Clssif the tringle. c + b Compre c nd + b Substitution 1681 = 1681 Simplif nd compre. Since c > + b, the tringle is right tringle. GuidedPrctice 4A. 11, 60, 61 4B., 4, 5 4C. 6., 1.8, 0 Check Your Understnding = Step-b-Step Solutions begin on pge R0. Emple 1 Find Emple 4. Use Pthgoren triple to find. Eplin our resoning. 1 5 connected.mcgrw-hill.com 545

20 Emple 5. MULTIPLE CHOICE The minsil of bot is shown. Wht is the length, in feet, of LN? A 5.5 C 7.5 B 65 D ft Emple 4 Determine whether ech set of numbers cn be the mesures of the sides of tringle. If so, clssif the tringle s cute, obtuse, or right. Justif our nswer. 45 ft 6. 15, 6, , 18, , 0, 4 Prctice nd Problem Solving Etr Prctice begins on pge 969. Emple 1 Find Emple Use Pthgoren Triple to find Emple 19. BASKETBALL The support for bsketbll gol forms right tringle s shown. Wht is the length of the horizontl portion of the support? 1 1 ft 1 ft 0. DRIVING The street tht Khlih usull uses to get to school is under construction. She hs been tking the detour shown. If the construction strts t the point where Khlih leves her norml route nd ends t the point where she re-enters her norml route, bout how long is the stretch of rod under construction? 1.8 mi 0.8 mi 546 Lesson 8- The Pthgoren Theorem nd Its Converse

21 Emple 4 Determine whether ech set of numbers cn be the mesures of the sides of tringle. If so, clssif the tringle s cute, obtuse, or right. Justif our nswer. 1. 7, 15, 1. 10, 1,. 4.5, 0, , 46, , 6.4, , 1, 14 B Find COORDINATE GEOMETRY Determine whether XYZ is n cute, right, or obtuse tringle for the given vertices. Eplin. 0. X(-, -), Y(-1, 0), Z(0, -1) 1. X(-7, -), Y(-, -5), Z(-4, -1). X(1, ), Y(4, 6), Z(6, 6). X(, 1), Y(, 7), Z(11, 1) 4. JOGGING Brett jogs in the prk three times week. Usull, he tkes _ -mile pth tht cuts through 4 the prk. Tod, the pth is closed, so he is tking the ornge route shown. How much frther will he jog on his lternte route thn he would hve if he hd followed his norml pth? 0.45 mi 4 mi C 5. PROOF Write prgrph proof of Theorem 8.5. PROOF Write two-column proof for ech theorem. 6. Theorem Theorem 8.7 Find the perimeter nd re of ech figure ALGEBRA The sides of tringle hve lengths, + 5, nd 5. If the length of the longest side is 5, wht vlue of mkes the tringle right tringle? 4. ALGEBRA The sides of tringle hve lengths, 8, nd 1. If the length of the longest side is, wht vlues of mke the tringle cute? 4 TELEVISION The screen spect rtio, or the rtio of the width to the length, of high-definition television is 16:9. The size of television is given b the digonl distnce cross the screen. If n HDTV is 41 inches wide, wht is its screen size? Screen Size 41 in. connected.mcgrw-hill.com 547

22 44. PLAYGROUND According to the Hndbook for Public Plground Sfet, the rtio of the verticl distnce to the horizontl distnce covered b slide should not be more thn bout 4 to 7. If the horizontl distnce llotted in slide design is 14 feet, pproimtel how long should the slide be? Find MULTIPLE REPRESENTATIONS In this problem, ou will investigte specil right tringles.. Geometric Drw three different isosceles right tringles tht hve whole-number side lengths. Lbel the tringles ABC, MNP, nd XYZ with the right ngle locted t verte A, M, nd X, respectivel. Lbel the leg lengths of ech side, nd find the ect length of the hpotenuse. b. Tbulr Cop nd complete the tble below. Tringle Length Rtio ABC BC AB _ BC AB MNP NP MN _ NP MN XYZ YZ XY YZ_ XY c. Verbl Mke conjecture bout the rtio of the hpotenuse to leg of n isosceles right tringle. cm cm cm H.O.T. Problems Use Higher-Order Thinking Skills 49. CHALLENGE Find the vlue of in the figure t the right. 50. REASONING True or flse? An two right tringles with the sme hpotenuse hve the sme re. Eplin our resoning OPEN ENDED Drw right tringle with side lengths tht form Pthgoren triple. If ou double the length of ech side, is the resulting tringle cute, right, or obtuse? if ou hlve the length of ech side? Eplin. 5. WRITING IN MATH Reserch incommensurble mgnitudes, nd describe how this phrse reltes to the use of irrtionl numbers in geometr. Include one emple of n irrtionl number used in geometr. 548 Lesson 8- The Pthgoren Theorem nd Its Converse

23 Stndrdized Test Prctice 5. Which set of numbers cnnot be the mesures of the sides of tringle? A 10, 11, 0 C 5, 45, 75 B 14, 16, 8 D 41, 55, A squre prk hs digonl wlkw from one corner to nother. If the wlkw is 10 meters long, wht is the pproimte length of ech side of the prk? F 60 m H 170 m G 85 m J 40 m 55. SHORT RESPONSE If the perimeter of squre is 00 units nd the perimeter of squre 1 is 150 units, wht is the perimeter of squre? 56. SAT/ACT In ABC, B is right ngle nd A is 0 greter thn C. Wht is the mesure of C? A 0 C 40 E 70 B 5 D 45 SPI , SPI Spirl Review Find the geometric men between ech pir of numbers. (Lesson 8-1) nd nd nd nd SCALE DRAWING Teodoro is creting scle model of sktebording rmp on 10-b-8-inch sheet of grph pper. If the rel rmp is going to be 1 feet b 8 feet, find n pproprite scle for the drwing nd determine the rmp s dimensions. (Lesson 7-7) Determine whether the tringles re similr. If so, write similrit sttement. If not, wht would be sufficient to prove the tringles similr? Eplin our resoning. (Lesson 7-) Write two-column proof. (Lesson 5-) Given: FG l FH is n nonperpendiculr segment from F to l. Prove: FH > FG 1 l Find ech mesure if m DGF = 5 nd m AGC = 40. (Lesson 4-) 66. m m 68. m 69. m 4 Find the distnce between ech pir of prllel lines with the given equtions. (Lesson -6) 70. = = - 7. = = = = Skills Review Find the vlue of. (Lesson 0-9) = = = = 4_ connected.mcgrw-hill.com 549

24 Geometr Lb Coordintes in Spce You hve used ordered pirs of two coordintes to describe the loction of point on the coordinte plne. Becuse spce hs three dimensions, point requires three numbers, or coordintes, to describe its loction in spce. A point in spce is represented b n ordered triple of rel numbers (,, z ). In the figure t the right, the ordered triple (,, 6) loctes point P. Notice tht rectngulr prism is used to show perspective. z P(,, 6) z O Tennessee Curriculum Stndrds Appl the midpoint nd distnce formuls to points nd segments to find midpoints, distnces, nd missing informtion in two nd three dimensions. Also ddresses The -, -, nd z-es re perpendiculr to ech other. Activit 1 Grph Rectngulr Solid Grph rectngulr solid tht hs two vertices, L(4, -5, ) nd the origin. Lbel the coordintes of ech verte. Step 1 Plot the -coordinte first. Drw segment from the origin 4 units in the positive direction. z Step To plot the -coordinte, drw segment five units in the negtive direction. K(0, -5, ) J(0, 0, ) Step Net, to plot the z-coordinte, drw segment two units long in the positive direction. Step 4 Lbel the coordinte L. L(4, -5, ) P(0, -5, 0) N(0, 0, 0) O M(4, 0, ) Step 5 Drw the rectngulr prism nd lbel ech verte: L(4, -5, ), K(0, -5, ), J(0, 0, ), M(4, 0, ) Q(4, -5, 0), P(0, -5, 0), N(0, 0, 0), nd R(4, 0, 0). Q(4, -5, 0) R(4, 0, 0) Finding the distnce between points nd the midpoint of segment in spce is similr to finding distnce nd midpoint in the coordinte plne. Ke Concept Distnce nd Midpoint Formuls in Spce If A hs coordintes A( 1, 1, z 1 ) nd B hs coordintes B(,, z ), then AB = ( - 1 ) + ( - 1 ) + (z - z 1 ). The midpoint M of AB hs coordintes M (_ 1 +,_ 1 +,_ z 1 + z ). A( 1, 1, z 1 ) z M B(,, z ) 550 Etend 8- Geometr Lb: Coordintes in Spce

25 Activit Distnce nd Midpoint Formuls in Spce Consider J(, 4, 9) nd K(-4, -5, 11).. Find JK. JK = ( - 1 ) + ( - 1 ) + (z - z 1 ) Distnce Formul in Spce = (-4 - ) + (-5-4) + (11-9) Substitution = 11 Simplif. = 11 Use clcultor. b. Determine the coordintes of the midpoint M of JK. M = (_ 1 +,_ 1 +,_ z 1 + z ) Midpoint Formul in Spce = (_ + (-4),_ 4 + (-5),_ ) Substitution = (-1, - 1_, 10 ) Simplif. Eercises Grph rectngulr solid tht contins the given point nd the origin s vertices. Lbel the coordintes of ech verte. 1. A(, 1, 5). P(-1, 4, ). C(-,, ) 4. R(, -4, 1) 5. P(4, 6, -) 6. G(4, 1, -) 7. K(-, -4, -4) 8. W(-1, -, -6) 9. W(,, 4) Determine the distnce between ech pir of points. Then determine the coordintes of the midpoint M of the segment joining the pir of points. 10. D(0, 0, 0) nd E(1, 5, 7) 11. G(-, -4, 6) nd H(5, -, -5) 1. K(,, 0) nd L(-, -, 0) 1. P(-, -5, 8) nd Q(, -, -1) 14. A(4, 7, 9) nd B(-, 8, -8) 15. W(-1, 8, 10) nd Z(-4, 1, -) 16. F _ (, 0, 4_ nd G(0,, 0) 5 5) 17. G(1, -1, 6) nd H ( 1_ 5, - _ 5, ) 18. B(,, ) nd C(-, 4, 4 ) 19. S(6, 4, 4 ) nd T(4, 5, ) 0. PROOF Write coordinte proof of the Distnce Formul in Spce. Given: A hs coordintes A( 1, 1, z 1 ), nd B hs coordintes B(,, z ). Prove: AB = ( - 1 ) + ( - 1 ) + (z - z 1 ) 1. WRITING IN MATH Compre nd contrst the Distnce nd Midpoint Formuls on the coordinte plne nd in three-dimensionl coordinte spce. A( 1, 1, z 1 ) z B(,, z ) connected.mcgrw-hill.com 551

