Maintaining Mathematical Proficiency

Transcription

1 Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression Solve the proportion = 8. = = = = = The Pythgoren Theorem sttes tht + =, where nd re legs of right tringle nd is the hypotenuse. re you le to simplify the Pythgoren Theorem further to sy tht + =? Eplin. opyright ig Ides Lerning, LL Geometry 41

2 Nme Dte 9.1 The Pythgoren Theorem For use with Eplortion 9.1 Essentil Question How n you prove the Pythgoren Theorem? 1 EXPLORTION: Proving the Pythgoren Theorem without Words Work with prtner.. Drw nd ut out right tringle with legs nd, nd hypotenuse.. Mke three opies of your right tringle. rrnge ll four tringles to form lrge squre s shown.. Find the re of the lrge squre in terms of,, nd y summing the res of the tringles nd the smll squre. d. opy the lrge squre. Divide it into two smller squres nd two eqully-sized retngles, s shown. e. Find the re of the lrge squre in terms of nd y summing the res of the retngles nd the smller squres. f. ompre your nswers to prts () nd (e). Eplin how this proves the Pythgoren Theorem. 4 Geometry opyright ig Ides Lerning, LL

3 Nme Dte 9.1 The Pythgoren Theorem (ontinued) EXPLORTION: Proving the Pythgoren Theorem Work with prtner.. onsider the tringle shown. h d D d. Eplin why, D, nd D re similr.. Write two-olumn proof using the similr tringles in prt () to prove tht + =. ommunite Your nswer 3. How n you prove the Pythgoren Theorem? 4. Use the Internet or some other resoure to find wy to prove the Pythgoren Theorem tht is different from Eplortions 1 nd. opyright ig Ides Lerning, LL Geometry 43

4 Nme Dte 9.1 Notetking with Voulry For use fter Lesson 9.1 In your own words, write the mening of eh voulry term. Pythgoren triple Theorems Theorem 9.1 Pythgoren Theorem In right tringle, the squre of the length of the hypotenuse is equl to the sum of the squres of the lengths of the legs. Notes: = + ore onepts ommon Pythgoren Triples nd Some of Their Multiples 3, 4, 5 5, 1, 13 8, 15, 17 7, 4, 5 6, 8, 10 10, 4, 6 16, 30, 34 14, 48, 50 9, 1, 15 15, 36, 39 4, 45, 51 1, 7, 75 3, 4, 5 5, 1, 13 8, 15, 17 7, 4, 5 The most ommon Pythgoren triples re in old. The other triples re the result of multiplying eh integer in old-fed triple y the sme ftor. Notes: 44 Geometry opyright ig Ides Lerning, LL

5 Nme Dte 9.1 Notetking with Voulry (ontinued) Theorems Theorem 9. onverse of the Pythgoren Theorem 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. If = +, then is right tringle. Notes: Theorem 9.3 Pythgoren Inequlities Theorem For ny, where is the length of the longest side, the following sttements re true. If < +, then is ute. If > +, then is otuse. Notes: < + > + opyright ig Ides Lerning, LL Geometry 45

6 Nme Dte 9.1 Notetking with Voulry (ontinued) Etr Prtie In Eerises 1 6, find the vlue of. Then tell whether the side lengths form Pythgoren triple From shool, you iked 1. miles due south nd then 0.5 mile due est to your house. If you hd iked home on the street tht runs diretly digonl from your shool to your house, how mny fewer miles would you hve iked? In Eerises 8 nd 9, verify tht the segment lengths form tringle. Is the tringle ute, right, or otuse? 8. 90, 16, nd , 1, nd 3 46 Geometry opyright ig Ides Lerning, LL

7 Nme Dte 9. Speil Right Tringles For use with Eplortion 9. Essentil Question Wht is the reltionship mong the side lengths of tringles? tringles? 1 EXPLORTION: Side Rtios of n Isoseles Right Tringle Go to igidesmth.om for n intertive tool to investigte this eplortion. Work with prtner.. Use dynmi geometry softwre to onstrut n isoseles right tringle with leg length of 4 units.. Find the ute ngle mesures. Eplin why this tringle is lled tringle.. Find the et rtios of the side lengths (using squre roots). = = = Smple Points ( 0, 4) ( 4, 0) 0, 0 ( ) Segments = 5.66 = 4 = 4 ngles m = 45 m = 45 d. Repet prts () nd () for severl other isoseles right tringles. Use your results to write onjeture out the rtios of the side lengths of n isoseles right tringle. opyright ig Ides Lerning, LL Geometry 47

