Essential Question How can you prove the Pythagorean Theorem?
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1 9.1 Te Pytgoren Teorem Essentil Question How n you prove te Pytgoren Teorem? Proving te Pytgoren Teorem witout Words Work wit prtner.. Drw nd ut out rigt tringle wit legs nd, nd ypotenuse.. Mke tree opies of your rigt tringle. rrnge ll four tringles to form lrge squre, s sown.. Find te re of te lrge squre in terms of,, nd y summing te res of te tringles nd te smll squre. d. opy te lrge squre. Divide it into two smller squres nd two eqully-sized retngles, s sown. e. Find te re of te lrge squre in terms of nd y summing te res of te retngles nd te smller squres. f. ompre your nswers to prts () nd (e). Eplin ow tis proves te Pytgoren Teorem. Proving te Pytgoren Teorem Work wit prtner.. Drw rigt tringle wit legs nd, nd ypotenuse, s sown. Drw te ltitude from to B. Lel te lengts, s sown. RESONING BSTRTLY To e profiient in mt, you need to know nd fleily use different properties of opertions nd ojets. d D. Eplin wy B, D, nd BD re similr. d B. Write two-olumn proof using te similr tringles in prt () to prove tt = 2. ommunite Your nswer 3. How n you prove te Pytgoren Teorem? 4. Use te Internet or some oter resoure to find wy to prove te Pytgoren Teorem tt is different from Eplortions 1 nd 2. Setion 9.1 Te Pytgoren Teorem 463
2 9.1 Lesson Wt You Will Lern ore Voulry Pytgoren triple, p. 464 Previous rigt tringle legs of rigt tringle ypotenuse Use te Pytgoren Teorem. Use te onverse of te Pytgoren Teorem. lssify tringles. Using te Pytgoren Teorem One of te most fmous teorems in mtemtis is te Pytgoren Teorem, nmed for te nient Greek mtemtiin Pytgors. Tis teorem desries te reltionsip etween te side lengts of rigt tringle. Teorem Teorem 9.1 Pytgoren Teorem In rigt tringle, te squre of te lengt of te ypotenuse is equl to te sum of te squres of te lengts of te legs. Proof Eplortions 1 nd 2, p. 463; E. 39, p = STUDY TIP You my find it elpful to memorize te si Pytgoren triples, sown in old, for stndrdized tests. Pytgoren triple is set of tree positive integers,, nd tt stisfy te eqution 2 = ore onept ommon Pytgoren Triples nd Some of Teir Multiples 3, 4, 5 6, 8, 10 9, 12, 15 3, 4, 5 5, 12, 13 10, 24, 26 15, 36, 39 5, 12, 13 8, 15, 1 16, 30, 34 24, 45, 51 8, 15, 1, 24, 25 14, 48, 50 21, 2, 5, 24, 25 Te most ommon Pytgoren triples re in old. Te oter triples re te result of multiplying e integer in old-fed triple y te sme ftor. Using te Pytgoren Teorem Find te vlue of. Ten tell weter te side lengts form Pytgoren triple. 2 = = = = 169 = 13 Pytgoren Teorem Sustitute. Multiply. dd. Find te positive squre root Te vlue of is 13. Beuse te side lengts 5, 12, nd 13 re integers tt stisfy te eqution 2 = 2 + 2, tey form Pytgoren triple. 464 pter 9 Rigt Tringles nd Trigonometry
3 Using te Pytgoren Teorem Find te vlue of. Ten tell weter te side lengts form Pytgoren triple. 2 = Pytgoren Teorem 14 2 = Sustitute. 196 = Multiply. 14 = 2 Sutrt 49 from e side. 14 = Find te positive squre root = Produt Property of Squre Roots 3 = Simplify. 14 Te vlue of is 3. Beuse 3 is not n integer, te side lengts do not form Pytgoren triple. Solving Rel-Life Prolem Te skysrpers sown re onneted y skywlk wit support ems. Use te Pytgoren Teorem to pproimte te lengt of e support em. E support em forms te ypotenuse of rigt tringle. Te rigt tringles re ongruent, so te support ems re te sme lengt. 