26 Specil Right Tringles Then You used properties of isosceles nd equilterl tringles. (Lesson 4-6) Now 1Use the properties of tringles. Use the properties of tringles. Wh? As prt of pcket for students ttending regionl student council meeting, Lnds orders tringulr highlighters. She wnts to bu rectngulr boes for the highlighters nd other items, but she is concerned tht the highlighters will not fit in the bo she hs chosen. If she knows the length of side of the highlighter, Lnds cn use the properties of specil right tringles to determine if it will fit in the bo. Tennessee Curriculum Stndrds CLE Develop the tools of right tringle trigonometr in the contetul pplictions, including the Pthgoren Theorem, Lw of Sines nd Lw of Cosines Appl properties of nd to determine side lengths of tringles. SPI Use properties of right tringles to solve problems. Also ddresses Properties of Tringles The digonl of squre forms two congruent isosceles right tringles. Since the bse ngles of n isosceles tringle re congruent, the mesure of ech cute ngle is 90 or 45. Such tringle is lso known s tringle. You cn use the Pthgoren Theorem to find reltionship mong the side lengths of right tringle. l + l = h Pthgoren Theorem l = h Simplif. l = h l = h Simplif. Tke the positive squre root of ech side. This lgebric proof verifies the following theorem. h 45 l 45 l Theorem Tringle Theorem In tringle, the legs l re congruent nd the length of the hpotenuse h is times the length of leg. Smbols In tringle, l = l nd h = l. 45 l l 45 l Emple 1 Find the Hpotenuse Length in Tringle Find b. 9 9 The cute ngles of right tringle The legs of this right tringle hve the re complementr, so the mesure sme mesure, so it is isosceles. Since of the third ngle is or 45. this is tringle, use Since this is tringle, Theorem 8.8. use Theorem 8.8. h = l Theorem 8.8 h = l Theorem 8.8 = 9 Substitution = 6 Substitution = 9 or 18 = 55 Lesson 8-

27 GuidedPrctice Find. 1A. 45 1B C You cn lso work bckwrd using Theorem 8.8 to find the lengths of the legs of tringle given the length of its hpotenuse. Emple Find the Leg Lengths in Tringle Find. The legs of this right tringle hve the sme mesure,, so it is tringle. Use Theorem 8.8 to find. Review Vocbulr rtionlizing the denomintor method used to eliminte rdicls from the denomintor of frction (Lesson 0-9) _ 1 h = l Tringle Theorem 1 = Substitution _ 1 = Divide ech side b. _ = Rtionlize the denomintor. _ 1 = Multipl. 6 = Simplif. 1 GuidedPrctice A. B Stud Tip Altitudes of Isosceles Tringles Notice tht n ltitude of n isosceles tringle is lso medin of the tringle. In the figure t the right, BD bisects AC. Properties of Tringles A tringle is nother specil right tringle or right tringle with side lengths tht shre specil reltionship. You cn use n equilterl tringle to find this reltionship. When n ltitude is drwn from n verte of n equilterl tringle, two congruent tringles re formed. In the figure shown, ABD CBD, so AD CD. If AD =, then CD = nd AC =. Since ABC is equilterl, AB = nd BC =. Use the Pthgoren Theorem to find, the length of the ltitude BD, which is lso the longer leg of BDC. + = () Pthgoren Theorem + = 4 = = Simplif. = Simplif. Subtrct from ech side. Tke the positive squre root of ech side connected.mcgrw-hill.com 55

28 Stud Tip Use Rtios The lengths of the sides of tringle re in rtio of 1 to to or 1 : :. This lgebric proof verifies the following theorem. Theorem Tringle Theorem In tringle, the length of the hpotenuse h is times the length of the shorter leg s, nd the length of the longer leg l is times the length of the shorter leg. s 60 s Smbols In tringle, h = s nd l = s. s 0 Remember, the shortest side of tringle is opposite the smllest ngle. So the shorter leg in tringle is opposite the 0 ngle, nd the longer leg is opposite the 60 ngle. Emple Find Lengths in Tringle Find nd. The cute ngles of right tringle re complementr, so the mesure of the third ngle in this tringle is or 0. This is tringle. Use Theorem 8.9 to find, the length of the shorter side. _ 15 l = s Theorem = Substitution _ 15 _ 15 = Divide ech side b. _ = Rtionlize the denomintor. _ 15 = Multipl. = = 5 = Simplif Now use Theorem 8.9 to find, the length of the hpotenuse. h = s Theorem 8.9 = (5 ) or 10 Substitution GuidedPrctice Find nd. A B C Lesson 8- Specil Right Tringles

29 You cn use the properties of nd tringles to solve rel-world problems. Rel-World Emple 4 Use Properties of Specil Right Tringles INVENTIONS A compn mkes crons tht do not roll off tbles b shping them s tringulr prisms with equilterl bses. Siteen of these crons fit into bo shped like tringulr 1 prism tht is 1_ inches wide. The crons stnd on end in the bo nd the bse of the bo is equilterl. Wht re the dimensions of ech cron? 1 1 in. Understnd You know tht 16 crons with equilterl tringulr bses fit into prism. You need to find the bse length nd height of ech cron. Problem-SolvingTip Guess nd Check When using the guess nd check strteg, it cn be helpful to keep list of those guesses tht ou hve lred tried nd know do not work. In Emple 4, suppose our first guess hd been tht the bo ws 5 crons wide. The sketch of this possibilit revels tht this leds to stck of 5, not 16 crons. Pln Guess nd check to determine the rrngement of 16 crons tht would stck to fill the bo. Find the width of one cron nd use the Tringle Theorem to find its ltitude. Solve Mke guess tht 4 equilterl crons will fit cross the bse of the bo. A sketch shows tht the totl number of crons it tkes to fill the bo using 4 crons cross the bse is in. 1 The width of the bo is 1_ inches, so the 1 width of one cron is 1_ 4 or _ inch. 8 Drw n equilterl tringle representing one cron. Its ltitude forms the longer leg of two tringles. Use Theorem 8.9 to find the pproimte length of the ltitude. 8 in in. longer leg length = shorter leg length 8 in. 16 in. 8 in. _ = or bout Ech cron is _ or bout 0.4 inch b bout 0. inch. 8 Check Find the height of the bo using the Tringle Theorem. Then divide b four, since the bo is four crons high. The result is cron height of bout 0. inch. GuidedPrctice 4. FURNITURE The top of the qurium coffee tble shown is n isosceles right tringle. The tble s longest side, AC, mesures 107 centimeters. Wht is the distnce from verte B to side AC? Wht re the lengths of the other two sides? # " $ connected.mcgrw-hill.com 555

30 Check Your Understnding = Step-b-Step Solutions begin on pge R0. Emples 1- Find Emple Find nd Emple 4 7. ART Pulo is miling piece of rt tht he hs rolled into tube. The dimeter of the roll is 1_ 4 inches. He hs miler tht is tringulr prism with 4-inch equilterl tringle bses s shown in the digrm. Will the rolled rt fit through the opening of the miler? Eplin. 4 in. Prctice nd Problem Solving Etr Prctice begins on pge 969. Emples 1- Find If tringle hs hpotenuse length of 9, find the leg length. 15. Determine the length of the leg of tringle with hpotenuse length of Wht is the length of the hpotenuse of tringle if the leg length is 6 centimeters? 17. Find the length of the hpotenuse of tringle with leg length of 8 centimeters. 556 Lesson 8- Specil Right Tringles

31 Emple Find nd An equilterl tringle hs n ltitude length of 18 feet. Determine the length of side of the tringle. 5. Find the length of the side of n equilterl tringle tht hs n ltitude length of 4 feet. Emple 4 6. PACKAGING Refer to the beginning of the lesson. Ech highlighter is n equilterl tringle with 9-centimeter sides. Will the highlighter fit in 10-inch b 7-inch rectngulr bo? Eplin. 9 cm 7. EVENT PLANNING Grce is hving prt, nd she wnts to decorte the gble of the house s shown. The gble is n isosceles right tringle nd she knows tht the height of the gble is 8 feet. Wht length of lights will she need to cover the gble below the roof line? 8 ft B Find nd QUILTS The quilt block shown is mde up of squre nd four isosceles right tringles. Wht is the vlue of? Wht is the side length of the entire quilt block? 6 in. connected.mcgrw-hill.com 557

32 5 ZIP LINE Suppose zip line is nchored in one corner of course shped like rectngulr prism. The other end is nchored in the opposite corner s shown. If the zip line mkes 60 ngle with post AF, find the zip line s length, AD. 60º 5 ft 0 ft 8.4 ft 6. GAMES Kei is building ben bg toss for the school crnivl. He is using -foot bck support tht is perpendiculr to the ground feet from the front of the bord. He lso wnts to use support tht is perpendiculr to the bord s shown in the digrm. How long should he mke the support? ft ft 7. Find,, nd z. 8. Ech tringle in the figure is tringle. Find. 60 z MACHINERY The dump truck shown hs 15-foot bed length. Wht is the height of the bed h when ngle is 0? 45? 60? 15 ft h C 40. Find,, nd z, nd the perimeter of trpezoid PQRS z 41. COORDINATE GEOMETRY XYZ is tringle with right ngle Z. Find the coordintes of X in Qudrnt I for Y(-1, ) nd Z(6, ). 4. COORDINATE GEOMETRY EFG is tringle with m F = 90. Find the coordintes of E in Qudrnt III for F(-, -4) nd G(-, ). FG is the longer leg. 4. COORDINATE GEOMETRY JKL is tringle with right ngle K. Find the coordintes of L in Qudrnt IV for J(-, 5) nd K(-, -). 558 Lesson 8- Specil Right Tringles

33 44. EVENT PLANNING Ev hs reserved gzebo t locl prk for prt. She wnts to be sure tht there will be enough spce for her 1 guests to be in the gzebo t the sme time. If the floor of the gzebo is regulr hegon nd ech side is 7 feet, will there be enough room for Ev nd her friends? Eplin. (Hint: Use the Polgon Interior Angle Sum Theorem nd the properties of specil right tringles.) 7 ft 45 MULTIPLE REPRESENTATIONS In this problem, ou will investigte rtios in right tringles.. Geometric Drw three similr right tringles with 50 ngle. Lbel one tringle ABC where ngle A is the right ngle nd B is the 50 ngle. Lbel second tringle MNP where M is the right ngle nd N is the 50 ngle. Lbel the third tringle XYZ where X is the right ngle nd Y is the 50 ngle. b. Tbulr Cop nd complete the tble below. Tringle Length Rtio ABC AC BC AC_ BC MNP MP NP _ MP NP XYZ XZ YZ _ XZ YZ c. Verbl Mke conjecture bout the rtio of the leg opposite the 50 ngle to the hpotenuse in n right tringle with n ngle mesuring 50. H.O.T. Problems Use Higher-Order Thinking Skills 46. ERROR ANALYSIS Crmen nd Audre wnt to find in the tringle shown. Is either of them correct? Eplin. Crmen = _ 6 = Audre = _ 6 = OPEN ENDED Drw rectngle tht hs digonl twice s long s its width. Then write n eqution to find the length of the rectngle. 48. CHALLENGE Find the perimeter of qudrilterl ABCD REASONING The rtio of the mesure of the ngles of tringle is 1::. The length of the shortest side is 8. Wht is the perimeter of the tringle? E WRITING IN MATH Eplin how ou cn find the lengths of the two legs of tringle in rdicl form if ou re given the length of the hpotenuse. connected.mcgrw-hill.com 559