8 Nme Dte 9. Speil Right Tringles (ontinued) EXPLORTION: Side Rtios of Tringle Go to igidesmth.om for n intertive tool to investigte this eplortion. Work with prtner.. Use dynmi geometry softwre to onstrut right tringle with ute ngle mesures of 30 nd 60 ( tringle ), where the shorter leg length is 3 units.. Find the et rtios of the side lengths (using squre roots). = = = Smple Points 0, 5.0 ( 3, 0) 0, 0 ( ) ( ) Segments = 6 = 3 = 5.0 ngles m = 30 m = 60. Repet prts () nd () for severl other tringles. Use your results to write onjeture out the rtios of the side lengths of tringle. ommunite Your nswer 3. Wht is the reltionship mong the side lengths of tringles? tringles? 48 Geometry opyright ig Ides Lerning, LL

9 Nme Dte 9. Notetking with Voulry For use fter Lesson 9. In your own words, write the mening of eh voulry term. isoseles tringle Theorems Theorem Tringle Theorem In tringle, the hypotenuse is times s long s eh leg. Notes: hypotenuse = leg Theorem Tringle Theorem In tringle, the hypotenuse is twie s long s the shorter leg, nd the longer leg is 3 times s long s the shorter leg. Notes: hypotenuse = shorter leg longer leg = shorter leg 3 opyright ig Ides Lerning, LL Geometry 49

10 Nme Dte 9. Notetking with Voulry (ontinued) Etr Prtie In Eerises 1 4, find the vlue of. Write your nswer in simplest form In Eerises 5 7, find the vlues of nd y. Write your nswers in simplest form y y 9 y 50 Geometry opyright ig Ides Lerning, LL

11 Nme Dte 9. Notetking with Voulry (ontinued) In Eerises 8 nd 9, sketh the figure tht is desried. Find the indited length. Round deiml nswers to the nerest tenth. 8. The length of digonl in squre is 9. n isoseles tringle with 30 se ngles 3 inhes. Find the perimeter of hs n ltitude of 3 meters. Find the the squre. length of the se of the isoseles tringle. 10. Find the re of DEF. Round deiml nswers to the nerest tenth. F 14 m D m E opyright ig Ides Lerning, LL Geometry 51

12 Nme Dte 9.3 Similr Right Tringles For use with Eplortion 9.3 Essentil Question How re ltitudes nd geometri mens of right tringles relted? 1 EXPLORTION: Writing onjeture Go to igidesmth.om for n intertive tool to investigte this eplortion. Work with prtner.. Use dynmi geometry softwre to onstrut right, s shown. Drw D so tht it is n ltitude from the right ngle to the hypotenuse of D Points ( 0, 5) ( 8, 0) ( 0, 0) D.5, 3.6 ( ) Segments = 9.43 = 8 = 5. The geometri men of two positive numers nd is the positive numer tht stisfies =. is the geometri men of nd. so tht D Write proportion involving the side lengths of D nd D is the geometri men of two of the other side lengths. Use similr tringles to justify your steps. 5 Geometry opyright ig Ides Lerning, LL

13 Nme Dte 9.3 Similr Right Tringles (ontinued) 1 EXPLORTION: Writing onjeture (ontinued). Use the proportion you wrote in prt () to find D. d. Generlize the proportion you wrote in prt (). Then write onjeture out how the geometri men is relted to the ltitude from the right ngle to the hypotenuse of right tringle. EXPLORTION: ompring Geometri nd rithmeti Mens Go to igidesmth.om for n intertive tool to investigte this eplortion. Work with prtner. Use spredsheet to find the rithmeti men nd the geometri men of severl pirs of positive numers. ompre the two mens. Wht do you notie? D rithmeti Men Geometri Men ommunite Your nswer 3. How re ltitudes nd geometri mens of right tringles relted? opyright ig Ides Lerning, LL Geometry 53

14 Nme Dte 9.3 Notetking with Voulry For use fter Lesson 9.3 In your own words, write the mening of eh voulry term. geometri men Theorems Theorem 9.6 Right Tringle Similrity Theorem If the ltitude is drwn to the hypotenuse of right tringle, then the two tringles formed re similr to the originl tringle nd to eh other. D D, D, nd D D. Notes: D D ore onepts Geometri Men The geometri men of two positive numers nd is the positive numer tht stisfies =. So, = nd =. Notes: 54 Geometry opyright ig Ides Lerning, LL

