Use a protractor to find the measures of the angles of each triangle
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1 Wht You ll Lern To ue the propertie of - -0 tringle To ue the propertie of tringle... An Wh 8- Specil Right Tringle To fin the itnce from home plte to econ e on oftll imon, in Emple A B C D E A B C D E A B C D E A B C D E A B C D E B C D E Uing Tringle Cliforni Content Stnr GEOM.0 Stuent ue the Pthgoren theorem to etermine itnce n fin miing length of ie of right tringle. Develop GEOM 0.0 Stuent know n re le to ue ngle n ie reltionhip in prolem with pecil right tringle, uch 0, 60, n 0 tringle n,, n 0 tringle. Introuce, Develop The cute ngle of n iocele right tringle re oth 8 ngle. Another nme for n iocele right tringle i tringle. If ech leg h length n the hpotenue h length, ou cn olve for in term of. + = Ue the Pthgoren Theorem. = Simplif.! = Tke the qure root of ech ie. You hve jut prove the following theorem. Ke Concept Theorem Tringle Theorem Tet-Tking Tip If ou forget the formul for - -0 tringle, ue the Pthgoren Theorem. The tringle i iocele, o the leg hve the me length. In tringle, oth leg re congruent n the length of the hpotenue i! time the length of leg. hpotenue =!? leg Fining the Length of the Hpotenue Fin the vlue of ech vrile... h h =!? hpotenue!? leg =!?! h =! Simplif. = Fin the length of the hpotenue of tringle with leg of length!. "6 Leon -6 Ue protrctor to fin the meure of the ngle of ech tringle.... Specil Nee L For Emple, hve tuent check the nwer cutting out 60-mm 60-mm qure. The fol it long it igonl, n meure the length of the igonl to the neret millimeter.,, 0 0, 60, 0 Leon 8- Specil Right Tringle,, 0 Below Level L In the igrm for Theorem 8-6, contruct 0 ngle jcent to the 0 ngle, uing leg one ie. Eten the e o tht it interect the new ie. Dicu wh thi form n equilterl tringle. 8-. Pln Cliforni Content Stnr GEOM.0 Stuent ue the Pthgoren theorem to etermine itnce n fin miing length of ie of right tringle. GEOM 0.0 Stuent know n re le to ue ngle n ie reltionhip in prolem with pecil right tringle, uch,, n 0 tringle n,, n 0 tringle. Cliforni Mth Bckgroun The rtio of the length of n two ie of right tringle i function of either cute ngle. Thi cn e prove uing imilrit theorem n i the i for the i trigonometric function. The fie ie-length rtio of - -0 n tringle, eil foun ppling the Pthgoren Theorem, provie enchmrk vlue for the trigonometric function ine, coine, n tngent of 0,, n 60 ngle. More Mth Bckgroun: p. C Leon Plnning n Reource See p. E for lit of the reource tht upport thi leon. CD, Online, or Trnprencie Bell Ringer Prctice Check Skill You ll Nee Ue tuent pge, trnprenc, or PowerPoint. Cliforni Stnr Dil Review Ue trnprenc 8.