34 Stndrdized Test Prctice 51. If the length of the longer leg in tringle is 5, wht is the length of the shorter leg? A C 5 B 5 D ALGEBRA Solve = 7. F -44 H 41 G -41 J SHORT RESPONSE XYZ is tringle with right ngle Y. Find the coordintes of X in Qudrnt III for Y(-, -) nd Z(-, 7). 54. SAT/ACT In the figure, below, squre ABCD is ttched to ADE s shown. If m EAD is 0 nd AE is equl to 4, then wht is the re of squre ABCD? A 8 B 16 C 64 D 7 E SPI , SPI Spirl Review 55. SPORTS Dln is mking rmp for bike jumps. The rmp support forms right ngle. The bse is 1 feet long, nd the height is 9 feet. Wht length of plwood does Dln need for the rmp? (Lesson 8-) Find,, nd z. (Lesson 8-1) 56. z z 1 4 z Find the mesures of the ngles of ech tringle. (Lesson 7-1) 59. The rtio of the mesures of the three ngles is :5:. 60. The rtio of the mesures of the three ngles is 6:9: The rtio of the mesures of the three ngles is 5:7:8. Use the Eterior Angle Inequlit Theorem to list ll of the ngles tht stisf the stted condition. (Lesson 5-) 6. mesures less thn m 5 6. mesures greter thn m mesures greter thn m mesures less thn m Skills Review Find. (Lesson 8-) Lesson 8- Specil Right Tringles

35 Grphing Technolog Lb Trigonometr You hve investigted ptterns in the mesures of specil right tringles. Trigonometr is the stud of the ptterns in ll right tringles. You cn use the Cbri Jr. ppliction on grphing clcultor to investigte these ptterns. Tennessee Curriculum Stndrds CLE Develop the tools of right tringle trigonometr in the contetul pplictions, including the Pthgoren Theorem, Lw of Sines nd Lw of Cosines. Also ddresses Activit Investigte Trigonometric Rtios Step 1 Use the line tool on the F menu to drw horizontl line. Lbel the points on the line A nd B. Step Press F nd choose the Perpendiculr tool to crete perpendiculr line through point B. Drw nd lbel point C on the perpendiculr line. Steps 1 nd Step Use the segment tool on the F menu to drw AC. Step 4 Find nd lbel the mesures of BC nd AC using the Distnce nd Length tool under Mesure on the F5 menu. Use the Angle tool to find the mesure of A. Step 5 Clculte nd displ the rtio _ BC using the AC Clculte tool on the F5 menu. Lbel the rtio s A/B. Steps through 5 Step 6 Press. Then use the rrow kes to move the cursor close to point B. When the rrow is cler, press nd hold the ke. Drg B nd observe the rtio. Anlze the Results 1. Discuss the effect on _ BC b drgging point B on BC, AC, nd A. AC. Use the clculte tool to find the rtios _ AB AC nd _ BC. Then drg B nd observe the rtios. AB. MAKE A CONJECTURE The sine, cosine, nd tngent functions re trigonometric functions bsed on ngle mesures. Mke note of m A. Eit Cbri Jr. nd use,, nd on the clcultor to find sine, cosine nd tngent for m A. Compre the results to the rtios ou found in the ctivit. Mke conjecture bout the definitions of sine, cosine, nd tngent. connected.mcgrw-hill.com 561

36 Trigonometr Then You used the Pthgoren Theorem to find missing lengths in right tringles. (Lesson 8-) Now 1Find trigonometric rtios using right tringles. Use trigonometric rtios to find ngle mesures in right tringles. Wh? The steepness of hiking tril is often epressed s percent of grde. The steepest prt of Bright Angel Tril in the Grnd Cnon Ntionl Prk hs bout 15.7% grde. This mens tht the tril rises or flls 15.7 feet over horizontl distnce of 100 feet. You cn use trigonometric rtios to determine tht this steepness is equivlent to n ngle of bout 9. New Vocbulr trigonometr trigonometric rtio sine cosine tngent inverse sine inverse cosine inverse tngent Tennessee Curriculum Stndrds Understnd how the similrit of right tringles llows the trigonometric functions sine, cosine, nd tngent to be defined s rtio of sides Find the sine, cosine nd tngent rtios of n cute ngle of right tringle given the side lengths. SPI Determine nd use the pproprite trigonometric rtio for right tringle to solve contetul problem. Also ddresses CLE nd SPI Trigonometric Rtios The word trigonometr comes from two Greek terms, trigon, mening tringle, nd metron, mening mesure. The stud of trigonometr involves tringle mesurement. A trigonometric rtio is rtio of the lengths of two sides of right tringle. One trigonometric rtio of ABC is _ AC AB. B AA Similrit, right tringle with given cute ngle mesure is similr to ever other right tringle with the sme cute ngle mesure. So, trigonometric rtios re constnt for given ngle mesure. ABC FGH JKL, so _ AC AB = FH_ FG = _ JL JK The nmes of the three most common trigonometric rtios re given below. Ke Concept Trigonometric Rtios Words If ABC is right tringle with cute A, then the sine of A (written sin A) is the rtio of the length of the leg opposite A (opp) to the length of the hpotenuse (hp). If ABC is right tringle with cute A, then the cosine of A (written cos A) is the rtio of the length of the leg djcent A (dj) to the length of the hpotenuse (hp). If ABC is right tringle with cute A, then the tngent of A (written tn A) is the rtio of the length of the leg opposite A (opp) to the length of the leg djcent A (dj). _ sin A = opp hp sin B = _ opp hp _ cos A = dj hp cos B = _ dj hp _ tn A = opp dj tn B = _ opp dj or _ c or b_ c 5 or b_ c or _ c or _ b or b_ Smbols b 5 5 c 56 Lesson 8-4

37 Emple 1 Find Sine, Cosine, nd Tngent Rtios Stud Tip Memorizing Trigonometric Rtios SOH-CAH-TOA is mnemonic device for lerning the rtios for sine, cosine, nd tngent using the first letter of ech word in the rtios. sin A = _ opp hp cos A = _ dj hp tn A = _ opp dj Epress ech rtio s frction nd s deciml to the nerest hundredth.. sin P b. cos P sin P = _ opp cos P = _ dj 15_ hp hp 8_ = or bout 0.88 = or bout c. tn P d. sin Q tn P = _ opp sin Q = _ opp _ dj hp = 15 8_ or bout 1.88 = or bout e. cos Q f. tn Q cos Q = _ dj tn Q = _ opp _ hp dj = 15 8_ or bout 0.88 = or bout GuidedPrctice 1. Find sin J, cos J, tn J, sin K, cos K, nd tn K. Epress ech rtio s frction nd s deciml to the nerest hundredth Specil right tringles cn be used to find the sine, cosine, nd tngent of 0, 60, nd 45 ngles. Emple Use Specil Right Tringles to Find Trigonometric Rtios Use specil right tringle to epress the tngent of 0 s frction nd s deciml to the nerest hundredth. Drw nd lbel the side lengths of right tringle, with s the length of the shorter leg. 0 The side opposite the 0 ngle hs mesure of. The side djcent to the 0 ngle hs mesure of. tn 0 = _ opp 60 Definition of tngent rtio dj = = _ 1_ Substitution Simplif nd rtionlize the denomintor. = or bout 0.58 Simplif nd use clcultor. GuidedPrctice. Use specil right tringle to epress the cosine of 45 s frction nd s deciml to the nerest hundredth. connected.mcgrw-hill.com 56

38 Rel-World Link The grde of tril often chnges mn times. Averge grde is the verge of severl consecutive running grdes of tril. Mimum grde is the smller section of tril tht eceeds the tril s tpicl running grde. Trils often hve mimum grdes tht re much steeper thn tril s verge running grde. Source: Federl Highw Administrtion Rel-World Emple Estimte Mesures Using Trigonometr HIKING A certin prt of hiking tril slopes upwrd t bout 5 ngle. After trveling horizontl distnce of 100 feet long this prt of the tril, wht would be the chnge in hiker s verticl position? Wht distnce hs the hiker trveled long the pth? Let m A = 5. The verticl chnge in the hiker s position is, the mesure of the leg opposite A. The horizontl distnce trveled is 100 feet, the mesure of the leg djcent to A. Since the length of the leg opposite nd the leg djcent to given ngle re involved, write n eqution using tngent rtio. tn A = _ opp Definition of tngent rtio dj tn 5 = _ Substitution tn 5 = Multipl ech side b 100. Use clcultor to find The hiker is bout 8.75 feet higher thn when he strted wlking. 5º ft 100 ft ft Stud Tip Grphing Clcultor Be sure our grphing clcultor is in degree mode rther thn rdin mode. The distnce trveled long the pth is the length of the hpotenuse, so ou cn use cosine rtio to find this distnce. cos A = _ dj Definition of cosine rtio hp cos 5 = _ 100 Substitution cos 5 = 100 Multipl ech side b. 100 = _ cos 5 Use clcultor to find. Divide ech side b cos The hiker hs trveled distnce of bout feet long the pth. GuidedPrctice Find to the nerest hundredth. A B C. ARCHITECTURE The front of the vction cottge shown is n isosceles tringle. Wht is the height of the cottge bove its foundtion? Wht is the length of the roof? Eplin our resoning. ft ft 65 ft 60º 564 Lesson 8-4 Trigonometr