15 Nme Dte 9.3 Notetking with Voulry (ontinued) Theorems Theorem 9.7 Geometri Men (ltitude) Theorem In right tringle, the ltitude from the right ngle to the hypotenuse divides the hypotenuse into two segments. The length of the ltitude is the geometri men of the lengths of the two segments of the hypotenuse. D D = D D Notes: Theorem 9.8 Geometri Men (Leg) Theorem In right tringle, the ltitude from the right ngle to the hypotenuse divides the hypotenuse into two segments. The length of eh leg of the right tringle is the geometri men of the lengths of the hypotenuse nd the segment of the hypotenuse tht is djent to the leg. Notes: D = D = D opyright ig Ides Lerning, LL Geometry 55

16 Nme Dte 9.3 Notetking with Voulry (ontinued) Etr Prtie In Eerises 1 nd, identify the similr tringles. 1. J. O P H K I M N In Eerises 3 nd 4, find the geometri men of the two numers. 3. nd nd 45 In Eerises 5 8, find the vlue of the vrile y t Geometry opyright ig Ides Lerning, LL

17 Nme Dte 9.4 The Tngent Rtio For use with Eplortion 9.4 Essentil Question How is right tringle used to find the tngent of n ute ngle? Is there unique right tringle tht must e used? Let e right tringle with ute. The tngent of (written s tn ) is defined s follows. tn length of leg opposite = = length of leg djent to djent opposite 1 EXPLORTION: lulting Tngent Rtio Go to igidesmth.om for n intertive tool to investigte this eplortion. Work with prtner. Use dynmi geometry softwre.. onstrut, s shown. onstrut segments perpendiulr to to form right tringles tht shre verte nd re similr to with verties, s shown K L M N O P Q J I H G F E D Smple Points (0, 0) (8, 6) (8, 0) ngle m = lulte eh given rtio to omplete the tle for the deiml vlue of tn for eh right tringle. Wht n you onlude? Rtio KD D LE E MF F NG G OH H PI I QJ J tn opyright ig Ides Lerning, LL Geometry 57

18 Nme Dte 9.4 The Tngent Rtio (ontinued) EXPLORTION: Using lultor Work with prtner. Use lultor tht hs tngent key to lulte the tngent of Do you get the sme result s in Eplortion 1? Eplin. ommunite Your nswer with verties ( ) ( ) nd ( ) 3. Repet Eplortion 1 for 0, 0, 8, 5, 8, 0. onstrut the seven perpendiulr segments so tht not ll of them interset t integer vlues of. Disuss your results. 4. How is right tringle used to find the tngent of n ute ngle? Is there unique right tringle tht must e used? 58 Geometry opyright ig Ides Lerning, LL

19 Nme Dte 9.4 Notetking with Voulry For use fter Lesson 9.4 In your own words, write the mening of eh voulry term. trigonometri rtio tngent ngle of elevtion ore onepts Tngent Rtio Let e right tringle with ute. The tngent of (written s tn ) is defined s follows. tn Notes: length of leg opposite = = length of leg djent to leg opposite hypotenuse leg djent to opyright ig Ides Lerning, LL Geometry 59

20 Nme Dte 9.4 Notetking with Voulry (ontinued) Etr Prtie In Eerises 1 3, find the tngents of the ute ngles in the right tringle. Write eh nswer s frtion nd s deiml rounded to four deiml ples. 1. R. L T 45 S J 5 K In Eerises 4 6, find the vlue of. Round your nswer to the nerest tenth In DE, E = 90 nd tn =. Find tn D? Write your nswer s frtion Geometry opyright ig Ides Lerning, LL

21 Nme Dte 9.4 Notetking with Voulry (ontinued) 8. n environmentlist wnts to mesure the width of river to monitor its erosion. From point, she wlks downstrem 100 feet nd mesures the ngle from this point to point to e 40.. How wide is the river? Round to the nerest tenth One yer lter, the environmentlist returns to mesure the sme river. From point, she gin wlks downstrem 100 feet nd mesures the ngle from this point to point to e now 51. y how mny feet hs the width of the river inresed? 9. oy flies kite t n ngle of elevtion of 18. The kite rehes its mimum height 300 feet wy from the oy. Wht is the mimum height of the kite? Round to the nerest tenth. 10. Find the perimeter of the figure opyright ig Ides Lerning, LL Geometry 61

