Pge 1 of. The onverse of the Pythgoren Theorem Gol Use the onverse of Pythgoren Theorem. Use side lengths to lssify tringles. Key Words onverse p. 13 grdener n use the onverse of the Pythgoren Theorem to mke sure tht the orners of grden ed form right ngles. In the photogrph, tringle with side lengths 3 feet, feet, nd feet ensures tht the ngle t one orner is right ngle. 3 THEOREM.8 The onverse of the Pythgoren Theorem Words If the squre of the length of the longest side of tringle is equl to the sum of the squres of the lengths of the other two sides, then the tringle is right tringle. Symols If 2 2 2, then T is right tringle. Student Help LOOK K For the definition of onverse, see p. 13. EXMPLE 1 Verify Right Tringle Is T right tringle? 1 Let represent the length of the longest side of the tringle. hek to see whether the side lengths stisfy the eqution 2 2 2. 2 2 2 ompre 2 with 2 2. 20 2 2 1 2 Sustitute 20 for, for, nd 1 for. 00 1 2 Multiply. 00 00 Simplify. NSWER It is true tht 2 2 2. So, T is right tringle. 20 200 hpter Tringle Reltionships
Pge 2 of lssifying Tringles You n determine whether tringle is ute, right, or otuse y its side lengths. LSSIFYING TRINGLES In T with longest side : Student Help If 2 2 2, then T is ute. STUDY TIP This is the onverse of the Pythgoren Theorem. If 2 2 2, then T is right. If 2 2 2, then T is otuse. Student Help STUDY TIP 3.9, so use 3 s the vlue of, the longest side length of the tringle. EXMPLE Show tht the tringle is n ute tringle. 2 ute Tringles ompre the side lengths. 2 2 2 ompre 2 with 2 2. ( 3 ) 2 2 2 Sustitute 3 for, for, nd for. 3 1 2 Multiply. 3 1 Simplify. 3 NSWER euse 2 2 2, the tringle is ute. EXMPLE 3 Otuse Tringles Show tht the tringle is n otuse tringle. 8 ompre the side lengths. 1 2 2 2 ompre 2 with 2 2. (1) 2 8 2 2 Sustitute 1 for, 8 for, nd for. 22 1 Multiply. 22 208 Simplify. NSWER euse 2 2 2, the tringle is otuse.. The onverse of the Pythgoren Theorem 201
Pge 3 of IStudent Help I L S S Z O N E. O M MORE EXMPLES More exmples t lsszone.om EXMPLE lssify the tringle s ute, right, or otuse. lssify Tringles ompre the squre of the length of the longest side with the sum of the squres of the lengths of the two shorter sides. 2 2 2 ompre 2 with 2 2. 8 8 2 2 2 Sustitute 8 for, for, nd for. 2 3 Multiply. 1 Simplify. NSWER euse 2 2 2, the tringle is otuse. EXMPLE lssify the tringle with the given side lengths s ute, right, or otuse..,,., 3, 3 lssify Tringles. 2 2 2. 2 2 2 2 2 2 3 2 2 3 2 9 1 3 139 1 2 9 2 139 139 The tringle is ute. The tringle is right. lssify Tringles lssify the tringle s ute, right, or otuse. Explin. 1. 2. 3. 1 8 1 2 Use the side lengths to lssify the tringle s ute, right, or otuse.., 2, 2., 2, 2., 2, 2 202 hpter Tringle Reltionships
Pge of. Exerises Guided Prtie Voulry hek 1. Write the onverse of the Pythgoren Theorem in your own words. Skill hek Determine whether the tringle is ute, right, or otuse. 2. 3.. 1 1 9 Mth the side lengths of tringle with the est desription.. 2, 10, 11. right. 8,,. ute.,,. otuse 8., 8, 10 D. equingulr Prtie nd pplitions Extr Prtie See p. 82. Verifying Right Tringles Show tht the tringle is right tringle. 9. 20 10. 11. 2 1 10 1 1 2 2 Verifying ute Tringles Show tht the tringle is n ute tringle.. 13. 1. 21 11 23 29 3 Homework Help Exmple 1: Exs. 9 11, 2 Exmple 2: Exs. 1 Exmple 3: Exs. 1 1 Exmple : Exs. 18 23, 3 38 Exmple : Exs. 2 3 Verifying Otuse Tringles Show tht the tringle is n otuse tringle. 1. 1. 1. 2 8 13 22 1. The onverse of the Pythgoren Theorem 203
Pge of lssifying Tringles lssify the tringle s ute, right, or otuse. 18. 20 19. 21 20. 1 9 23 39 3 1 History ERLY MTHEMTIS This photogrph shows prt of ylonin ly tlet mde round 30.. The tlet ontins tle of numers. 21. 2 22. 23. 2 20 18 1 2. Erly Mthemtis The ylonin tlet shown t the left ontins severl sets of tringle side lengths, suggesting tht the ylonins my hve een wre of the reltionships mong the side lengths of right tringles. The side lengths in the tle elow show severl sets of numers from the tlet. Use lultor to verify tht eh set of side lengths stisfies the Pythgoren Theorem. 8 0 119 19,800,01,9 13,00,09 18,1 1 1 10 lssifying Tringles lssify the tringle with the given side lengths s ute, right, or otuse. 2. 20, 99, 101 2. 21, 28, 3 2. 2, 10, 1 28., 10, 11 29.,, 9 30. 1 3,, 31. 8, 9, 32. 10, 11, 1 33.,, 3. 1, 1, 1 3. 10, 9, 0 3.,,. ir Trvel In Exerises 3 nd 38, use the mp elow. 3. Use the distnes given on the mp to tell whether the tringle formed y the three ities is right tringle. 38. ininnti is diretly west of Wshington, D.. Is Tllhssee diretly south of ininnti? Explin your nswer. ininnti 99 mi Tllhssee 03 mi 1 mi Wshington, D.. W N S E 20 hpter Tringle Reltionships
Pge of You e the Judge 39. lssmte tells you if you find three side lengths tht form right tringle nd doule eh of them, the sides will form n otuse tringle. Is your lssmte orret? Explin. Stndrdized Test Prtie hllenge Grph points P, Q, nd R. onnet the points to form TPQR. Deide whether TPQR is ute, right, or otuse. 0. P( 3, ), Q(, 0), R(, 2) 1. P( 1, 2), Q(, 1), R(0, 1) 2. Multi-Step Prolem doule ply ours in sell when two outs re mde on single ply. In the digrm shown, the ll is hit to the plyer t point. doule ply is mde when the plyer t point throws the ll to the plyer t point who in turn throws it to the plyer t point. 2 ft 90 ft. Use the digrm to determine wht kind of tringle is formed y points,, nd.. Wht kind of tringle is formed y points,, nd if the distne etween points nd is 99 feet?. ritil Thinking Find vlues for nd tht would mke T in the digrm right tringle if 90 feet. Mixed Review Finding Mesures Find the vlue of x. (Lesson.3) 3... 8 3x x x 2x lger Skills Multiplying Frtions Multiply. Write the nswer s frtion or mixed numer in simplest form. (Skills Review, p. 9). 1 2. 3 8 3 8. 3 11 1 1 9. 3 9 0. 3 1. 8 1 3 2. 1 1 3 9 3. 1 2 3. The onverse of the Pythgoren Theorem 20