39 Use Inverse Trigonometric Rtios In Emple, ou found tht tn It follows tht if the tngent of n cute ngle is 0.58, then the ngle mesures pproimtel 0. Reding Mth Inverse Trigonometric Rtios The epression sin -1 is red the inverse sine of nd is interpreted s the ngle with sine. Be creful not to confuse this nottion with the nottion for negtive eponents sin -1 1_ sin. Insted, this nottion is similr to the nottion for n inverse function, f -1 (). If ou know the sine, cosine, or tngent of n cute ngle, ou cn use clcultor to find the mesure of the ngle, which is the inverse of the trigonometric rtio. Words Ke Concept Inverse Trigonometric Rtios If A is n cute ngle nd the sine of A is, then the inverse sine of is the mesure of A. Smbols If sin A =, then sin -1 = m A. Words If A is n cute ngle nd the cosine of A is, then the inverse cosine of is the mesure of A. Smbols If cos A =, then cos -1 = m A. Words If A is n cute ngle nd the tngent of A is, then the inverse tngent of is the mesure of A. Smbols If tn A =, then tn -1 = m A. So if tn , then tn Emple 4 Find Angle Mesures Using Inverse Trigonometric Rtios Stud Tip Grphing Clcultor The second functions of the,, nd kes re usull the inverses. Use clcultor to find the mesure of A to the nerest tenth. The mesures given re those of the leg opposite A nd the hpotenuse, so write n eqution using the sine rtio. sin A = _ 18 or _ sin A = _ opp 7 hp If sin A = _, then _ sin-1 = m A. Use clcultor. KEYSTROKES: [SIN] So, the mesure of A is pproimtel GuidedPrctice Use clcultor to find the mesure of A to the nerest tenth A. 4B connected.mcgrw-hill.com 565

40 When ou use given mesures to find the unknown ngle nd side mesures of right tringle, this is known s solving right tringle. To solve right tringle, ou need to know two side lengths or one side length nd the mesure of one cute ngle. Emple 5 Solve Right Tringle Solve the right tringle. Round side mesures to the nerest tenth nd ngle mesures to the nerest degree. Stud Tip Alterntive Methods Right tringles cn often be solved using different methods. In Emple 5, m Y could hve been found using tngent rtio, nd m X nd sine rtio could hve been used to find XY. Step 1 Find m X b using tngent rtio. tn X = _ 9 tn X = _ opp 5 dj tn -1 _ 9 = m X Definition of inverse tngent m X Use clcultor. So the mesure of X is bout 61. Step Find m Y using Corollr 4.1, which sttes tht the cute ngles of right tringle re complementr. m X + m Y = 90 Corollr m Y 90 m X 61 m Y 9 Subtrct 61 from ech side. So the mesure of Y is bout 9. Step Find XY b using the Pthgoren Theorem. (XZ) + (ZY) = (XY) Pthgoren Theorem Wtch Out! Approimtion If using clculted mesures to find other mesures in right tringle, be creful not to round vlues until the lst step. So in the following -1 eqution, use tn 9_ 5 insted of its pproimte vlue, 61. XY = 9_ sin X = 9_ sin ( tn-1 9_ 10. 5) GuidedPrctice = (XY) Substitution 106 = (XY ) Simplif. 106 = XY Tke the positive squre root of ech side. 10. XY Use clcultor. So the mesure of XY is bout 10.. Solve ech right tringle. Round side mesures to the nerest tenth nd ngle mesures to the nerest degree. 5A B C Lesson 8-4 Trigonometr

41 Check Your Understnding = Step-b-Step Solutions begin on pge R0. Emple 1 Epress ech rtio s frction nd s deciml to the nerest hundredth. 1. sin A. tn C. cos A tn A 5. cos C 6. sin C 16 Emple 7. Use specil right tringle to epress sin 60 s frction nd s deciml to the nerest hundredth. Emple Find. Round to the nerest hundredth SPORTS Dvid is building bike rmp. He wnts the ngle tht the rmp mkes with the ground to be 0. If the bord he wnts to use for his rmp is 1_ feet long, bout how tll will the rmp need to be t the highest point? 0º 1 ft Emple 4 Use clcultor to find the mesure of Z to the nerest tenth Emple Solve the right tringle. Round side mesures to the nerest tenth nd ngle mesures to the nerest degree. 9 6 Prctice nd Problem Solving Etr Prctice begins on pge 969. Emple 1 Find sin J, cos J, tn J, sin L, cos L, nd tn L. Epress ech rtio s frction nd s deciml to the nerest hundredth connected.mcgrw-hill.com 567

42 Emple Use specil right tringle to epress ech trigonometric rtio s frction nd s deciml to the nerest hundredth.. tn 60. cos 0 4. sin sin 0 6. tn cos 60 Emple Find. Round to the nerest tenth GYMNASTICS The springbord tht Eric uses in his gmnstics clss hs 6-inch coils nd forms n ngle of 14.5 with the bse. About how long is the springbord? 6 in ROLLER COASTERS The ngle of scent of the first hill of roller coster is 55. If the length of the trck from the beginning of the scent to the highest point is 98 feet, wht is the height of the roller coster when it reches the top of the first hill? 98 ft 55 Emple 4 Use clcultor to find the mesure of T to the nerest tenth Lesson 8-4 Trigonometr

43 Emple 5 Solve ech right tringle. Round side mesures to the nerest tenth nd ngle mesures to the nerest degree BACKPACKS Rmón hs rolling bckpck tht is _ feet tll when the hndle is etended. 4 When he is pulling the bckpck, Rmon s hnd is feet from the ground. Wht ngle does his bckpck mke with the floor? Round to the nerest degree. 4 ft ft B COORDINATE GEOMETRY Find the mesure of ech ngle to the nerest tenth of degree using the Distnce Formul nd n inverse trigonometric rtio. 47 K in right tringle JKL with vertices J(-, -), K(-7, -), nd L(-, 4) 48. Y in right tringle XYZ with vertices X(4, 1), Y(-6, ), nd Z(-, 7) 49. A in in right tringle ABC with vertices A(, 1), B(, -), nd C(8, -) 50. SCHOOL SPIRIT Hn is mking pennnt for ech of the 18 girls on her bsketbll tem. Wht is the totl length of ornge felt tht Hn needs to bu in order to hve enough for the borders of ll of the pennnts? 7 in. 70º TVH Vikings Bsketbll 70º Find the perimeter nd re of ech tringle. Round to the nerest hundredth in cm 48.5 ft 54. Find the tngent of the greter cute ngle in tringle with side lengths of, 4, nd 5 centimeters. 55. Find the cosine of the smller cute ngle in tringle with side lengths of 10, 4, nd 6 inches. 56. ESTIMATION Ethn nd Triq wnt to estimte the re of the field tht their tem will use for soccer prctice. The know tht the field is rectngulr, nd the hve pced off the width of the field s shown. The used the fence posts t the corners of the field to estimte tht the ngle between the length of the field nd the digonl is bout 40. If the ssume tht ech of their steps is bout 18 inches, wht is the re of the prctice field in squre feet? Round to the nerest squre foot. 80 steps 40 connected.mcgrw-hill.com 569

44 C Find nd. Round to the nerest tenth COORDINATE GEOMETRY Show tht the slope of line t 5 from the -is is equl to the tngent of MULTIPLE REPRESENTATIONS In this problem, ou will investigte n lgebric reltionship between the sine nd cosine rtios.. Geometric Drw three right tringles tht re not similr to ech other. Lbel the tringles ABC, MNP, nd XYZ, with the right ngles locted t vertices B, N, nd Y, respectivel. Mesure nd lbel ech side of the three tringles. b. Tbulr Cop nd complete the tble below. Tringle Trigonometric Rtios Sum of Rtios Squred ABC MNP XYZ cos A sin A (cos A) + (sin A) = cos C sin C (cos C) + (sin C) = cos M sin M (cos M) + (sin M) = cos P sin P (cos P) + (sin P) = cos X sin X (cos X) + (sin X) = cos Z sin Z (cos Z) + (sin Z) = c. Verbl Mke conjecture bout the sum of the squres of the cosine nd sine of n cute ngle of right tringle. d. Algebric Epress our conjecture lgebricll for n ngle X. e. Anlticl Show tht our conjecture is vlid for ngle A in the figure t the right using the trigonometric functions nd the Pthgoren Theorem. r H.O.T. Problems Use Higher-Order Thinking Skills 6. CHALLENGE Solve ABC. Round to the nerest whole number. (7-1) 6. REASONING After solving severl right tringle trigonometr problems, Tristn noticed tht the vlues of sine nd cosine for the cute ngles of (4 + 1) ( - ) right tringle were less thn 1. Will the vlues of sine nd cosine for n cute ngle lws be - 1 less thn 1? Eplin. 64. OPEN ENDED Drw right ABC with the right ngle t verte B. Use ruler to mesure ech side, nd write the rtios for sin A nd cos C. How re the two rtios relted? 65. WRITING IN MATH Eplin how ou cn use rtios of the side lengths to find the ngle mesures of the cute ngles in right tringle. 570 Lesson 8-4 Trigonometr

45 Stndrdized Test Prctice 66. Wht is the vlue of tn? A tn = _ 1 5 B tn = 1_ C tn = 5 _ 1 D tn = 5 _ ALGEBRA Which of the following hs the sme vlue s -1? F -6 H -9 G 4-9 J GRIDDED RESPONSE If AC = 1 nd AB = 5, wht is the mesure of B to the nerest tenth? SAT/ACT The re of right tringle is 40 squre inches. If the bse is 0 inches long, how mn inches long is the hpotenuse? A 5 D 41 B 8 E 4 C 16 SPI , SPI 10.1., SPI Spirl Review Find nd. (Lesson 8-) Determine whether ech set of numbers cn be the mesures of the sides of tringle. If so, clssif the tringle s cute, obtuse, or right. Justif our nswer. (Lesson 8-) 7. 8, 15, , 1, , 0, , 4, , 5., , 14, MAPS The scle on the mp of New Meico is centimeters = 160 miles. The width of New Meico through Albuquerque on the mp is 4.1 centimeters. How long would it tke to drive cross New Meico if ou drove t n verge of 60 miles per hour? (Lesson 7-7) Albuquerque ALGEBRA Find nd. (Lesson 7-4) Roswell Ls Cruces Skills Review Solve ech proportion. Round to the nerest tenth if necessr. (Lesson 0-5) = _ = = _ = _ = _ = _ connected.mcgrw-hill.com 571

46 Grphing Technolog Lb Secnt, Cosecnt, nd Cotngent OBJECTIVE Investigte the other trigonometric rtios using technolog. In the previous lesson, ou used the trigonometric functions sine, cosine, nd tngent to find ngle reltionships in right ngles. In this ctivit, ou will use the reciprocls of those functions, cosecnt, secnt, nd cotngent, to eplore ngle nd side reltionships in right tringles. Ke Concept Reciprocl Trigonometric Rtios Words The cosecnt of A (written csc A) is the reciprocl of sin A. The secnt of A (written sec A) is the reciprocl of cos A. The cotngent of A (written cot A) is the reciprocl of tn A. Smbols csc A = 1_ sin A or _ c sec A = cot A = 1_ cos A or c _ b 1_ tn A or _ b b c Activit Find Trigonometric Vlues Step 1 Drw nd lbel right tringle with the dimensions shown t the right. Step Use our grphing clcultor to find the vlues for sin A, cos A, nd tn A Step Net, find the vlue for csc A b dividing 1 b A. Repet step to find sec A nd cot A. Step 4 Cop the tble below nd record our results. Net, find the vlue of ech trigonometric function for ngle C. Angle sin cos tn csc sec cot A C Eercises 1. Find the vlues of the si trigonometric functions for 45 ngle in tringle with legs tht re 4 cm.. In FGH, tn F = _ 5. Find cot F nd sin F if G is right ngle. 1. Find the vlues of the si trigonometric functions for ngle T in RST if m R = 6 nd r = 1. Round to the nerest hundredth. s r t 57 Etend 8-4 Grphing Technolog Lb: Secnt, Cosecnt, nd Cotngent

