Section 2.1 Special Right Triangles
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1 Se..1 Speil Rigt Tringles 49 Te --90 Tringle Setion.1 Speil Rigt Tringles Te --90 tringle (or just ) is so nme euse of its ngle mesures. Te lengts of te sies, toug, ve very speifi pttern to tem wi we will fin useful in te stuy of trigonometry. Equilterl = Equingulr n equilterl (equl-sie-mesure) tringle is lso known s n equingulr (equl-ngle-mesure) tringle euse ll tree ngles ve te sme mesure:. Wy?* To rete tringle from n equilterl tringle, follow tese steps on : 1. In, mrk,, n s.. In, rw n ngle isetor from te top vertex,, to, n lel te point of intersetion D.. euse of symmetry in te equilterl tringle, D is ot n ltitue n segment isetor: ) D n D re rigt ngles, n ) D isets. 4. Ientify te lengts of D n D. 5. t rigt, rw D n lel te sie n ngle mesures; lel D wit. *euse no one sie is longer tn noter, none of te opposite ngles n e lrger tn noter, so tey must ll e te sme mesure: 180 = Se..1 Speil Rigt Tringles 49 Roert H. Prior, 018 opying is proiite.
2 50 Se..1 Speil Rigt Tringles Wt you soul ve isovere on te previous pge is tt we n estlis te tringle y iseting n equilterl tringle. We n o tis on our own if we ever forget te reltionsips of te sies of tringle. D D Te Rtio of te Sorter Leg to te Hypotenuse Te se,, of te equilterl tringle s te sme lengt,, s te ypotenuse of e of te tringles. euse te se of te equilterl tringle ws isete, te new ses of e of te two rigt tringles is lf of te originl se, 1, or. In tringle, te sortest sie wi is lf of te ypotenuse is lwys opposite te smllest ngle, te ngle. D Exmple 1: In te tringle t rigt, given te vlue of, fin. ) = 8 ) = 5 ) = 4 nswer: is lf of. ) = 8 = 4 ) = 5 ) = 4 = Exmple : In te tringle t rigt, given te vlue of, fin. ) = 5 ) = 7 ) = 6 nswer: is twie : = ) = 5 = 10 ) = 7 = 7 ) = 6 = 1 Se..1 Speil Rigt Tringles 50 Roert H. Prior, 018 opying is proiite.
3 Se..1 Speil Rigt Tringles 51 Te Rtio of te Longer Leg to te Sorter Leg Te longer leg of te tringle is opposite te ngle. It is te seon longest of te tree sies, so it must e less tn te ypotenuse,, n more tn te sortest leg,. We n fin te lengt of tis longer leg using te Pytgoren Teorem. onsier, for exmple, tringle wit ypotenuse of lengt ines. Tis mens te sorter leg is 1 in long, n te oter leg must ve lengt, x, etween 1 in n ines. (Te sme is true if we use feet, meters or miles inste of ines.) Tis mens tt 1 < x <. Let s put te Pytgoren Teorem to work: x 1 + x = x = seems resonle euse: 1 + x = 4 1 < < 4 1 x = 1 < < 4 x = ± 1 < < x = only y te wy, 1.7 From tis exmple, we see tt te longer leg is times s long s te sorter leg. Exmple : In te tringle t rigt, given te vlue of, fin. ) = 5 ) = 7 ) = 6 nswer: Multiply te sorter leg y. = ) = 5 ) = 7 or 7 ) = 6 = 6 9 = 6 = 18 Se..1 Speil Rigt Tringles 51 Roert H. Prior, 018 opying is proiite.
4 5 Se..1 Speil Rigt Tringles Exmple 4: In te tringle t rigt, given te vlue of, fin. Rtionlize te enomintor if neessry. ) = 4 ) = 6 ) = nswer: Divie te longer leg y ; = ) = 4 = 4 ) = 6 = 6 = ) = = Fining ll of te Sies of --90 Tringle In tringle, if you know one sie ten we n use te reltionsips esrie in tis setion to fin te oter two sies. Te reltionsips inlue: ) ftor of etween te ypotenuse n te sortest leg, n ) ftor of etween te two legs. Note: Wen fining te sie lengt of tringle, o not go iretly etween te longer leg n te ypotenuse; lwys use te sortest leg s go-etween; Strting t te ypotenuse: Strting t te longer leg: x x Se..1 Speil Rigt Tringles 5 Roert H. Prior, 018 opying is proiite.
5 Se..1 Speil Rigt Tringles 5 Exmple 5: In te tringle t rigt, given te vlue of, fin n. ) = 10 ) = 7 nswer: Fin te sorter leg,, first; ten fin. ) = 10 = 5 ) = 7 = 5 = 5 = 7 = 7 Exmple 6: In te tringle t rigt, given te vlue of, fin n. ) = 4 ) = 6 nswer: Fin te sorter leg,, first; ten fin. ) = 4 = 4 ) = 6 = 6 = = 4 = 8 = = 4 Te --90 Tringle Just like te --90 tringle, te --90 is very importnt in te stuy of trigonometry. n, no mtter te size, every tringle is similr* to every oter tringle. Te tringle is not only rigt tringle, it is n isoseles tringle. euse te tringle is isoseles, te legs re ongruent to e oter. Tis mens tt knowing te lengt of one of te legs utomtilly tells you te lengt of te oter leg. Te ypotenuse n e foun y pplying te Pytgoren Teorem. *Similr tringles ve te sme ngle mesures n te sme proportion etween onseutive sies. In oter wors, no mtter te size of tringle, te proportion etween ny pir of sies is te sme witin e tringle. Se..1 Speil Rigt Tringles 5 Roert H. Prior, 018 opying is proiite.