2 = (23.26) 2 + (4.5) 2 Pytgoren Teorem = (23.26) 2 + (4.5) 2 Find te positive squre root m Use lultor to pproimte. Te lengt of e support em is out meters m support ems 4.5 m Monitoring Progress Help in Englis nd Spnis t BigIdesMt.om Find te vlue of. Ten tell weter te side lengts form Pytgoren triple n nemometer is devie used to mesure wind speed. Te nemometer sown is tted to te top of pole. Support wires re tted to te pole 5 feet ove te ground. E support wire is 6 feet long. How fr from te se of te pole is e wire tted to te ground? 6 ft d 5 ft Setion 9.1 Te Pytgoren Teorem 465
4 Using te onverse of te Pytgoren Teorem Te onverse of te Pytgoren Teorem is lso true. You n use it to determine weter tringle wit given side lengts is rigt tringle. Teorem Teorem 9.2 onverse of te Pytgoren Teorem If te squre of te lengt of te longest side of B tringle is equl to te sum of te squres of te lengts of te oter two sides, ten te tringle is rigt tringle. If 2 = 2 + 2, ten B is rigt tringle. Proof E. 39, p. 40 Verifying Rigt Tringles Tell weter e tringle is rigt tringle. USING TOOLS STRTEGILLY Use lultor to determine tt is te lengt of te longest side in prt () Let represent te lengt of te longest side of te tringle. ek to see weter te side lengts stisfy te eqution 2 = ( 113 ) 2 =? =? = 113 Te tringle is rigt tringle.. ( 4 95 ) 2 =? ( 95 ) 2 =? =? Te tringle is not rigt tringle. Monitoring Progress Help in Englis nd Spnis t BigIdesMt.om Tell weter te tringle is rigt tringle pter 9 Rigt Tringles nd Trigonometry
5 lssifying Tringles Te onverse of te Pytgoren Teorem is used to determine weter tringle is rigt tringle. You n use te teorem elow to determine weter tringle is ute or otuse. Teorem Teorem 9.3 Pytgoren Inequlities Teorem For ny B, were is te lengt of te longest side, te following sttements re true. If 2 < 2 + 2, ten B is ute. If 2 > 2 + 2, ten B is otuse. REMEMBER Te Tringle Inequlity Teorem (Teorem 6.11) on pge 339 sttes tt te sum of te lengts of ny two sides of tringle is greter tn te lengt of te tird side. lssifying Tringles Verify tt segments wit lengts of 4.3 feet, 5.2 feet, nd 6.1 feet form tringle. Is te tringle ute, rigt, or otuse? 2 < Proof Es. 42 nd 43, p. 40 B Step 1 Use te Tringle Inequlity Teorem (Teorem 6.11) to verify tt te segments form tringle. 2 > B >? >? >? > > > 4.3 Te segments wit lengts of 4.3 feet, 5.2 feet, nd 6.1 feet form tringle. Step 2 lssify te tringle y ompring te squre of te lengt of te longest side wit te sum of te squres of te lengts of te oter two sides ompre 2 wit Sustitute Simplify < is less tn Te segments wit lengts of 4.3 feet, 5.2 feet, nd 6.1 feet form n ute tringle. Monitoring Progress Help in Englis nd Spnis t BigIdesMt.om 6. Verify tt segments wit lengts of 3, 4, nd 6 form tringle. Is te tringle ute, rigt, or otuse?. Verify tt segments wit lengts of 2.1, 2.8, nd 3.5 form tringle. Is te tringle ute, rigt, or otuse? Setion 9.1 Te Pytgoren Teorem 46
6 9.1 Eerises Dynmi Solutions ville t BigIdesMt.om Voulry nd ore onept ek 1. VOBULRY Wt is Pytgoren triple? 2. DIFFERENT WORDS, SME QUESTION Wi is different? Find ot nswers. Find te lengt of te longest side. Find te lengt of te ypotenuse. 3 Find te lengt of te longest leg. 4 Find te lengt of te side opposite te rigt ngle. Monitoring Progress nd Modeling wit Mtemtis In Eerises 3 6, find te vlue of. Ten tell weter te side lengts form Pytgoren triple. (See Emple 1.) In Eerises 10, find te vlue of. Ten tell weter te side lengts form Pytgoren triple. (See Emple 2.) ERROR NLYSIS In Eerises 11 nd 12, desrie nd orret te error in using te Pytgoren Teorem (Teorem 9.1) = = = ( + 24) 2 2 = 31 2 = 31 2 = = = = 6 = pter 9 Rigt Tringles nd Trigonometry