47 Mid-Chpter Quiz Lessons 8-1 through 8-4 Tennessee Curriculum Stndrds SPI Find the geometric men between ech pir of numbers. (Lesson 8-1) 1. 1 nd. 6 nd nd nd 10 Write similrit sttement identifing the three similr tringles in ech figure. (Lesson 8-1) Find,, nd z. (Lesson 8-1) 6. Find. (Lesson 8-) DESIGN Jmie designed pinwheel to put in her grden. In the pinwheel, the blue tringles re congruent equilterl tringles, ech with n ltitude of 4 inches. The red tringles re congruent isosceles right tringles. The hpotenuse of red tringle is congruent to side of the blue tringle. (Lesson 8-) 9 16 z 1 8. PARKS There is smll prk in corner mde b two perpendiculr streets. The prk is 100 ft b 150 ft, with digonl pth, s shown below. Wht is the length of pth AC? (Lesson 8-) 150 ft. If ngles 1,, nd re congruent, find the mesure of ech ngle. b. Find the perimeter of the pinwheel. Find nd. (Lesson 8-) ft 0 14 Find. Round to the nerest hundredth. (Lesson 8-) Epress ech rtio s frction nd s deciml to the nerest hundredth. (Lesson 8-4) 17. tn M 18. cos M cos N 0. sin N Solve the right tringle. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth. (Lesson 8-4) 11. MULTIPLE CHOICE Which of the following sets of numbers is not Pthgoren triple? (Lesson 8-) A 9, 1, 15 C 15, 6, 9 B 1, 7, 75 D 8, 1, connected.mcgrw-hill.com 57

48 Angles of Elevtion nd Depression Then Now Wh? You used similr tringles to mesure distnces indirectl. 1 Solve problems involving ngles of elevtion nd depression. Use ngles of elevtion nd depression to find the distnce between two objects. To mke field gol, kicker must kick the bll with enough force nd t n pproprite ngle of elevtion to ensure tht the bll will rech the gol post t level high enough to mke it over the horizontl br. This ngle must chnge depending on the initil plcement of the bll w from the bse of the golpost. (Lesson 7-) NewVocbulr ngle of elevtion ngle of depression 1 Angles of Elevtion nd Depression An ngle of elevtion is the ngle formed b horizontl line nd n observer s line of sight to n object bove the horizontl line. An ngle of depression is the ngle formed b horizontl line nd n observer s line of sight to n object below the horizontl line. line of sight ngle of elevtion Tennessee Curriculum Stndrds CLE Develop the tools of right tringle trigonometr in the contetul pplictions, including the Pthgoren Theorem, Lw of Sines nd Lw of Cosines Define, illustrte, nd ppl ngles of elevtion nd ngles of depression in rel-world situtions. SPI Determine nd use the pproprite trigonometric rtio for right tringle to solve contetul problem. Also ddresses SPI ngle of depression Horizontl lines re prllel, so the ngle of elevtion nd the ngle of depression in the digrm re congruent b the Alternte Interior Angles Theorem. Emple 1 Angle of Elevtion VACATION Leh is meeting friends t the cstle in the center of n musement prk. She sights the top of the cstle t n ngle of elevtion of 8. From the prk s brochure, she knows tht the cstle is 190 feet tll. If Leh is 5.5 feet tll, bout how fr is she from the cstle to the nerest foot? Mke sketch. # 190 ft " 5.5 ft 8 $ Since Leh is 5.5 feet tll, BC = or feet. Let represent the distnce from Leh to the cstle, AC. BC tn A = _ AC _ tn 8 = =_ tn 8 tn = opposite _ djcent m A = 8, BC = 184.5, AC = Solve for. 6.1 Use clcultor. Leh is bout 6 feet from the cstle. 574 Lesson 8-5

49 GuidedPrctice 1. FOOTBALL The cross br of golpost is 10 feet high. If field gol ttempt is mde 5 rds from the bse of the golpost tht clers the gol b 1 foot, wht is the smllest ngle of elevtion t which the bll could hve been kicked to the nerest degree? Emple Angle of Depression Wtch Out! Angles of Elevtion nd Depression To void mislbeling, remember tht ngles of elevtion nd depression re lws formed with horizontl line nd never with verticl line. EMERGENCY A serch nd rescue tem is irlifting people from the scene of boting ccident when the observe nother person in need of help. If the ngle of depression to this other person is 4 nd the helicopter is 18 feet bove the wter, wht is the horizontl distnce from the rescuers to this person to the nerest foot? Mke sketch of the sitution. Since AB nd DC re prllel, m BAC = m ACD b the Alternte Interior Angles Theorem. Let represent the horizontl distnce from the rescuers to the person DC. tn C = AD_ DC tn 4 = 18 _ tn = _ opposite djcent C = 4, AD = 18, nd DC = tn 4 = 18 Multipl ech side b. 18 = _ tn 4 Divide ech side b tn Use clcultor. 18 ft 4º ngle of depression 4º Note: Art not drwn to scle. The horizontl distnce from the rescuers to the person is 0.0 feet. Mth Histor Link Ertosthenes ( B.C.) Ertosthenes ws mthemticin nd stronomer who ws born in Crene, which is now Lib. He used the ngle of elevtion of the Sun t noon in the cities of Alendri nd Sene (now Egpt) to mesure the circumference of Erth. Source: Encclopedi Britnnic GuidedPrctice. LIFEGUARDING A lifegurd is wtching bech from line of sight 6 feet bove the ground. She sees swimmer t n ngle of depression of 8. How fr w from the tower is the swimmer? Two Angles of Elevtion or Depression Angles of elevtion or depression to two different objects cn be used to estimte the distnce between those objects. Similrl, the ngles from two different positions of observtion to the sme object cn be used to estimte the object s height. 8º connected.mcgrw-hill.com 575

50 Emple Use Two Angles of Elevtion or Depression TREE REMOVAL To estimte the height of tree she wnts removed, Mrs. Long sights the tree s top t 70 ngle of elevtion. She then steps bck 10 meters nd sights the top t 6 ngle. If Mrs. Long s line of sight is 1.7 meters bove the ground, how tll is the tree to the nerest meter? 1.7 m D 6º 10 m A 70º C B Understnd ABC nd ABD re right tringles. The height of the tree is the sum of Mrs. Long s height nd AB. Pln Since her initil distnce from the tree is not given, write nd solve sstem of equtions using both tringles. Let AB = nd CB =. So DB = + 10 nd the height of the tree is Rel-World Link In the United Sttes, lumber volume is mesured in bordfeet, which is defined s piece of wood contining 144 cubic inches. Woodlnd owners often estimte the lumber volume of trees the own to determine how mn to cut nd sell. Source: The Ohio Stte Universit School of Nturl Resources Solve Use ABC. tn 70 = opposite tn = ; m ACB = 70 djcent tn 70 = Multipl ech side b. Use ABD. tn 6 = _ + 10 _ tn = opposite ; m D = 6 djcent ( + 10) tn 6 = Multipl ech side b Substitute the vlue for from ABD in the eqution for ABC nd solve for. tn 70 = tn 70 = ( + 10) tn 6 tn 70 = tn tn 6 tn 70 - tn 6 = 10 tn 6 (tn 70 - tn 6 ) = 10 tn 6 10 tn 6 = tn 70 - tn 6 Use clcultor to find tht.16. Using the eqution from ABC, =.16 tn 70 or bout 5.9. Stud Tip Indirect Mesurement When using the ngles of depression to two different objects to clculte the distnce between them, it is importnt to remember tht the two objects must lie in the sme horizontl plne. In other words, one object cnnot be higher or lower thn the other. The height of the tree is or 7.6, which is bout 8 meters. Check Substitute the vlue for in the eqution from ABD. GuidedPrctice = ( ) tn 6 or bout 5.9. This is the sme vlue found using the eqution from ABC.. SKYSCRAPERS Two buildings re sited from top 00-meter skscrper. Building A is sited t 5 ngle of depression, while Building B is sighted t 6 ngle of depression. How fr prt re the two buildings to the nerest meter? 576 Lesson 8-5 Angles of Elevtion nd Depression

51 Check Your Understnding = Step-b-Step Solutions begin on pge R0. Emple 1 1. BIKING Lenor wnts to build the bike rmp shown. Find the length of the bse of the rmp. Emple. BASEBALL A fn is seted in the upper deck of stdium 00 feet w from home plte. If the ngle of depression to the field is 6, t wht height is the fn sitting? Emple 10 ft 0. DANCES Annbelle nd Rich re setting up decortions for their school dnce. Rich is stnding 5 feet directl in front of Annbelle under disco bll. If the ngle of elevtion from Annbelle to the bll is 40 nd Rich to the bll is 50, how high is the disco bll? 40º 5 ft Prctice nd Problem Solving Emple 1 50º Etr Prctice begins on pge HOCKEY A hocke pler tkes shot t distnce of 0 feet w from 5-foot gol. If the puck trvels t 15 ngle of elevtion towrd the center of the gol, will the pler score? 15º 5 MOUNTAINS Find the ngle of elevtion to the pek of mountin for n observer who is 155 meters from the mountin if the observer s ee is 1.5 meters bove the ground nd the mountin is 50 meters tll. Emple 6. PLANTS Alici hngs plnt in her bedroom window. If the sunlight comes in with 46 ngle of depression, nd the length of the bsket chin is 9 inches, how fr is the plnt from the window? 9 in. 46º 7. AVIATION Due to storm, pilot fling t n ltitude of 58 feet hs to lnd erl. If he hs horizontl distnce of 000 feet to lnd, t wht ngle of depression should he lnd? Emple 8. PYRAMIDS Miko nd Tler re visiting the Gret Prmid in Egpt. From where Miko is stnding, the ngle of elevtion to the top of the prmid is From Tler s position, the ngle of elevtion is 50. If the re stnding 0 feet prt, how tll is the prmid? ft connected.mcgrw-hill.com 577