22 Nme Dte 9.5 The Sine nd osine Rtios For use with Eplortion 9.5 Essentil Question How is right tringle used to find the sine nd osine of n ute ngle? Is there unique right tringle tht must e used? Let e right tringle with ute. The sine of nd osine of (written s sin nd os, respetively) re defined s follows. sin os length of leg opposite = = length of hypotenuse length of leg djent to = = length of hypotenuse hypotenuse opposite djent 1 EXPLORTION: lulting Sine nd osine Rtios Go to igidesmth.om for n intertive tool to investigte this eplortion. Work with prtner. Use dynmi geometry softwre.. onstrut, s shown. onstrut segments perpendiulr to to form right tringles tht shre verte nd re similr to with verties, s shown K L M N O P Q J I H G F E D Smple Points ( 0, 0) ( 8, 6) 8, 0 ( ) ngle m = Geometry opyright ig Ides Lerning, LL

23 Nme Dte 9.5 The Sine nd osine Rtios (ontinued) 1 EXPLORTION: lulting Sine nd osine Rtios (ontinued). lulte eh given rtio to omplete the tle for the deiml vlues of sin nd os for eh right tringle. Wht n you onlude? Sine rtio KD K LE L MF M NG N OH O PI P QJ Q sin osine rtio D K E L F M G N H O I P J Q os ommunite Your nswer. How is right tringle used to find the sine nd osine of n ute ngle? Is there unique right tringle tht must e used? 3. In Eplortion 1, wht is the reltionship etween nd in terms of their mesures? Find sin nd os. How re these two vlues relted to sin nd os? Eplin why these reltionships eist. opyright ig Ides Lerning, LL Geometry 63

24 Nme Dte 9.5 Notetking with Voulry For use fter Lesson 9.5 In your own words, write the mening of eh voulry term. sine osine ngle of depression ore onepts Sine nd osine Rtios Let e right tringle with ute. The sine of nd osine of (written s sin nd os ) re defined s follows. sin os Notes: length of leg opposite = = length of hypotenuse length of leg djent to = = length of hypotenuse leg opposite hypotenuse leg djent to 64 Geometry opyright ig Ides Lerning, LL

25 Nme Dte 9.5 Notetking with Voulry (ontinued) Sine nd osine of omplementry ngles The sine of n ute ngle is equl to the osine of its omplement. The osine of n ute ngle is equl to the sine of its omplement. Let nd e omplementry ngles. Then the following sttements re true. Notes: ( ) ( ) ( ) ( ) sin = os 90 = os sin = os 90 = os os = sin 90 = sin os = sin 90 = sin Etr Prtie In Eerises 1 3, find sin F, sin G, os F, nd os G. Write eh nswer s frtion nd s deiml rounded to four ples. 1. G. E 3. G F G 97 F E E 5 F In Eerises 4 6, write the epression in terms of osine. 4. sin 9 5. sin sin 77 opyright ig Ides Lerning, LL Geometry 65

26 Nme Dte 9.5 Notetking with Voulry (ontinued) In Eerises 7 9, write the epression in terms of sine. 7. os15 8. os os 45 In Eerises 10 13, find the vlue of eh vrile using sine nd osine. Round your nswers to the nerest tenth y m 7 n d mer tthed to kite is filming the dmge used y rush fire in losed-off re. The mer is diretly ove the enter of the losed-off re.. person is stnding 100 feet wy from the enter of the losed-off re. The ngle of depression from the mer to the person flying the kite is 5. How long is the string on the kite?. If the string on the kite is 00 feet long, how fr wy must the person flying the kite stnd from the enter of the losed-off re, ssuming the sme ngle of depression of 5, to film the dmge? 66 Geometry opyright ig Ides Lerning, LL

27 Nme Dte 9.6 Solving Right Tringles For use with Eplortion 9.6 Essentil Question When you know the lengths of the sides of right tringle, how n you find the mesures of the two ute ngles? 1 EXPLORTION: Solving Speil Right Tringles Go to igidesmth.om for n intertive tool to investigte this eplortion. Work with prtner. Use the figures to find the vlues of the sine nd osine of nd. Use these vlues to find the mesures of nd. Use dynmi geometry softwre to verify your nswers opyright ig Ides Lerning, LL Geometry 67

