UNIT 8 Module 27 Right Tringles ontents M9-2.G.SRT.8 M9-2.G.SRT.6 M9-2.G.O.2 27- The Pthgoren Theorem.............................. 820 27-2 ppling Speil Right Tringles......................... 828 Tsk 27-2 Grph Irrtionl Numers............................ 835 Red to Go On? Module Quiz......................... 836 MTHEMTIL The ommon ore Georgi Performne Stndrds for Mthemtil Prtie PRTIES desrie vrieties of epertise tht ll students should seek to develop. Opportunities to develop these prties re integrted throughout this progrm. Mke sense of prolems nd persevere in solving them. 2 Reson strtl nd quntittivel. 3 onstrut vile rguments nd ritique the resoning of others. Model with mthemtis. 5 Use pproprite tools strtegill. 6 ttend to preision. 7 Look for nd mke use of struture. 8 Look for nd epress regulrit in repeted resoning. 88 Module 27 Right Tringles
Unpking the Stndrds Understnding the stndrds nd the voulr terms in the stndrds will help ou know etl wht ou re epeted to lern in this hpter. Multilingul Glossr M9-2.G.SRT.8 Use the Pthgoren Theorem to solve right tringles in pplied prolems. Ke Voulr Pthgoren Theorem (Teorem de Pitágors) If right tringle hs legs of lengths nd nd hpotenuse of length, then 2 + 2 = 2. right tringle (triángulo retángulo) tringle with one right ngle. Wht It Mens For You You n use the reltionship etween the side lengths of right tringle to solve rel-world prolems. EXMPLE The digrm shows the reommended position for pling ldder. Given the length L of the ldder, ou n use the Pthgoren Theorem to find, the distne from the se of the wll to ple the foot of the ldder. L 2 = 2 + ( ) 2 L 2 = 7 2 _ L 2 7 = 2 L_ 7 = L PhotoDis/Gett Imges Unpking the Stndrds 89
27- The Pthgoren Theorem Essentil Question: How n ou use side lengths to determine whether tringle is ute, right, or otuse? Ojetives Use the Pthgoren Theorem nd its onverse to solve prolems. Use Pthgoren inequlities to lssif tringles. Voulr Pthgoren triple Wh lern this? You n use the Pthgoren Theorem to determine whether ldder is in sfe position. (See Emple 2.) The Pthgoren Theorem is prol the most fmous mthemtil reltionship. The theorem sttes tht in right tringle, the sum of the squres of the lengths of the legs equls the squre of the length of the hpotenuse. The Pthgoren Theorem is nmed for the Greek mthemtiin Pthgors, who lived in the sith entur..e. However, this reltionship ws known to erlier people, suh s the lonins, Egptins, nd hinese. There re mn different proofs of the Pthgoren Theorem. The one elow uses re nd lger. The re of squre with side length s is given the formul = s 2. The re of tringle with se nd height h is given the formul = 2 h. PROOF Pthgoren Theorem Given: right tringle with leg lengths nd nd hpotenuse of length Prove: 2 + 2 = 2 Proof: rrnge four opies of the tringle s shown. The sides of the tringles form two squres. The re of the outer squre is ( + ) 2. The re of the inner squre is 2. The re of eh lue tringle is 2. re of outer squre = re of lue tringles + re of inner squre ( + ) 2 = ( _ 2 ) + 2 Sustitute the res. 2 + 2 + 2 = 2 + 2 2 + 2 = 2 Simplif. Sutrt 2 from oth sides. The Pthgoren Theorem gives ou w to find unknown side lengths when ou know tringle is right tringle. Dnn Lehmn/ORIS 820 Module 27 Right Tringles