52 9 DIVING Austin is stnding on the high dive t the locl pool. Two of his friends re in the wter on the opposite side of the pool. If the ngle of depression to one of his friends is 40, nd 0 to his other friend who is 5 feet beond the first, how tll is the pltform? 0º 40º 5 ft 10. BASKETBALL Clire nd Mris re both witing to get rebound during bsketbll gme. If the height of the bsketbll hoop is 10 feet, the ngle of elevtion between Clire nd the gol is 5, nd the ngle of elevtion between Mris nd the gol is 5, how fr prt re the stnding? 5º 5º 10 ft 11. RIVERS Hugo is stnding in the top of St. Louis Gtew Arch, looking down on the Mississippi River. The ngle of depression to the closer bnk is 45 nd the ngle of depression to the frther bnk is 18. The rch is 60 feet tll. Estimte the width of the river t tht point. 60 ft 18º 45º B 1. VOLCANOES The Unzen Volcno in Jpn hs mgm reservoir tht is locted 15 kilometers beneth the Chijiw B. The mgm rises estwrd t n ngle of elevtion of 40. Wht length of mgm is below se level? 1. BRIDGES Suppose ou re stnding in the middle of the pltform of the world s longest suspension bridge, the Akshi Kiko Bridge. If the height from the top of the pltform holding the suspension cbles is 97 meters, nd the length from the pltform to the center of the bridge is 995 meters, wht is the ngle of depression from the pltform to the center of the bridge? 97 m 995 m 14. LIGHTHOUSES To id in nvigtion, Little Gull Islnd Lighthouse shines light from height of 91 feet with 6 ngle of depression. Plum Islnd Lighthouse, 1800 feet w, shines light from height of 4 feet with ngle of depression. Which light will rech bot tht sits ectl between Little Gull Islnd Lighthouse nd Plum Islnd Lighthouse? 15. TOURISM Hkeem is on tour bus in Pris with his fmil. He sees L rc de Triomphe t n ngle of 4. If the rc is 16 feet tll, how fr w is the bus? Round to the nerest tenth. 16 ft 4º 578 Lesson 8-5 Angles of Elevtion nd Depression

53 16. MAINTENANCE Two telephone repir workers rrive t loction to restore electricit fter power outge. One of the workers climbs up the telephone pole while the other worker stnds 10 feet to left of the pole. If the terminl bo is locted 0 feet bove ground on the pole nd the ngle of elevtion from the truck to the repir worker is 7, how fr is the worker on the ground stnding from the truck? 0 ft 7 10 ft 17 PHOTOGRAPHY A digitl cmer with pnormic lens is described s hving view with n ngle of elevtion of 8. If the cmer is on -foot tripod imed directl t 14-foot-tll monument, how fr from the monument should ou plce the tripod to see the entire monument in our photogrph? 18. BALLOONING As prt of their wether unit, Anoki s science clss took hot ir blloon ride. As the pssed over fenced field, the ngle of depression of the closer side of the fence ws, nd the ngle of depression of the frther side of the fence ws 7. If the height of the blloon ws 800 feet, estimte the width of the field ft 19. MARATHONS The Bdwter Ultrmrthon is rce tht begins t the lowest point in North Americ, Deth Vlle, nd ends t the highest point on the continent, Mount Whitne. The rce strts t depth of 86 meters from se level nd ends 441 meters bove se level.. Determine the ngle of elevtion to Mount Whitne if the length from the bse to the pek is 100 meters. b. If the ngle of depression to Deth Vlle is 8, wht is the horizontl distnce from se level? 8 86 m 441 m 100 m C 0. AMUSEMENT PARKS Indi, Enrique, nd Trin went to n musement prk while visiting Jpn. The went on Ferris wheel tht ws 100 meters in dimeter nd on n 80-meter cliff-dropping slide.. When Enrique nd Tin re t the topmost point on the Ferris wheel shown below, how fr re the from Indi? 57 b. If the cliff-dropping ride hs n ngle of depression of 46, how long is the slide? 46 connected.mcgrw-hill.com 579

54 1 DARTS Kelse nd José re pling ver competitive gme of drts. The center of the bull s-ee on the drtbord is 5.7 feet from the floor, nd the throw from distnce of 8.5 feet. José throws from height of 6 feet from the floor, nd Kelse throws 5 feet from the floor. Wht re the ngles of elevtion or depression from which ech must throw to get bull s-ee? 5.7 ft 8.5 ft. MULTIPLE REPRESENTATIONS In this problem, ou will investigte reltionships between the sides nd ngles of tringles.. Geometric Drw three tringles. Mke one cute, one obtuse, nd one right. Lbel one tringle ABC, second MNP, nd the third XYZ. Lbel the side lengths nd ngle mesures of ech tringle. b. Tbulr Cop nd complete the tble below. Tringle Rtios ABC _ sin A BC = _ sin B CA = _ sin C AB = MNP _ sin M NP = _ sin N PM = _ sin P MN = XYZ _ sin X YZ = _ sin Y ZX = _ sin Z XY = c. Verbl Mke conjecture bout the rtio of the sine of n ngle to the length of the leg opposite tht ngle for given tringle. H.O.T. Problems Use Higher-Order Thinking Skills. ERROR ANALYSIS Terrence nd Rodrigo re tring to determine the reltionship between ngles of elevtion nd depression. Terrence ss tht if ou re looking up t someone with n ngle of elevtion of 5, then the re looking down t ou with n ngle of depression of 55, which is the complement of 5. Rodrigo disgrees nd ss tht the other person would be looking down t ou with n ngle of depression equl to our ngle of elevtion, or 5. Is either of them correct? Eplin. 4. CHALLENGE Find the vlue of. Round to the nerest tenth REASONING Clssif the sttement below s true or flse. Eplin. 580 Lesson 8-5 Angles of Elevtion nd Depression As person moves closer to n object he or she is sighting, the ngle of elevtion increses. 6. WRITE A QUESTION A clssmte finds the ngle of elevtion of n object, but she is tring to find the ngle of depression. Write question to help her solve the problem. 7. WRITING IN MATH Describe w tht ou cn estimte the height of n object without using trigonometr b choosing our ngle of elevtion. Eplin wh our method will work.

55 Stndrdized Test Prctice 8. Rn wnted to know the height of cell-phone tower neighboring his propert. He wlked 80 feet from the bse of the tower nd mesured the ngle of elevtion to the top of the tower t 54. If Rn is 5 feet tll, wht is the height of the cell-phone tower? A 5 C 110 B 6 D SHORT RESPONSE A serchlight is 6500 feet from wether sttion. If the ngle of elevtion to the spot of light on the clouds bove the sttion is 45, how high is the cloud ceiling? 0. ALGEBRA Wht is the solution of this sstem of equtions? - 4 = = 8 F (4, 4) H (-4, -4) G (-4, 1) J (1, -4) 1. SAT/ACT A tringle hs sides in the rtio of 5 : 1 : 1. Wht is the mesure of the tringle s smllest ngle in degrees? A 1.4 D 4.71 B.6 E 67.8 C 4.14 SPI , SPI Spirl Review Epress ech rtio s frction nd s deciml to the nerest hundredth. (Lesson 8-4). sin C. tn A 4. cos C 5. tn C 6. cos A 7. sin A LANDSCAPING Imni needs to determine the height of tree. Holding drfter s 45 tringle so tht one leg is horizontl, she sights the top of the tree long the hpotenuse, s shown t the right. If she is 6 rds from the tree nd her ees re 5 feet from the ground, find the height of the tree. (Lesson 8-) PROOF Write two-column proof. (Lesson 7-5) 9. Given: CD bisects ACB. B construction, AE CD. Prove: AD_ DB = _ AC BC Given: JF bisects EFG. EH FG, EF HG Prove: EK_ KF = _ GJ JF 5 ft 45 6 d COORDINATE GEOMETRY Find the coordintes of the centroid of ech tringle. (Lesson 5-) 41. A(, ), B(7, 8), C(1, ) 4. X(-, -), Y(1, -1), Z(-7, -7) 4. A(-1, 11), B(-5, 1), C(-9, 6) 44. X(4, 0), Y(-, 4), Z(0, 6) Skills Review Solve ech proportion. (Lesson 7-1) 45. 1_ 5 = _ 46. _ = _ _ 4 16 = _ _ 1 = _ connected.mcgrw-hill.com 581

56 The Lw of Sines nd Lw of Cosines Then You used trigonometric rtios to solve right tringles. (Lesson 8-4) Now 1Use the Lw of Sines to solve tringles. Use the Lw of Cosines to solve tringles. Wh? You hve lerned tht the height or length of tree cn be clculted using right tringle trigonometr if ou know the ngle of elevtion to the top of the tree nd our distnce from the tree. Some trees, however, grow t n ngle or len due to wether dmge. To clculte the length of such trees, ou must use other forms of trigonometr. New Vocbulr Lw of Sines Lw of Cosines Tennessee Curriculum Stndrds CLE Develop the tools of right tringle trigonometr in the contetul pplictions, including the Pthgoren Theorem, Lw of Sines nd Lw of Cosines Use the Lw of Sines (ecluding the mbiguous cse) nd the Lw of Cosines to find missing side lengths nd/or ngle mesures in non-right tringles. SPI Determine nd use the pproprite trigonometric rtio for right tringle to solve contetul problem. Also ddresses SPI Lw of Sines In Lesson 8-4, ou used trigonometric rtios to find ngle nd side mesures in right tringles. The Lw of Sines cn be used for nonright tringles. Theorem 8.10 Lw of Sines If ABC hs lengths, b, nd c, representing the lengths of the sides opposite the ngles with mesures A, B, nd C, then sin A_ = sin B_ b = sin _ C c. You will prove one of the proportions for Theorem 8.10 in Eercise 45. You cn use the Lw of Sines to solve tringle if ou know the mesures of two ngles nd n side (AAS or ASA). If given ASA, use the Tringle Angle Sum Theorem to first find the mesure of the third ngle. Emple 1 Lw of Sines (AAS or ASA) Find. Round to the nerest tenth. We re given the mesures of two ngles nd nonincluded side, so use the Lw of Sines to write proportion. _ sin A = _ sin c C Lw of Sines _ sin 97 = _ sin 1 16 m A = 97, = 16, m C = 1, c = sin 97 = 16 sin 1 Cross Products Propert 16 sin 1 = _ Divide ech side b sin 97. sin Use clcultor. c b GuidedPrctice 1A. 1B You cn lso use the Lw of Sines to solve tringle if ou know the mesures of two sides nd nonincluded ngle (SSA). 58 Lesson 8-6