2 . Tech Guie Intruction Technolog Tip Point out tht uing mentl mth i much fter thn uing clcultor for prt. The clcultor nwer lo woul e inect, where quring the qure root of numer i lw ect. Teching Tip Hve everl tuent eplin lou to the cl how to rtionlize enomintor. CD, Online, or Trnprencie Aitionl Emple Fin the length of the hpotenue of - -0 tringle with leg of length "6. 0" Fin the length of leg of - -0 tringle with hpotenue of length. " The itnce from one corner to the oppoite corner of qure plgroun i 6 ft. To the neret foot, how long i ech ie of the plgroun? 68 ft Differentiting Intruction Sugget tht tuent itinguih etween the - -0 n the Tringle Theorem uing the rtio igrm elow Error Prevention Whenever the length of hpotenue or longer leg of tringle i given, encourge tuent to fin the length of the horter leg firt. Fining the Length of Leg Multiple Choice Wht i the vlue of?! 6 6! 6 =!? hpotenue!? leg = 6! Divie ech ie!. = 6! 6!? =!! Multipl form of. =! Simplif. The correct nwer i B. Fin the length of leg of tringle with hpotenue of length 0. " When ou ppl the Tringle Theorem to rel-life emple, ou cn ue clcultor to evlute qure root. Rel-Worl Connection Softll A high chool oftll imon i qure. The itnce from e to e i 60 ft. To the neret foot, how fr oe ctcher throw the ll from home plte to econ e? The itnce from home plte to econ 60 ft e i the length of the hpotenue of tringle. = 60! hpotenue!? leg = 8.88 Uing Tringle Ue clcultor. 6 On high chool oftll imon, the ctcher throw the ll out 8 ft from home plte to econ e. A qure gren h ie 00 ft long. You wnt to uil rick pth long igonl of the qure. How long will the pth e? Roun our nwer to the neret foot. ft Another tpe of pecil right tringle i tringle. Ke Concept Theorem Tringle Theorem 6 Chpter 8 Right Tringle n Trigonometr In tringle, the length of the hpotenue i twice the length of the horter leg. The length of the longer leg i! time the length of the horter leg. hpotenue =? horter leg longer leg =!? horter leg Avnce Lerner L After tuent lern n ppl Theorem 8-, hve them write formul for the re of n iocele right tringle whoe hpotenue h length. Englih Lerner EL Ak tuent to complete ech ttement: The hortet ie of tringle i lw oppoite the mllet ngle. In tringle, the hortet ie i lw oppoite the 0 ngle. 6
3 Proof To prove Theorem 8-6, rw tringle uing n equilterl tringle. Proof of Theorem 8-6 For #WXY in equilterl #WXZ, WY i the perpeniculr iector of XZ. Thu, XY = XZ = XW, or XW = XY =. Alo, XY + YW = XW Ue the Pthgoren Theorem. + YW = () Sutitute for XY n for XW. YW = - Sutrct from ech ie. YW = Simplif. YW =! Fin the qure root of ech ie. The Tringle Theorem, like the - -0 Tringle Theorem, let ou fin two ie of right tringle when ou know the length of the thir ie. Uing the Length of One Sie Alger Fin the vlue of ech vrile. =! longer leg Á? horter leg =!!? = Solve for. "! f = hpotenue? horter leg! 0!! f =? = Sutitute for. Fin the vlue of ech vrile. =, = " Rel-Worl Connection Ro Sign The mooe wrning ign t the left i n equilterl tringle. The height of the ign i m. Fin the length of ech ie of the ign to the neret tenth of meter. The tringle i equilterl, o the ltitue ivie the tringle into two tringle hown in the igrm. The ltitue lo iect the e, o the horter leg of ech tringle i. =!Q R longer leg Á? horter leg!. Solve for. Simplif. Ue clcultor. Ech ie of the ign i out. m long. X Y Z 8 f m W Mth Tip Stuent cn ue the Pthgoren Theorem to check their work. The longer leg of tringle h length 8. Fin the length of the horter leg n the hpotenue. horter leg: 6 "; hpotenue: " A gren hpe like rhomu h perimeter of 00 ft n 60 ngle. Fin the perpeniculr height etween two e..7 ft Reource Dil Notetking Guie 8- L Dil Notetking Guie 8- Apte Verion L Cloure In qurilterl ABCD, AD = DC n AC = 0. Fin the re of ABCD. Leve our nwer in implet ricl form. B CD, Online, or Trnprencie Aitionl Emple 0 A 00 ± 0" 7 C D If the ie of the ign re m long, wht i the height? out 0. m Leon 8- Specil Right Tringle 7 7