EXMPLE M9-2.G.SRT.8 Online Video Tutor Using the Pthgoren Theorem Find the vlue of. Give our nswer in simplest rdil form. 6-5 2 + 2 = 2 Pthgoren Theorem 6 2 + 2 = 2 Sustitute 6 for, for, nd for. 52 = 2 Simplif. 52 = Find the positive squre root. = () (3) = 2 3 Simplif the rdil. 2 + 2 = 2 5 2 + ( - ) 2 = 2 25 + 2-2 + = 2-2 + 26 = 0 26 = 2 = 3 Pthgoren Theorem Sustitute 5 for, - for, nd for. Multipl. omine like terms. dd 2 to oth sides. Divide oth sides 2. Find the vlue of. Give our nswer in simplest rdil form.. 8. + 2 EXMPLE M9-2.G.SRT.8 Online Video Tutor 2 Sfet pplition To prevent ldder from shifting, sfet eperts reommend tht the rtio of : e :. How fr from the se of the wll should ou ple the foot of 0-foot ldder? Round to the nerest inh. Let e the distne in feet from the foot of the ldder to the se of the wll. Then is the distne in feet from the top of the ldder to the se of the wll. 2 + 2 = 2 Pthgoren Theorem () 2 + 2 = 0 2 Sustitute. 7 2 = 00 Multipl nd omine like terms. 2 = _ 00 7 Divide oth sides 7. = _ 00 7 2 ft 5 in. Find the positive squre root nd round it. 2. Wht if...? ording to the reommended rtio, how high will 30-foot ldder reh when pled ginst wll? Round to the nerest inh. set of three nonzero whole numers,, nd suh tht 2 + 2 = 2 is lled Pthgoren triple. ommon Pthgoren Triples 3,, 5 5, 2, 3, 8, 5, 7 7, 2, 25 27- The Pthgoren Theorem 82
EXMPLE M9-2..REI. 3 Identifing Pthgoren Triples Find the missing side length. Tell if the side lengths form Pthgoren triple. Eplin. 5 Online Video Tutor 2 2 + 2 = 2 Pthgoren Theorem 2 2 + 2 = 5 2 Sustitute 2 for nd 5 for. 2 = 8 Multipl nd sutrt from oth sides. = 9 Find the positive squre root. The side lengths re nonzero whole numers tht stisf the eqution 2 + 2 = 2, so the form Pthgoren triple. 5 9 2 + 2 = 2 Pthgoren Theorem 9 2 + 5 2 = 2 Sustitute 9 for nd 5 for. 306 = 2 Multipl nd dd. = 306 = 3 3 Find the positive squre root nd simplif. The side lengths do not form Pthgoren triple euse 3 3 is not whole numer. Find the missing side length. Tell if the side lengths form Pthgoren triple. Eplin. 3. 8 0 3. 2 26 3. 3d. 2. 30 6 The onverse of the Pthgoren Theorem gives ou w to tell if tringle is right tringle when ou know the side lengths. Theorems 27-- onverse of the Pthgoren Theorem THEOREM HYPOTHESIS ONLUSION If the sum of the squres of the lengths of two sides of tringle is equl to the squre of the length of the third side, then the tringle is right tringle. is right tringle. 2 + 2 = 2 You will prove Theorem 27-- in Eerise 5. 822 Module 27 Right Tringles
You n lso use side lengths to lssif tringle s ute or otuse. Theorems 27--2 Pthgoren Inequlities Theorem In, is the length of the longest side. If 2 > 2 + 2, then is n otuse tringle. If 2 < 2 + 2, then is n ute tringle. To understnd wh the Pthgoren inequlities re true, onsider. If 2 = 2 + 2, then is right tringle the onverse of the Pthgoren Theorem. So m = 90. If 2 > 2 + 2, then hs inresed. the onverse of the Hinge Theorem, m hs lso inresed. So m > 90. If 2 < 2 + 2, then hs deresed. the onverse of the Hinge Theorem, m hs lso deresed. So m < 90. EXMPLE M9-2..ED.3 Online Video Tutor lssifing Tringles Tell if the mesures n e the side lengths of tringle. If so, lssif the tringle s ute, otuse, or right. 8,, 3 Step Determine if the mesures form tringle. the Tringle Inequlit Theorem, 8,, nd 3 n e the side lengths of tringle. Step 2 lssif the tringle. 2 2 + 2 ompre 2 to 2 + 2. 3 2 8 2 + 2 Sustitute the longest side length for. 69 6 + 2 Multipl. 69 < 85 dd nd ompre. Sine 2 < 2 + 2, the tringle is ute. 5.8, 9.3, 5.6 Step Determine if the mesures form tringle. Sine 5.8 + 9.3 = 5. nd 5. 5.6, these nnot e the side lengths of tringle. Tell if the mesures n e the side lengths of tringle. If so, lssif the tringle s ute, otuse, or right.. 7, 2, 6., 8, 3. 3.8,., 5.2 27- The Pthgoren Theorem 823