57 Emple Lw of Sines (SSA) Find. Round to the nerest degree. _ sin H = _ sin K Lw of Sines h k _ sin 45 = _ sin m H = 45, h = 8, m K =, k = sin 45 = 8 sin Cross Products Propert _ 10 sin 45 = sin Divide ech side b 8. 8 = sin-1 _ 10 sin 45 Use the inverse sine rtio. 8 6 Use clcultor GuidedPrctice A B Stud Tip Lw of Cosines Notice tht the ngle referenced in ech eqution of the Lw of Cosines corresponds to the side opposite tht ngle on the other side of the eqution. = b + c - bc cos A b = + c - c cos B c = + b - b cos C Lw of Cosines When the Lw of Sines cnnot be used to solve tringle, the Lw of Cosines m ppl. Theorem 8.11 Lw of Cosines If ABC hs lengths, b, nd c, representing the lengths of the sides opposite the ngles with mesures A, B, nd C, then = b + c - bc cos A, b = + c - c cos B, nd c = + b - b cos C. b c You will prove one of the equtions for Theorem 8.11 in Eercise 46. You cn use the Lw of Cosines to solve tringle if ou know the mesures of two sides nd the included ngle (SAS). Emple Lw of Cosines (SAS) Find. Round to the nerest tenth. Wtch Out! Order of opertions Remember to follow the order of opertions when simplifing epressions. Multiplictions or divisions must be performed before dditions or subtrctions. So, cos 8 cnnot be simplified to 4 cos 8. We re given the mesures of two sides nd their included ngle, so use the Lw of Cosines. c = + b - b cos C = (9)(11) cos 8 = cos 8 Lw of Cosines Substitution Simplif. = cos 8 Tke the squre root of ech side. 5. Use clcultor connected.mcgrw-hill.com 58

58 Stud Tip Obtuse Angles There re lso vlues for sin A, cos A, nd tn A when A 90. Vlues of the rtios for these ngles will be found using the trigonometric functions on our clcultor GuidedPrctice Find. Round to the nerest tenth. A B You cn lso use the Lw of Cosines if ou know three side mesures (SSS). Emple 4 Lw of Sines (AAS or ASA) Find. Round to the nerest degree. m = p + q - pq cos M Lw of Cosines 8 = (6)() cos Substitution 64 = 45-6 cos Simplif = -6 cos Subtrct 45 from ech side. _ 19 = cos Divide ech side b = cos -1 _ (- 19 Use the inverse cosine rtio. 6) 1 Use clcultor. GuidedPrctice 4A B You cn use the Lw of Sines nd Lw of Cosines to solve direct nd indirect mesurement problems. Rel-World Emple 5 Indirect Mesurement Rel-World Creer The first gme of bsketbll ws pled t YMCA in Springfield, Msschusetts, on December 1, Jmes Nismith, phsicl eduction instructor, invented the sport using soccer bll nd two hlf-bushel pech bskets, which is how the nme bsketbll cme bout. Source: Encclopedi Britnnic. BASKETBALL Drew nd Hunter re pling bsketbll. Drew psses the bll to Hunter when he is 6 feet from the gol nd 4 feet from Hunter. How fr is Hunter from the gol if the ngle from the gol to Drew nd then to Hunter is 4? Hunter Drw digrm. Since we 4 ft know two sides of tringle ft nd the included ngle, use 4 the Lw of Cosines. gol 6 ft Drew = (4)(6) cos 4 Lw of Cosines = cos 4 Simplif nd tke the positive squre root of ech side. 15 Use clcultor. Hunter is bout 15 feet from the gol when he tkes his shot. Hunter Drew 584 Lesson 8-6 The Lw of Sines nd Lw of Cosines

59 GuidedPrctice 5. LANDSCAPING At 10 feet w from the bse of tree, the ngle the top of tree mkes with the ground is 61. If the tree grows t n ngle of 78 with respect to the ground, how tll is the tree to the nerest foot? When solving right tringles, ou cn use sine, cosine, or tngent. When solving other tringles, ou cn use the Lw of Sines or the Lw of Cosines, depending on wht informtion is given. Reding Mth Solve Tringle Remember tht to solve tringle mens to find ll of the missing side mesures nd/or ngle mesures. Emple 6 Solve Tringle Solve tringle ABC. Round to the nerest degree. Since , this is not right tringle. Since the mesures of ll three sides re given (SSS), begin b using the Lw of Cosines to find m A = b + c - bc cos A Lw of Cosines 1 15 = (1)(1) cos A = 15, b = 1, nd c = 1 5 = 1-1 cos A Simplif. -88 = -1 cos A Subtrct 1 from ech side. _-88 = cos A -1 Divide ech side b -1. m A = cos -1 _ 88 1 Use the inverse cosine rtio. m A 74 Use clcultor. Wtch Out Rounding When ou round numericl solution nd then use it in lter clcultions, our nswers m be inccurte. Wit until fter ou hve completed ll of our clcultions to round. Use the Lw of Sines to find m B. _ sin A = _ sin B b _ sin 74 _ sin B 15 1 Lw of Sines m A 74, = 15, nd b = 1 1 sin 74 = 15 sin B Cross Products Propert _ 1 sin 74 = sin B Divide ech side b m B = sin-1 _ 1 sin 74 Use the inverse sine rtio. 15 m B 50 Use clcultor. B the Tringle Angle Sum Theorem, m C ( ) or 56. Therefore m A 74, m B 50, nd m C 56. GuidedPrctice Solve tringle ABC using the given informtion. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth. 6A. b = 10., c = 9., m A = 6 6B. = 6.4, c = 5.8, m C = 46 b c connected.mcgrw-hill.com 585

60 Concept Summr Solving Tringle To solve... Given Begin b using... right tringle: n tringle: leg-leg (LL) hpotenuse-leg (HL) cute ngle-hpotenuse (AH) cute ngle-leg (AL) ngle-ngle-side (AAS or ASA) side-side-ngle (SSA) ngle-side-ngle (SAS) side-side-side (SSS) tngent rtio sine or cosine rtio sine or cosine rtio sine, cosine, or tngent rtios Lw of Sines Lw of Sines Lw of Cosines Lw of Cosines Check Your Understnding = Step-b-Step Solutions begin on pge R0. Emple 1 4 Find. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth Emple Emple 5 7. SAILING Determine the length of the 47 bottom edge, or foot, of the sil. 55 ft 6 ft foot Emple 6 Solve ech tringle. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth Solve DEF if DE = 16, EF = 1.6, FD = Lesson 8-6 The Lw of Sines nd Lw of Cosines

61 Prctice nd Problem Solving Etr Prctice begins on pge 947. Emples 1 Find. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth ASTRONOMY Angelin is looking t the Big Dipper through telescope. From her view, the cup of the constelltion forms tringle tht hs mesurements shown on the digrm t the right. Use the Lw of Sines to determine distnce between A nd C in. Emples 4 Find. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth HIKING A group of friends who re cmping decide to go on hike. According to the mp shown t the right, wht is the mesure of the ngle between Tril 1 nd Tril? connected.mcgrw-hill.com 587

62 Emple 5 9. TORNADOES Find the width 0. TRAVEL A pilot flies 90 miles of the mouth of the torndo from Memphis, Tennessee, to shown below. Tupelo, Mississippi, to Huntsville, Albm, nd finll bck to Memphis. How fr is Memphis from mouth 44 Huntsville? 6 00 ft AR Memphis TN 90 mi 15.9º Huntsville Tupelo 1 mi MS AL Emple 6 Solve ech tringle. Round ngle mesures to the nerest degree nd side mesures to the nerest tenth m Solve JKL if JK =, KL = 56, LJ = Solve ABC if m B = 119, m C = 6, CA = Solve XYZ if XY = 190, YZ = 184, ZX = 75. B 4. GARDENING Crstl hs n orgnic vegetble grden. She wnts to dd nother tringulr section so tht she cn strt growing tomtoes. If the grden nd neighboring spce hve the dimensions shown, find the perimeter of the new grden. 0 ft ft 5º 18 ft 588 Lesson 8-6 The Lw of Sines nd Lw of Cosines

63 44. FIELD HOCKEY Alss nd Nri re pling field hocke. Alss is stnding 0 feet from one post of the gol nd 5 feet from the opposite post. Nri is stnding 45 feet from one post of the gol nd 8 feet from the other post. If the gol is 1 feet wide, which pler hs greter chnce to mke shot? Wht is the mesure of the pler s ngle? 45 ft 5 ft 8 ft 0 ft 1 ft 45. PROOF Justif ech sttement for the derivtion of the Lw of Sines. Given: CD is n ltitude of ABC. sin A Prove: _ = _ sin B b Proof: b h Sttements. sin A = h_, sin B = h_ b.? b. b sin A = h, sin B = h b.? c. b sin A = sin B c.? _ d. sin A = _ sin B b d.? Resons 46. PROOF Justif ech sttement for the derivtion of the Lw of Cosines. Given: h is n ltitude of ABC. c h b Prove: c = + b - b cos C Proof: - Sttements. c = ( - ) + h.? b. c = h b.? c. + h = b c.? d. c = - + b d.? e. cos C = _ e.? b f. b cos C = f.? g. c = - (b cos C) + b g.? h. c = + b - b cos C h.? Resons Find the perimeter of ech figure. Round to the nerest tenth connected.mcgrw-hill.com 589

64 51 MODELS Vito is working on model cstle. Find the length of the missing side (in inches) using the digrm t the right. 8 in in. C 5. COORDINATE GEOMETRY Find the mesure of the lrgest ngle in ABC with coordintes A(-, 6), B(4, ), nd C(-5, 1). Eplin our resoning. 5. MULTIPLE REPRESENTATIONS In this problem, ou will use trigonometr to find the re of tringle.. Geometric Drw n cute, sclene ABC including n ltitude of length h originting t verte A. b. Algebric Use trigonometr to represent h in terms of m B. c. Algebric W rite n eqution to find the re of ABC using trigonometr. d. Numericl If m B is 47, AB = 11.1, BC = 14.1, nd CA = 10.4, find the re of ABC. Round to the nerest tenth. e. Anlticl Write n eqution to find the re of ABC using trigonometr in terms of different ngle. H.O.T. Problems Use Higher-Order Thinking Skills 54. ERROR ANALYSIS Colleen nd Mike re plnning prt. Colleen wnts to sew tringulr decortions nd needs to know the perimeter of one of the tringles to bu enough trim. The tringles re isosceles with ngle mesurements of 64 t the bse nd side lengths of 5 inches. Colleen thinks the perimeter is 15.7 inches nd Mike thinks it is 15 inches. Is either of them correct? 5 in CHALLENGE Find the vlue of in the figure t the right. 56. REASONING Are there n sets of three mesures tht cnnot be used to solve tringle? Eplin. 57. OPEN ENDED Drw nd lbel tringle tht cn be solved:. using onl the Lw of Sines. b. using onl the Lw of Cosines WRITING IN MATH Suppose ou use clcultor nd determine the nswer to Eercise 0 to be bout 57 miles. Could this be correct? Eplin our resoning. Wht does this emple tell our bout the limittions of using clcultors to solve problems? 590 Lesson 8-6 The Lw of Sines nd Lw of Cosines