4 . Prctice Aignment Guie Guie Prolem Solving Enrichment Reteching Apte Prctice Prctice Peron Euction, Inc., pulihing Peron Prentice Hll. All right reerve. AB -8,,, 6 AB -0, - C Chllenge 7-8 Multiple Choice Prctice, 0 Mie Review - Homework Quick Check To check tuent unertning of ke kill n concept, go over Eercie,, 8,,. Eercie 7 Ak volunteer to ring choptick to cl n emontrte how to ue them. Eercie 7 Stuent houl firt fin n ie length tht cn e erive uing given ie. After the firt length i foun, the other length often fll into plce. Eercie 0 Ech of thee eercie require contructing n ltitue to form rectngle. Nme Cl Dte Prctice 8- Fin the vlue of ech vrile. Leve our nwer in implet ricl form z 0. Fin the length to the neret centimeter of the igonl of qure 0 cm on ie.. The hpotenue of n iocele right tringle i 8. in. Fin the length of ie to the neret tenth of n inch.. In tringle, the horter leg i 6 ft long. Fin the length to the neret tenth of foot of the other two ie.. Ech ie of rhomu i in. long. Two of the ie form 60 ngle. Fin the re of the rhomu. Roun our nwer to the neret qure inch. Alger Fin the vlue of ech vrile. Leve our nwer in implet ricl form w 8 8 z 8 r 6 q Specil Right Tringle c p L L L L L A B EXERCISES Stnr Prctice Prctice Emple GO for Help Emple (pge ) Emple n (pge 6) Eercie 7 Emple n (pge 7) 7. 7; ; c 7; 7" 8. 6; 6 "; c "; 6. 0 "; "; c ; Appl Your Skill Fin the vlue of ech vrile. If our nwer i not n integer, leve it in implet ricl form. "; 60". 8; 8".. 8 For more eercie, ee Etr Skill, Wor Prolem, n Proof Prctice. 8 Chpter 8 Right Tringle n Trigonometr. ; " GEOM.0, 0.0 "0 7. Dinnerwre Deign You re eigning innerwre. Wht i the length of ie of the mllet qure plte on which 0-cm choptick cn fit long igonl without n overhng? Roun our nwer to the neret tenth of centimeter.. cm 8. Helicopter The four le of helicopter meet t right ngle n re ll the me length. The itnce etween the tip of two jcent le i 6 ft. How long i ech le? Roun our nwer to the neret tenth.. ft Alger Fin the vlue of ech vrile. If our nwer i not n integer, leve it in implet ricl form ; 0" "; ; "... ; ; " ; 8. Architecture An ecltor lift people to the econ floor, ft ove the firt floor. The ecltor rie t 0 ngle. How fr oe peron trvel from the ft ottom to the top of the ecltor? 0 ft 6. Cit Plnning Jefferon Prk it on one qure cit lock 00 ft on ech ie. Siewlk join oppoite corner. Aout how long i ech igonl iewlk? ft 7. See ove left. Alger Fin the vlue of ech vrile. Leve our nwer in implet ricl form c c c 60 8
5 Viit: PHSchool.com We Coe: ce-080. Rik; Snr mrke the horter leg oppoite the 608 ngle. Eercie C 7. " unit. " unit c. " unit Chllenge Multiple Choice Prctice GEOM ; ; 7 ; 6". Error Anli Snr rew the tringle t the right. Rik i tht the length couln t e correct. With which tuent o ou gree? Eplin our nwer Open-Ene Write rel-life prolem tht 0 ou cn olve uing tringle with -ft hpotenue. Show our olution. See mrgin.. Frming A conveor elt crrie le of h from the groun to the rn loft GPS ft ove the groun. The elt mke 608 ngle with the groun.. How fr oe le of h trvel from one en of the conveor elt to the other? Roun our nwer to the neret foot. 8 ft. The conveor elt move t 00 ft/min. How long oe it tke for le of h to go from the groun to the rn loft? 0.8 min 6. Houe Repir After hev win mge frmhoue, worker plce 6-m rce gint it ie t 8 ngle. Then, t the me pot on the groun, the plce econ, longer rce to mke 08 ngle with the ie of the houe. 6 m. How long i the longer rce? Roun our nwer to the neret tenth of meter. 