THINK ND DISUSS. How do ou know whih numers to sustitute for,, nd when using the Pthgoren Inequlities? 2. Eplin how the figure t right demonstrtes the Pthgoren Theorem. 3. List the onditions tht set of three numers must stisf in order to form Pthgoren triple.. GET ORGNIZED op nd omplete the grphi orgnizer. In eh o, summrize the Pthgoren reltionship. Pthgoren Theorem Pthgoren Reltionships onverse of the Pthgoren Theorem M.MP. Pthgoren Inequlities Theorem MTHEMTIL PRTIES 27- Eerises Homework Help GUIDED PRTIE. Voulr Do the numers 2.7, 3.6, nd.5 form Pthgoren triple? Eplin wh or wh not. SEE EXMPLE Find the vlue of. Give our nswer in simplest rdil form. 2. 3 9 SEE EXMPLE 2 5. omputers The size of omputer monitor is usull given the length of its digonl. monitor s spet rtio is the rtio of its width to its height. This monitor hs digonl length of 9 inhes nd n spet rtio of 5 :. Wht re the width nd height of the monitor? Round to the nerest tenth of n inh. 3. 7. - 2 9 in. 8 SEE EXMPLE 3 Find the missing side length. Tell if the side lengths form Pthgoren triple. Eplin. 6. 5 7. 2 20 8..5.7 SEE EXMPLE Multi-Step Tell if the mesures n e the side lengths of tringle. If so, lssif the tringle s ute, otuse, or right. 9. 7, 0, 2 0. 9,, 5. 9, 0, 2. _ 2, _ 3, 3 _ 82 Module 27 Right Tringles 3. 5.9, 6, 8.., 3, 7 6
Independent Prtie For See Eerises Emple 5 7 8 2 9 2 3 22 27 Online Etr Prtie PRTIE ND PROLEM SOLVING Find the vlue of. Give our nswer in simplest rdil form. 8 5. 6. 3 7. 7 6 9 + 8. Sfet The sfet rules for plground stte tht the height of the slide nd the distne from the se of the ldder to the front of the slide must e in rtio of 3 : 5. If slide is out 8 feet long, wht re the height of the slide nd the distne from the se of the ldder to the front of the slide? Round to the nerest inh. 8 ft 5 ft 3 ft Find the missing side length. Tell if the side lengths form Pthgoren triple. Eplin. 9. 20. 2. 7 6.5 2.5 20 5 2 Surveing Multi-Step Tell if the mesures n e the side lengths of tringle. If so, lssif the tringle s ute, otuse, or right. 22. 0, 2, 5 23. 8, 3, 23 2. 9,, 7 25. _ 2, 2, 2 _ 2 26. 0.7,.,.7 27. 7, 2, 6 5 28. Surveing It is elieved tht surveors in nient Egpt lid out right ngles using rope divided into twelve setions eleven equll sped knots. How ould the surveors use this rope to mke right ngle? nient Egptin surveors were referred to s rope-strethers. The stndrd surveing rope ws 00 rol uits. uit is 52. m long. 29. / ERROR NLYSIS / elow re two solutions for finding. Whih is inorret? Eplin the error. 2 + 2 = 3 2 2 = 69-6 = 53 2. + 3 2. 9. ( + 3) 2 + 2 = 3 2 2 + 9 + 6 = 69 2 = = 2 + 3 3 (tr), Peter Vn Steen/ HMH Photo; (l), Erih Lessing/rt Resoure Find the vlue of. Give our nswer in simplest rdil form. 30. 5 9 3. 0 7 6 25 2 33. 3. 35. 3 3 20 30 3 5 32. 5 7 8 27- The Pthgoren Theorem 825
