Nme PP Something found t sld r 4.7 Notes RIGHT TRINGLE hs extly one right ngle. To solve right tringle, you n use things like SOH-H-TO nd the Pythgoren Theorem. n OLIQUE TRINGLE hs no right ngles. To solve n olique tringle, you will hve to use things like the Lw of Sines or the Lw of osines. LW OF SINES: sin sin = sin = Plese note tht the Lw of Sines onsists of the following three formuls: sin sin sin sin sin sin ) = ) = 3) = Proof: (The se where ll 3 ngles re ute) The Lw of Sines n e used to solve the following types of tringles: Pge
) Solve E. Round to the nerest tenth, if neessry. 3 63 = = = = 4 E E = e = ) Two fire-lookout sttions re 0 miles prt, with Sttion diretly est of sttion. oth sttions spot fire on mountin to the north. The ering from Sttion to the fire is 50 est of north. The ering from sttion to the fire is 36 west of north. How fr, to the nerest tenth of mile, is the fire from Sttion? From Geometry, you know tht the mesures of two sides nd non-inluded ngle,, do not neessrily define unique tringle. Suh se is lled the se euse the given informtion my result in one tringle, two tringles, or no tringle t ll. Pge
THE MIGUOUS SE (SS) Find ll solutions for the given tringle, if possile. If no solution exists, write no solution. Round to the nerest tenth. 3) = 6, = 8, = Find ll solutions to the given eqution suh tht 0 < < 80. Round to the nerest tenth, if neessry. 4) sin = 5) sin = 0. 967 Pge 3
Find ll solutions for the given tringle, if possile. If no solution exists, write no solution. Round to the nerest tenth. 6) = 39, =, = 7 7) = 05, = 73, = 55 Pge 4
LW OF OSINES: = + os = + os = + os Proof: (The se where = + os ) The Lw of osines n e used to solve the following types of tringles: Tips for Solving SS Tringle. Use the Lw of osines to find the side opposite the given ngle.. Use the Lw of Sines to find the ngle opposite the shorter of the two given sides. This ngle is ute. 3. Find the third ngle y sutrting the sum of the two known ngles from 80. Tips for Solving SSS Tringle. Use the Lw of osines to find the ngle opposite the longest side.. Use the Lw of Sines to find either of the two remining ute ngles. 3. Find the third ngle y sutrting the sum of the two known ngles from 80. Pge 5
8) Solve. Round sides to the nerest tenth nd ngles to the nerest degree. = = = = = = 9) Solve EN if the rdius of irle is 5, the rdius of irle E is 4, nd the rdius of irle N is 3.5. Round to the nerest tenth. N E Pge 6
re of Tringle: The re of tringle equls one-hlf the produt of the lengths of ny two sides nd the (Given SS) sine of the ngle etween them. re = sin re = sin re = sin Proof: ( re = sin ) 0) Foots nd Dody wnted to know the re of tringulr plot of lnd for their soyen rop. lnd surveyor determined tht two of the sides hve lengths of 400 feet nd 600 feet nd tht the mesure of the ngle etween them is 46.3. ) Find the re of the plot of lnd. Round to the nerest squre foot. ) Foots nd Dody wnt to put fene round the perimeter of their field to keep the soyen retures out. If the verge ost of wooden fene is $ per liner foot, how muh will they hve to spend? 400 feet 600 feet 46.3 Pge 7
Heron of lexndri (0 D 70 D) developed formul whih n find the re of tringle given the lengths of its sides. The proof of Heron s Formul (lso known s Hogie s Formul, I men Hero s Formul) is rther time onsuming, so we will not write it out. He lso reted method for itertively omputing the squre root of numer. esides his mthemtil tlents, Heron lso hd gret del of engineering skill. In vrious temples his mehnil inventions produed effets tht seemed supernturl! Heron s Formul: This formul will lulte the re of tringle when ll three sides re known. re = s ( s )( s )( s ) s = ( + ) ( s = semiperimeter ) where + ) tringulr field hs sides of lengths 375 yrds, 480 yrds, nd 675 yrds. To the nerest whole numer, pproximte the numer of res in the field. (One re is equivlent to 4840 squre yrds.) 480 yrds 375 yrds 675 yrds ) pproximte the re of if = 7 m, = 4 m, nd = 5. Round to the nerest tenth. Lw of Sines Lw of osines Heron's re Formul SS re Formul S SS SSS SS S SS (0,, or Solutions) SSS Pge 8