7 13. MODELING WITH MTHEMTIS Te fire espe forms rigt tringle, s sown. Use te Pytgoren Teorem (Teorem 9.1) to pproimte te distne etween te two pltforms. (See Emple 3.) In Eerises 21 28, verify tt te segment lengts form tringle. Is te tringle ute, rigt, or otuse? (See Emple 5.) , 11, nd , 8, nd ft , 16, nd , 20, nd , 6., nd , 8.2, nd , 30, nd , 15, nd ft 14. MODELING WITH MTHEMTIS Te kord of te sketll oop forms rigt tringle wit te supporting rods, s sown. Use te Pytgoren Teorem (Teorem 9.1) to pproimte te distne etween te rods were tey meet te kord. 29. MODELING WITH MTHEMTIS In sell, te lengts of te pts etween onseutive ses re 90 feet, nd te pts form rigt ngles. Te plyer on first se tries to stel seond se. How fr does te ll need to trvel from ome plte to seond se to get te plyer out? 30. RESONING You re mking nvs frme for pinting using streter rs. Te retngulr pinting will e 10 ines long nd 8 ines wide. Using ruler, ow n you e ertin tt te orners of te frme re 90? 13.4 in. 9.8 in. In Eerises 15 20, tell weter te tringle is rigt tringle. (See Emple 4.) In Eerises 31 34, find te re of te isoseles tringle m 10 m 16 m 1 m 10 m ft 32 ft 20 ft m 50 m 50 m m Setion 9.1 Te Pytgoren Teorem 469
8 35. NLYZING RELTIONSHIPS Justify te Distne Formul using te Pytgoren Teorem (Tm. 9.1). 36. HOW DO YOU SEE IT? How do you know is rigt ngle? PROBLEM SOLVING You re mking kite nd need to figure out ow mu inding to uy. You need te inding for te perimeter of te kite. Te inding omes in 12 in. pkges of two yrds. How mny pkges sould you uy? 38. PROVING THEOREM Use te Pytgoren Teorem (Teorem 9.1) to prove te Hypotenuse-Leg (HL) ongruene Teorem (Teorem 5.9). 39. PROVING THEOREM Prove te onverse of te Pytgoren Teorem (Teorem 9.2). (Hint: Drw B wit side lengts,, nd, were is te lengt of te longest side. Ten drw rigt tringle wit side lengts,, nd, were is te lengt of te ypotenuse. ompre lengts nd.) 40. THOUGHT PROVOKING onsider two positive integers m nd n, were m > n. Do te following epressions produe Pytgoren triple? If yes, prove your nswer. If no, give ounteremple. 8 2mn, m 2 n 2, m 2 + n 2 B 15 in. 12 in. 20 in. 41. MKING N RGUMENT Your friend lims 2 nd 5 nnot e prt of Pytgoren triple euse does not equl positive integer squred. Is your friend orret? Eplin your resoning. 42. PROVING THEOREM opy nd omplete te proof of te Pytgoren Inequlities Teorem (Teorem 9.3) wen 2 < Given In B, 2 < 2 + 2, were is te lengt of te longest side. PQR s side lengts,, nd, were is te lengt of te ypotenuse, nd R is rigt ngle. Prove B is n ute tringle. STTEMENTS B Q R 1. In B, 2 < 2 + 2, were is te lengt of te longest side. PQR s side lengts,, nd, were is te lengt of te ypotenuse, nd R is rigt ngle. P RESONS = < < 4. Tke te positive squre root of e side. 5. m R = m < m R 6. onverse of te Hinge Teorem (Teorem 6.13). m < is n ute ngle B is n ute tringle PROVING THEOREM Prove te Pytgoren Inequlities Teorem (Teorem 9.3) wen 2 > (Hint: Look k t Eerise 42.) Mintining Mtemtil Profiieny Simplify te epression y rtionlizing te denomintor. (Skills Review Hndook) Reviewing wt you lerned in previous grdes nd lessons pter 9 Rigt Tringles nd Trigonometry
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