36. Spe Eplortion The Interntionl Spe Sttion orits t n ltitude of out 250 miles ove Erth s surfe. The rdius of Erth is pproimtel 3963 miles. How fr n n stronut in the spe sttion see to the horizon? Round to the nerest mile. 37. ritil Thinking In the proof of the Pthgoren Theorem on the first pge of this lesson, how do ou know the outer figure is squre? How do ou know the inner figure is squre? 3963 mi mi 250 mi Not drwn to sle Multi-Step Find the perimeter nd the re of eh figure. Give our nswer in simplest rdil form. 38. 39. 0. 2 8 8 8 6 8 7 2. 5 2. 5 2 5 3. 5 5 3 6 8. Write out It When ou ppl oth the Pthgoren Theorem nd its onverse, ou use the eqution 2 + 2 = 2. Eplin in our own words how the two theorems re different. Q 5. Use this pln to write prgrph proof of the onverse of the Pthgoren Theorem. Given: with 2 + 2 = 2 Prove: is right tringle. P R Pln: Drw PQR with R s the right ngle, leg lengths of nd, nd hpotenuse of length. the Pthgoren Theorem, 2 + 2 = 2. Use sustitution to ompre nd. Show tht PQR nd thus is right ngle. 6. omplete these steps to prove the Distne Formul. Given: J (, ) nd K ( 2, 2 ) with 2 nd 2 Prove: JK = ( 2 - ) 2 + ( 2 - ) 2. Lote L so tht JK is the hpotenuse of right JKL. Wht re the oordintes of L?. Find JL nd LK.. the Pthgoren Theorem, JK 2 = JL 2 + LK 2. Find JK. J(, ) K( 2, 2 ) L Rel-World onnetions 7. The figure shows n irline s routes etween four ities.. trveler wnts to go from Snk (S) to Mnitou (M). To minimize the totl numer of miles trveled, should she first fl to King it (K) or to Rie Lke (R)?. The irline deides to offer diret flight from Snk (S) to Mnitou (M). Given tht the length of this flight is more thn 360 mi, wht n ou s out m SRM? 500 mi K S 300 mi R 390 mi M Trnstok In./lm Imges 826 Module 27 Right Tringles
TEST PREP 8. Gridded Response KX, LX, nd MX re the perpendiulr isetors of GHJ. Find GJ to the nerest tenth of unit. 9. Whih numer forms Pthgoren triple with 2 nd 25? 7 26 9 50. The lengths of two sides of n otuse tringle re 7 meters nd 9 meters. Whih ould NOT e the length of the third side? meters 5 meters meters 2 meters 5. Etended Response The figure shows the first si tringles in pttern of tringles.. Find P, P, P, PD, PE, nd PF in simplest rdil form.. If the pttern ontinues, wht would e the length of the hpotenuse of the ninth tringle? Eplin our nswer.. Write rule for finding the length of the hpotenuse of the nth tringle in the pttern. Eplin our nswer. HLLENGE ND EXTEND 52. lger Find ll vlues of k so tht (-, 2), (-0, 5), nd (-, k) re the verties of right tringle. 53. ritil Thinking Use digrm of right tringle to eplin wh + > 2 + 2 for n positive numers nd. 5. In right tringle, the leg lengths re nd, nd the length of the ltitude to the hpotenuse is h. Write n epression for h in terms of nd. (Hint: Think of the re of the tringle.) 55. ritil Thinking Suppose the numers,, nd form Pthgoren triple. Is eh of the following lso Pthgoren triple? Eplin.. +, +, +. 2, 2, 2. 2, 2, 2 d.,, G P K F H 6 X M E L D J MTHEMTIL PRTIES FOUS ON MTHEMTIL PRTIES 56. Resoning Joe rides his ile 9 loks north, then 22 loks est, then 3 loks north, nd then 2 loks est. The loks re squre.. How mn loks north did Joe ride? How mn loks est?. Eh lok is 0.5 mile long. To the nerest tenth of mile, how fr is Joe from his strting point? 57. Prolem Solving dog pen in the shpe of right isoseles tringle will e pled in the orner of rd. The owner wnts the pen to hve n re of 200 squre feet. out how muh fening will the owner need? 58. nlsis = 20 nd the perimeter of D is 58.. Find.. Eplin wh XD is right tringle.. Find the perimeter of XD. 65 X D 27- The Pthgoren Theorem 827