65 Stndrdized Test Prctice 59. For ABC, m A = 4, m B = 74, nd =, wht is the vlue of b? A 4. C.1 B.8 D ALGEBRA Which inequlit best describes the grph below? 61. SHORT RESPONSE Wht is the perimeter of the tringle shown below? Round to the nerest tenth SAT/ACT If sin = 0.6 nd AB = 1, wht is the re of ABC? SPI 10..8, SPI F - + H - + G + J + A 9.6 units D 4.6 units B 8.8 units E 4. units C 1. units Spirl Review 6. HIKING A hiker is on top of mountin 50 feet bove se level with 68 ngle of depression. She cn see her cmp from where she is stnding. How fr is her cmp from the top of the mountin? (Lesson 8-5) Use clcultor to find the mesure of J to the nerest degree. (Lesson 8-4) ft Determine whether the polgons re lws, sometimes, or never similr. Eplin our resoning. (Lesson 7-) 66. right tringle nd n isosceles tringle 67. n equilterl tringle nd sclene tringle Nme the missing coordintes of ech tringle. (Lesson 4-8) 68. (?, b) 69. (,?) 70. (?,?) (0, 0) (, 0) (0, 0) (?,?) (0, 0) (, 0) Skills Review Find the distnce between ech pir of points. Round to the nerest tenth. (Lesson 1-) 71. A(5, 1) nd C(-, -) 7. J(7, 11) nd K(-1, 5) 7. W(, 0) nd X(8, 6) connected.mcgrw-hill.com 591

66 Geometr Softwre Lb The Ambiguous Cse of the Lws of Sines In Lesson 8-6, 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 (AAS or ASA). You cn lso solve tringle using the Lw of Sines if ou know the mesures of two sides nd n ngle opposite one of the sides (SSA). When ou use SSA to solve tringle, nd the given ngle is cute, sometimes it is possible to find two different tringles. You cn use the Geometer s Sketchpd to eplore this cse, clled the mbiguous cse, of the Lw of Sines. Activit Step 1 Construct AB nd AC. Construct circle with center t B so tht it intersects AC t two points. Then, construct n rdius BD. Step Find the mesures of BD, AB, nd A. Step Use the rotte tool to move D so tht it lies on one of the intersection points of circle B nd AC. In ABD, find the mesures of ABD, BDA, nd AD. Step 4 Using the rotte tool, move D to the other intersection point of circle B nd AC. Step 5 Note the mesures of ABD, BDA, nd AD in ABD. = 1.8 cm =.40 cm m = 9.65 Steps 1 m = m = =.47 cm = 1.8 cm =.40 cm m = 9.65 m = m = = 0.69 cm Steps 4 5 Anlze the Results 1. Which mesures re the sme in both tringles?. Repet the ctivit using different mesures for A, BD, nd AB. How do the results compre to the erlier results? Mke Conjecture. Compre our results with those of our clssmtes. How do the results compre? 4. Wht would hve to be true bout circle B in order for there to be one unique solution? Test our conjecture b repeting the ctivit. 5. WRITING IN MATH Is it possible, given the mesures of BD, AB, nd A, to hve no solution? Test our conjecture nd eplin. 59 Etend 8-6 Geometr Softwre Lb: The Ambiguous Cse of the Lws of Sines

67 Vectors Then You used trigonometr to find side lengths nd ngle mesures of right tringles. (Lesson 8-4) Now 1Find mgnitudes nd directions of vectors. Add nd subtrct vectors. Wh? A pilot must correct for environmentl fctors such s wind to keep plne on course. The cn use vectors to tke hed or til winds into ccount. New Vocbulr vector mgnitude direction stndrd position component form resultnt Tennessee Curriculum Stndrds CLE Eplore vectors s numeric sstem, focusing on grphic representtions nd the properties of the opertion Add vectors grphicll nd lgebricll Multipl vector b sclr grphicll nd lgebricll. SPI Perform opertions on vectors in vrious representtions. Also ddresses SPI Describe Vectors A vector is quntit tht hs both mgnitude nd direction. The mgnitude of vector is the length of the vector from its initil point to its terminl point. The direction of vector is the ngle tht is formed with the positive -is or n other horizontl line. mgnitude of AB = AB direction of AB = Vectors cn be represented s directed line segments. The vector shown, which cn be clled AB or v, hs n initil point A nd terminl point B. When plced on the coordinte plne, vector is in stndrd position when it hs its initil point t the origin. Vector w is in stndrd position. The component form of vector is described in terms of ts horizontl chnge nd verticl chnge from its initil point to its terminl point. The component form of w is (, ). Emple 1 Write Vectors in Component Form w v (, ) Write the component form of CD. Use component nottion nd find the chnge in -vlues nd the chnge in -vlues. CD = ( - 1, - 1 ) Component form of vector = (4-0, 1 - (-1)) 1 = 0, = 4, 1 = -1, = 1 = (4, ) Simplif. (-1, ) (4, 1) (-, -1) (0, -1) GuidedPrctice 1. Write the component form of PQ. The Distnce Formul cn be used to find the mgnitude of vector. You cn use trigonometric rtios to find the direction of vector. connected.mcgrw-hill.com 59

68 Emple Find the Mgnitude nd Direction of Vector Find the mgnitude nd direction of RT for R(, 4) nd T(-, -). Stud Tip Alternte Method You cn lso use the Pthgoren Theorem to find the mgnitude of the vector. Step 1 Use the Distnce Formul to find the vector s mgnitude. RT = ( - 1 ) + ( - 1 ) Distnce Formul = (- - ) + (- - 4) 1 =, = -, 1 = 4, nd = - = 61 or bout 7.8 Simplif. Step Use trigonometr to find the vector s direction. Grph RT. Drw right tringle with RT s its hpotenuse. tn R = _ tn = _ opp dj ; length cnnot be negtive. = _- - 4 or _ m R = tn -1 6 _ 5 1 =, = -, 1 = 4, = - Definition of inverse tngent (-, -) (, 4) Stud Tip Direction Angles Vectors tht lie in the third or fourth qudrnts when plced in stndrd position will hve direction ngles greter thn Use clcultor. When repositioned so tht its initil point is t the origin, RT lies in the third qudrnt nd forms n ngle with the negtive -is equl to m R. The direction of RT is the ngle it mkes with the positive -is, which is m R or bout Therefore, RT hs mgnitude of bout 7.8 units nd direction of bout 0.. GuidedPrctice. Find the mgnitude nd direction for FG for F(-1, 5) nd G(, -). Vector Addition Three tpes of vectors re described below. Ke Concept Equl, Prllel, nd Opposite Vectors Equl Vectors Two vectors re equl if nd onl if the hve the sme mgnitude nd direction. Emple b = c Nonemple b e Prllel Vectors Two vectors re prllel if nd onl if the hve the sme or opposite direction. Emple d Nonemple d f Opposite Vectors Two vectors re opposites if the hve the sme mgnitude nd opposite directions. Emple b nd e Nonemple d nd f c b f d e 594 Lesson 8-7 Vectors

69 The sum of two vectors is clled the resultnt. The resultnt of two vectors cn be found geometricll b using two different methods. Stud Tip Tringle Method The tringle method m lso be clled the tip-to-til method. Ke Concept Vector Addition Prllelogrm Method Step 1 Plce both vectors t the sme initil point. Step Complete the prllelogrm. The resultnt is the digonl of the prllelogrm. Tringle Method Step 1 Plce the initil point of the second vector t the terminl point of the first. Step The resultnt connects the initil point of the first vector nd the terminl point of the second. + + Emple Vector Addition nd Subtrction Cop the vectors to find - b. Subtrcting vector is equivlent to dding its opposite. Method 1 Use the prllelogrm method. b Stud Tip Opposite Vectors A vector hs the sme mgnitude nd direction tht is 180 more thn its opposite vector. Step 1 Step Step Cop nd b with the sme initil point. Complete the prllelogrm. Drw the digonl of the prllelogrm from the initil point. - b - b - b - b Method Use the tringle method. Step 1 Step Step Cop. Plce the initil point of Drw the vector from the - b t the terminl point initil point of to the of. terminl point of b. -b - b - b Compre the resultnts from the prllelogrm method nd the tringle method. The re equl. GuidedPrctice A. Find + b. B. Find b -. connected.mcgrw-hill.com 595

70 Vectors cn lso be dded or subtrcted lgebricll b dding or subtrcting their horizontl nd verticl components. v + w = v + w ( 1, 1 ) (, ) ( 1 +, 1 + ) Rel-World Emple 4 Algebric Vectors EVENT PLANNING Cheerleders re using n ir luncher to project T-shirts into the stnds. The wind is blowing in the sme direction tht shirts re lunched. If shirt leves the luncher t 40 feet per second t n ngle of 45 nd the wind is blowing 7 feet per second prllel to the horizontl, wht is the resultnt velocit nd direction of the T-shirt? Step 1 Drw digrm to represent ech direction vector. T-shirt 40 ft/s 45 7 ft/s Wind Rel-World Link Lunchers re used t sporting events to project T-shirts nd other soft prizes into the stnds. The lunchers mke it possible for the prizes to trvel frther thn if the were hnd-thrown b incresing their speed nd chnging their ngles of elevtion. Step Write the direction vector for the T-shirt in component form. cos 45 = _ cos = _ dj nd sin = _ opp sin 45 = _ 40 hp hp cos 45 = Multipl ech side b sin 45 = 8. Use clcultor. 8. The component form for the T-shirt s vector is (8., 8.). Since the wind is blowing prllel to the horizontl, the vector will onl hve horizontl component. The vector describing the wind is (7, 0). Use the lgebric method to dd the vectors for the shirt nd the wind. (8., 8.) + (7, 0) = (8. + 7, ) or (5., 8.) Drw digrm. Let v represent the velocit of the shirt, nd let represent the ngle of direction. Determine the vector s mgnitude. v = + v (5.) + (8.) v 45. Pthgoren Theorem Substitute. Use clcultor. Use trigonometr to find the vector s direction. tn = _ opp tn = _ dj tn _ 8. Substitute. 5. tn-1 _ 8. or bout 8.7 Definition of inverse tngent 5. The resultnt velocit of the T-shirt is 45. feet per second t n ngle of 8.7 with the horizontl. v 5. ft/s 8. ft/s GuidedPrctice 4. Wht would be the resultnt mgnitude nd direction of the shirt if the wind is blowing in the opposite direction? 596 Lesson 8-7 Vectors