6 54 Se..1 Speil Rigt Tringles Te Rtio of te Hypotenuse to Leg To emonstrte te reltionsip etween te lengts of te ypotenuse n one of te legs, let s ritrrily oose to ve te lengt of one of te legs e ines. Of ourse, te oter leg is lso ines, so let s fin te ypotenuse. + = = 18 = ± 18 = = Tis suggests tt te ypotenuse is times te lengt of leg, eiter leg. Tis is true for every tringle. To emonstrte tis onsisteny, let s oose te lengt of e leg to e mu more ritrry n let its vlue just e. Tis les to te following: + = = ± = = = or gin, te ypotenuse is times s long s eiter leg. Tis is onsistent wit wt we foun efore. Se..1 Speil Rigt Tringles 54 Roert H. Prior, 018 opying is proiite.
7 Se..1 Speil Rigt Tringles 55 Fining ll of te Sies of --90 Tringle In tringle, if you know one sie ten we n use te reltionsips esrie ove to fin te oter two sies. Te reltionsips inlue: ) Te two legs re ongruent, n x ) ftor of etween te ypotenuse n eiter leg. Exmple 7: In te tringle t rigt, given te vlue of, fin n. ) = 4 ) = 5 nswer: Fin te oter leg, ; ten fin. ) = 4 ) = 5 = 4 = 4 = 5 = 5 = 10 Exmple 8: In te tringle t rigt, given te vlue of, fin n. ) = ) = 8 nswer: Fin eiter leg, let s sy, first; ten fin te oter leg,. ) = = ) = 8 = 8 = 4 = = 4 Se..1 Speil Rigt Tringles 55 Roert H. Prior, 018 opying is proiite.
8 56 Se..1 Speil Rigt Tringles Seprting Tringles Sometimes igrm will inlue more tn one tringle, s emonstrte in Exmple 9. One ppro to fining te sies of te tringle to re-rw te igrm s two seprte tringles. Wen oing so, it is importnt to lel te tringles orretly. (We sw te seprtion of tringles t te eginning of te setion wen we split prt te two tringles.) Exmple 9: Given te lengt of one sie, fin te oter four sie mesures in tese tringles. Simplify ompletely. y x = 8 Proeure: First seprte te two tringles n orient tem in fmilir wy, s sown elow. Lel e tringles oring to te originl igrm. Note: Te ypotenuse of te tringle is lele x, n te ypotenuse of te tringle is lele m. x m p m x m y p euse we re given te vlue of x, we must strt in te tringle n fin ot y n m, in tt orer: y = 1 x n m = y One we know te vlue of m, we n fin te vlues of p n. = m n p =. nswer: y = 1 8 = 4 so m = 4... n = 4 = 4 6 = 6 so p = 6 Se..1 Speil Rigt Tringles 56 Roert H. Prior, 018 opying is proiite.
9 Se..1 Speil Rigt Tringles 57 Setion.1 Fous Exerises For #1-1, e tringle is lele seprtely, so oul e te smller ngle or te lrger ngle; in e se, is te rigt ngle. Given te lengt of, fin te lengts of n. 1. = 9. = 7. = 4 4. = 5 Given te lengt of, fin te lengts of n. 5. = 1 6. = 7. = = Se..1 Speil Rigt Tringles 57 Roert H. Prior, 018 opying is proiite.
10 58 Se..1 Speil Rigt Tringles Given te lengt of, fin te lengts of n. 9. = = = 1. = 5 For #1-0, in e tringle, is te rigt ngle. Given te lengt of, fin te lengts of n. 1. = = = = 7 6 Se..1 Speil Rigt Tringles 58 Roert H. Prior, 018 opying is proiite.
11 Se..1 Speil Rigt Tringles 59 Given te lengt of, fin te lengts of n. 17. = = = = 5 Given one of te vlues of, m, p, x, n y, fin te oter four vlues sown in te igrm. Simplify ompletely. Use te igrm t rigt for #1-4. p 1. m = 8 x y m. y = 6. p = 0 4. = 9 Se..1 Speil Rigt Tringles 59 Roert H. Prior, 018 opying is proiite.
12 60 Se..1 Speil Rigt Tringles Use te igrm t rigt for #5-8. m 5. m = y x p 6. p = y = 6 8. x = 15 Use te igrm t rigt for # p = 6 x y m p 0. y = 1. m = 1. x = 1 Se..1 Speil Rigt Tringles 60 Roert H. Prior, 018 opying is proiite.
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