27-2 ppling Speil Right Tringles Essentil Question: Wht re the proportions of the side lengths in 30-60 -90 tringles nd 5-5 -90 tringles? Ojetives Justif nd ppl properties of 5-5 -90 tringles. Justif nd ppl properties of 30-60 -90 tringles. nimted Mth Who uses this? You n use properties of speil right tringles to lulte the orret size of ndn for our dog. (See Emple 2.) digonl of squre divides it into two ongruent isoseles right tringles. Sine the se ngles of n isoseles tringle re ongruent, the mesure of eh ute ngle is 5. So nother nme for n isoseles right tringle is 5-5 -90 tringle. 5-5 -90 tringle is one tpe of speil right tringle. You n use the Pthgoren Theorem to find reltionship mong the side lengths of 5-5 -90 tringle. 5 5 2 + 2 = 2 2 + 2 = 2 2 2 = 2 Simplif. 2 2 = 2 2 = Simplif. Pthgoren Theorem Sustitute the given vlues. Find the squre root of oth sides. Theorem 27-2- 5-5 -90 Tringle Theorem In 5-5 -90 tringle, oth legs re ongruent, nd the length of the hpotenuse is the length of leg times 2. = = l = l 2 2 l 5 l 5 l EXMPLE M9-2.G.SRT.6 Finding Side Lengths in 5-5 -90 Tringle Find the vlue of. Give our nswer in simplest rdil form. 5 7 Online Video Tutor 828 Module 27 Right Tringles the Tringle Sum Theorem, the mesure of the third ngle of the tringle is 5. So it is 5-5 -90 tringle with leg length of 7. = 7 2 Hpotenuse = leg 2 Ti/Gett Imges
Find the vlue of. Give our nswer in simplest rdil form. 3 The tringle is n isoseles right tringle, whih is 5-5 -90 tringle. The length of the hpotenuse is 3. 3 = 2 Hpotenuse = leg 2 3_ Divide oth sides 2. 2 = _ 3 2 2 = Rtionlize the denomintor. Find the vlue of. Give our nswer in simplest rdil form.. 0 2. 5 6 5 EXMPLE M9-2.G.SRT.8 Online Video Tutor 2 rft pplition 8 m 32 m 8 m Tess wnts to mke ndn for her dog folding squre of loth into 5-5 -90 tringle. Her dog s nek hs irumferene of out 32 m. The folded l ndn needs to e n etr 6 m long so Tess n tie it round her dog s nek. Wht should the side length of the squre e? Round to the nerest entimeter. l Tess needs 5-5 -90 tringle with hpotenuse of 8 m. 8 = l 2 Hpotenuse = leg 2 l = 8 _ 2 3 m Divide 2 nd round. 2. Wht if...? Tess s other dog is wering squre ndn with side length of 2 m. Wht would ou epet the irumferene of the other dog s nek to e? Round to the nerest entimeter. 30-60 -90 tringle is nother speil right tringle. You n use n equilterl tringle to find reltionship etween its side lengths. Drw n ltitude in PQR. Sine PQS RQS, PS RS. Lel the side lengths in terms of, nd use the Pthgoren Theorem to find. P 2 Q S 2 R 2 + 2 = 2 2 + 2 = (2 ) 2 Pthgoren Theorem Sustitute for, for, nd 2 for. 2 = 3 2 Multipl nd omine like terms. HMH Photo 2 = 3 2 = 3 Find the squre root of oth sides. Simplif. 27-2 ppling Speil Right Tringles 829
Theorem 27-2-2 30-60 -90 Tringle Theorem In 30-60 -90 tringle, the length of the hpotenuse is 2 times the length of the shorter leg, nd the length of the longer leg is the length of the shorter leg times 3. = s = 2s = s 3 2s s 3 s EXMPLE M9-2.G.SRT.6 3 Finding Side Lengths in 30-60 -90 Tringle Find the vlues of nd. Give our nswers in simplest rdil form. 6 Online Video Tutor If two ngles of tringle re not ongruent, the shorter side lies opposite the smller ngle. 6 = 2 Hpotenuse = 2 (shorter leg) 8 = Divide oth sides 2. = 3 Longer leg = (shorter leg) 3 = 8 3 Sustitute 8 for. = 3 Longer leg = (shorter leg) 3 _ 3 Divide oth sides 3. _ 3 = 3 = 2 = 2 (_ 3 3 ) 22 = _ 3 3 Rtionlize the denomintor. Hpotenuse = 2 (shorter leg) Sustitute _ 3 for. 3 Simplif. Find the vlues of nd. Give our nswers in simplest rdil form. 3. 3. 2 8 3 3. 3d. 9 5 830 Module 27 Right Tringles