8. m. Aout how much higher on the houe oe the longer rce rech thn the horter rce?. m 7. Geometr in Dimenion Fin the length, in implet ricl form, of the igonl of cue with ie of the given length. See left.. unit. unit c. unit 8. Contruction Contruct tringle given egment tht i. the horter leg.. the hpotenue. c. the longer leg. See ck of ook. For Cliforni Stnr n CAHSEE Tutoril, viit PHSchool.com. We Coe: cq-0. A ler ret gint pech tree t 60 ngle in the picture t the right. If the ler i feet long, how long i, the itnce from the ler to the e of the tree? A 7. ft 7." ft 0 ft 0 ft ft. Ae & Retech CD, Online, or Trnprencie Leon Quiz Ue kabc for Eercie. A C 8. If m&a =, fin AC n AB. AC 8; AB 8". If m&a = 0, fin AC n AB. AC 8 "; AB 6. If m&a = 60, fin AC n AB. AC 6 "; AB ". Fin the ie length of - -0 tringle with -cm hpotenue. " N.8 cm. Two -mm ie of tringle form 0 ngle. Fin the length of the thir ie. " N 0.8 mm Alterntive Aement Hve tuent ue comp n trightege to contruct lrge equilterl tringle with one ltitue. Then hve them eplin how the three ie of one of the right tringle re relte. Cliforni Reource Stnr Mter Cliforni Dil Review Trnprencie Cliforni Review n Prctice Workook B nline Viit: PHSchool.com, We Coe: c-080. Anwer m vr. Smple: A rmp up to oor i ft long. It h n incline of 0. How high off the groun i the oor? ol.: 6 ft
6 Ue thi Checkpoint Quiz to check tuent unertning of the kill n concept of Leon 8- through 8-. Reource Gr & Go Checkpoint Quiz Mie Review GO for Help CAHSEE 7AF. Leon 8-0. Which epreion i equivlent to 8 c 0? D For Eercie n, leve our nwer in implet ricl form.. A right tringle h 6-in. hpotenue n -in. leg. Fin the length of the other leg. " in.. An iocele tringle h 0-cm leg n 6-cm e. Fin the length of the ltitue to the e. " cm 7 c 8 7 Leon 6- Determine whether ech qurilterl mut e prllelogrm. If not, provie counteremple.. The igonl re congruent n perpeniculr to ech other. See mrgin.. no Leon - 6. e; AAS Thm.. Two oppoite ngle re right ngle n two oppoite ie re cm long. e. One pir of ie i congruent n the other pir of ie i prllel. no; n iocele trpezoi Cn ou conclue tht #TRY O#ANG from the given conition? If o, nme the potulte or theorem tht jutifie our concluion. no 6. &A > &T, &Y > &G, TR > AN 7. &T > &A, &R > &N, &Y > &G 8. &R > &N, TR > AN, TY > AG no. &G > &Y, &N > &R, RY > NG e; ASA Pot. Leon 8- through 8- Alger Fin the vlue of ech vrile. Leve our nwer in implet ricl form... = 0;. = 0! 0 =!; =.. 6. =!; = 6 =!; = 0! The length of the ie of tringle re given. Clif ech tringle cute, otue, or right. 7. 7, 8, cute 8., 6, right. 0,, 6 otue 0. A qure h 0-cm igonl. How long i ech ie of the qure? Roun our nwer to the neret tenth of centimeter. 8. cm 0 Chpter 8 Right Tringle n Trigonometr 0
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9.5 Strt Thinking In Lesson 9.4, we discussed the tngent rtio which involves the two legs of right tringle. In this lesson, we will discuss the sine nd cosine rtios, which re trigonometric rtios for cute
See margin. See margin. 5 } = 0.8 }3 5 } = tan A } 4 6. sin B 3 } }3 5 } = cos B } 4 8. tan B 5 } = 0.8 }3 4 } = 0.
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