30-60 -90 Tringles Mrus Miello Johnson High Shool To rememer the side reltionships in 30-60 -90 tringle, I drw simple -2-3 tringle like this. 2 3 2 = 2 (), so hpotenuse = 2 (shorter leg). 3 = 3 (), so longer leg = 3 (shorter leg). EXMPLE M9-2.G.SRT.8 Online Video Tutor Using the 30-60 -90 Tringle Theorem The frme of the lok shown is n equilterl tringle. The length of one side of the frme is 20 m. Will the lok fit on shelf tht is 8 m elow the shelf ove it? Step Divide the equilterl tringle into two 30-60 -90 tringles. The height of the frme is the length of the longer leg. Step 2 Find the length of the shorter leg. 20 = 2 Hpotenuse = 2(shorter leg) 0 = Divide oth sides 2. Step 3 Find the length h of the longer leg. h = 0 3 7.3 m Longer leg = (shorter leg) 3 The frme is pproimtel 7.3 entimeters tll. So the lok will fit on the shelf. h 20 m. Wht if? mnufturer wnts to mke lrger lok with height of 30 entimeters. Wht is the length of eh side of the frme? Round to the nerest tenth. (tl), Stokte/Gett Imges; (r), Sm Dudgeon/HMH Photo THINK ND DISUSS. Eplin wh n isoseles right tringle is 5-5 -90 tringle. 2. Desrie how finding in tringle I is different from finding in tringle II. I. II. 8 8 3. GET ORGNIZED op nd omplete the grphi orgnizer. In eh o, sketh the speil right tringle nd lel its side lengths in terms of s. M.MP.6 Speil Right Tringles 5 5 90 tringle MTHEMTIL PRTIES 90 tringle 27-2 ppling Speil Right Tringles 83
27-2 Eerises GUIDED PRTIE Homework Help SEE EXMPLE Find the vlue of. Give our nswer in simplest rdil form.. 5 2. 2 3. 5 9 2 SEE EXMPLE 2. Trnsporttion The two rms of the rilrod sign re perpendiulr isetors of eh other. In Pennslvni, the lengths mrked in red must e 9.5 inhes. Wht is the distne leled d? Round to the nerest tenth of n inh. 9.5 in. RIL ROD ROSSING d SEE EXMPLE 3 Find the vlues of nd. Give our nswers in simplest rdil form. 5. 6. 7. 6 5 SEE EXMPLE 8. Entertinment Regultion illird lls re 2 inhes in dimeter. The rk used to group 5 illird lls is in the shpe of n equilterl tringle. Wht is the pproimte height of the tringle formed the rk? Round to the nerest qurter of n inh. 7 3 Independent Prtie For See Eerises Emple 9 2 2 3 5 3 6 Online Etr Prtie PRTIE ND PROLEM SOLVING Find the vlue of. Give our nswer in simplest rdil form. 9. 0.. 5 5 2 5 5 2. Design This tletop is n isoseles right tringle. The length of the front edge of the tle is 8 inhes. Wht is the length w of eh side edge? Round to the nerest tenth of n inh. Find the vlue of nd. Give our nswers in simplest rdil form. 3.. 5. 2 0 3 w w 8 2 8 in. 2 (r), oris Imges/Punhstok.om; (r), Sm Dudgeon/HMH Photo 832 Module 27 Right Tringles
6. Pets dog wlk is used in dog gilit ompetitions. In this dog wlk, eh rmp mkes n ngle of 30 with the ground.. How long is one rmp?. How long is the entire dog wlk, inluding oth rmps? 30 2 ft.5 ft 30 Multi-Step Find the perimeter nd re of eh figure. Give our nswers in simplest rdil form. 7. 5-5 -90 tringle with hpotenuse length 2 inhes 8. 30-60 -90 tringle with hpotenuse length 28 entimeters 9. squre with digonl length 8 meters 20. n equilterl tringle with side length feet 2. n equilterl tringle with height 30 rds 22. Estimtion The tringle loom is mde from wood strips shped into 5-5 -90 tringle. Pegs re pled ever inh long 2 the hpotenuse nd ever inh long eh leg. Suppose ou mke loom with n 8-inh hpotenuse. pproimtel how mn pegs will ou need? 23. ritil Thinking The ngle mesures of tringle re in the rtio : 2 : 3. re the side lengths lso in the rtio : 2 : 3? Eplin our nswer. Find the oordintes of point P under the given onditions. Give our nswers in simplest rdil form. 2. PQR is 5-5 -90 tringle with verties Q (, 6) nd R (-6, -), nd m P = 90. P is in Qudrnt II. 25. PST is 5-5 -90 tringle with verties S (, -3) nd T (-2, 3), nd m S = 90. P is in Qudrnt I. 26. PWX is 30-60 -90 tringle with verties W (-, -) nd X (, -), nd m W = 90. P is in Qudrnt II. 27. PYZ is 30-60 -90 tringle with verties Y (-7, 0) nd Z (5, 0), nd m Z = 90. P is in Qudrnt IV. 28. Write out It Wh do ou think 30-60 -90 tringles nd 5-5 -90 tringles re lled speil right tringles? (r), HMH; (l), Trnstok In./lm Imges; Rel-World onnetions 29. The figure shows n irline s routes mong four ities. The irline offers one frequent-flier mile for eh mile flown (rounded to the nerest mile). How mn frequent-flier miles do ou ern for eh flight?. Nelson (N) to elton (). Idri (I) to Nelson (N). elton () to Idri (I) N 5 320 mi I L 27-2 ppling Speil Right Tringles 833
TEST PREP 30. Whih is true sttement? = 2 = 3 = 3 = 2 3. n 8-foot pole is roken during storm. The top of the pole touhes the ground 2 feet from the se of the pole. How tll is the prt of the pole left stnding? 5 feet 3 feet 6 feet 22 feet 2 ft 32. The length of the hpotenuse of n isoseles right tringle is 2 inhes. Wht is the length of one leg of the tringle, rounded to the nerest tenth of n inh? 3.9 inhes 33.9 inhes 7.0 inhes.6 inhes 33. Gridded Response Find the re of the retngle to the nerest tenth of squre inh. 32 in. HLLENGE ND EXTEND Multi-Step Find the vlue of in eh figure. 3. 35. 36. Eh edge of the ue hs length e.. Find the digonl length d when e =, e = 2, nd e = 3. Give the nswers in simplest rdil form.. Write formul for d for n positive vlue of e. 37. Write prgrph proof to show tht the ltitude to the hpotenuse of 30-60 -90 tringle divides the hpotenuse into two segments, one of whih is 3 times s long s the other. e d e e MTHEMTIL PRTIES FOUS ON MTHEMTIL PRTIES 38. Numer Sense The lengths of the sides of tringle, rounded to the nerest ten, re 60 m, 00 m, nd 20 m. ould the tringle e speil right tringle? If so, whih one ould it e? 39. Mke onjeture Three loks with identil heights re shped like isoseles tringles. The hve se ngles of 30, 5, nd 60. Order the lengths of the ses from lest to gretest. Mke onjeture out the reltionship etween the mesures of the se ngles nd the lengths of the ses of isoseles tringles with equl heights. 0. Resoning The perimeter of 5-5 -90 tringle is P. Write n epression for the length of one leg in terms of P. 83 Module 27 Right Tringles
27-2 Use with ppling Speil Right Tringles tivit Grph Irrtionl Numers Numers suh s 2 nd 3 re irrtionl. Tht is, the nnot e written s the rtio of two integers. In deiml form, the re infinite nonrepeting deimls. You n round the deiml form to estimte the lotion of these numers on numer line, or ou n use right tringles to onstrut their lotions etl. MTHEMTIL PRTIES Use pproprite tools strtegill. Drw line. Mrk two points ner the left side of the line nd lel them 0 nd. The distne from 0 to is unit. 0 3 onstrut perpendiulr to the line through. 2 Set our ompss to unit nd mrk inrements t 2, 3,, nd 5 units to onstrut numer line. 0 2 3 5 Using our ompss, mrk unit up from the numer line nd then drw right tringle. The legs oth hve length, so the Pthgoren Theorem, the hpotenuse hs length of 2. 0 2 3 5 0 2 3 5 5 Set our ompss to the length of the hpotenuse. Drw n r entered t 0 tht intersets the numer line t 2. 6 Repet Steps 3 through 5, strting t 2, to onstrut segment of length 3. 3 0 2 2 3 5 0 2 3 5 2 Tr This. Sketh the two right tringles from Step 6. Lel the side lengths nd use the Pthgoren Theorem to show wh the onstrution is orret. 2. onstrut nd verif tht it is equl to 2. 3. onstrut 5 through 9 nd verif tht 9 is equl to 3.. Set our ompss to the length of the segment from 0 to 2. Mrk off nother segment of length 2 to show tht 8 is equl to 2 2. Geometr Tsk Grph Irrtionl Numers 835
MODULE 27 QUIZ Red to Go On? ssessment nd Intervention 27- The Pthgoren Theorem. Find the vlue of. 2. Find the missing side length. 5 9 Give the nswer in Tell if the side lengths form simplest rdil form. Pthgoren triple. Eplin. 3. Tell if the mesures 0, 2, nd 6 n e the side lengths of tringle. If so, lssif the tringle s ute, otuse, or right.. lndsper wnts to ple stone wlkw from one orner of the retngulr lwn to the opposite orner. Wht will e the length of 50 ft the wlkw? Round to the nerest inh. 9 80 ft Wlkw Find the missing side length. Tell if the sides form Pthgoren triple. Eplin. 5. 6..5 7.5 2 32 Tell if the mesures n e the side lengths of tringle. If so, lssif the tringle s ute, otuse, or right. 7. 9, 2, 6 8.,, 27 9..5, 3.6, 3.9 0. 2, 3.7,. 27-2 ppling Speil Right Tringles Find the vlues of the vriles. Give our nswers in simplest rdil form.. 8 5 2. 5 22 3. 5 3. 26 5. 5 2 6. 6 2 5 5 7. 8 8. 6 9. 836 Module 27 Red to Go On?
PR ssessment Rediness Seleted Response. Find the vlue of. Epress our nswer in simplest rdil form. 6. Eh tringle is 5-5 - 90 tringle. Find the vlue of. 3 3 6 = 3 5 = 3 3 = 9 5 = 5 3 2. Find the vlue of. Epress our nswer in simplest rdil form. = 8 = 2 65 = 2 33 = 2 3. Tell if the mesures 9,, nd 5 n e side lengths of tringle. If so, lssif the tringle s ute, right, or otuse. Yes; ute tringle Yes; right tringle Yes; otuse tringle No; the mesures nnot e side lengths of tringle.. The length of the hpotenuse of right tringle is three times the length of the shorter leg. The length of the longer leg is 2. Wht is the length of the shorter leg? 2 3 3 2. 6 2 5. Wht is n epression in simplest form for the perimeter of the 30-60 -90 tringle shown? 22 22 + 22 2 3 = _ 3 2 2 = _ 3 2 = 3 2 = _ 3 3 2 7. The size of TV sreen is given the length of its digonl. The sreen spet rtio is the rtio of its width to its height. The sreen spet rtio of stndrd TV sreen is :3. Wht re the width nd height of 27 TV sreen? Mini-Tsk Height 27 Width width: 2.6 in., height: 6.2 in. width: 6.2 in., height: 2.6 in. width: 2.6 in., height: 5. in. width: 5. in., height: 2.6 in. 8. The ield sign hs the shpe of n equilterl tringle with side length of 36 inhes. Wht is the height of the sign? Will retngulr metl sheet 36 inhes wide nd 32 inhes tll e ig enough to mke one sign? 36 in. YIELD h 33 + 22 3 33 + 3 837