28 Nme Dte 9.6 Solving Right Tringles (ontinued) EXPLORTION: Solving Right Tringles Go to igidesmth.om for n intertive tool to investigte this eplortion. Work with prtner. You n use lultor to find the mesure of n ngle when you know the vlue of the sine, osine, or tngent of the ngle. Use the inverse sine, inverse osine, or inverse tngent feture of your lultor to pproimte the mesures of nd to the nerest tenth of degree. Then use dynmi geometry softwre to verify your nswers ommunite Your nswer 3. When you know the lengths of the sides of right tringle, how n you find the mesures of the two ute ngles? 4. ldder lening ginst uilding forms right tringle with the uilding nd the ground. The legs of the right tringle (in meters) form Pythgoren triple. Find the mesures of the two ute ngles to the nerest tenth of degree. 68 Geometry opyright ig Ides Lerning, LL

29 Nme Dte 9.6 Notetking with Voulry For use fter Lesson 9.6 In your own words, write the mening of eh voulry term. inverse tngent inverse sine inverse osine solve right tringle ore onepts Inverse Trigonometri Rtios Let e n ute ngle. 1 1 Inverse Tngent If tn =, then tn = m. tn = m 1 1 Inverse Sine If sin = y, then sin y = m. sin = 1 1 Inverse osine If os = z, then os z = m. os = m m Notes: opyright ig Ides Lerning, LL Geometry 69

30 Nme Dte 9.6 Notetking with Voulry (ontinued) Solving Right Tringle To solve right tringle mens to find ll unknown side lengths nd ngle mesures. You n solve right tringle when you know either of the following. two side lengths one side length nd the mesure of one ute ngle Notes: Etr Prtie In Eerises 1 nd, determine whih of the two ute ngles hs the given trigonometri rtio. 1. The osine of the ngle is 4.. The sine of the ngle is out G G E 4 F E 14 F In Eerises 3 6, let mesure of H e n ute ngle. Use lultor to pproimte the H to the nerest tenth of degree. 3. sin H = tn H = 1 5. os H = sin H = Geometry opyright ig Ides Lerning, LL

31 Nme Dte 9.6 Notetking with Voulry (ontinued) In Eerises 7 10, solve the right tringle. Round deiml nswers to the nerest tenth E D L N 10. Z M X 18 Y 11. ot is pulled in y winh on dok 1 feet ove the dek of the ot. When the winh is fully etended to 5 feet, wht is the ngle of elevtion from the ot to the winh? 1 5 opyright ig Ides Lerning, LL Geometry 71

32 Nme Dte 9.7 Lw of Sines nd Lw of osines For use with Eplortion 9.7 Essentil Question Wht re the Lw of Sines nd the Lw of osines? 1 EXPLORTION: Disovering the Lw of Sines Go to igidesmth.om for n intertive tool to investigte this eplortion. Work with prtner.. omplete the tle for the tringle shown. Wht n you onlude? Smple Segments = 3.16 = 6.3 = 5.10 ngles m = 9.74 m = m = m sin m sin m sin. Use dynmi geometry softwre to drw two other tringles. omplete tle for eh tringle. Use your results to write onjeture out the reltionship etween the sines of the ngles nd the lengths of the sides of tringle. m sin m sin m sin m sin m sin m sin 7 Geometry opyright ig Ides Lerning, LL

33 Nme Dte 9.7 Lw of Sines nd Lw of osines (ontinued) EXPLORTION: Disovering the Lw of osines Go to igidesmth.om for n intertive tool to investigte this eplortion. Work with prtner.. omplete the tle for the tringle in Eplortion 1(). Wht n you onlude? m + os. Use dynmi geometry softwre to drw two other tringles. omplete tle for eh tringle. Use your results to write onjeture out wht you oserve in the ompleted tles. m + os m + os ommunite Your nswer 3. Wht re the Lw of Sines nd the Lw of osines? 4. When would you use the Lw of Sines to solve tringle? When would you use the Lw of osines to solve tringle? opyright ig Ides Lerning, LL Geometry 73

34 Nme Dte 9.7 Notetking with Voulry For use fter Lesson 9.7 In your own words, write the mening of eh voulry term. Lw of Sines Lw of osines ore onepts re of Tringle The re of ny tringle is given y one-hlf the produt of the lengths of two sides times the sine of their inluded ngle. For shown, there re three wys to lulte the re re = sin re = sin re = sin Notes: 74 Geometry opyright ig Ides Lerning, LL

35 Nme Dte 9.7 Notetking with Voulry (ontinued) Theorems Theorem 9.9 Lw of Sines The Lw of Sines n e written in either of the following forms for with sides of length,, nd. sin sin sin = = = = sin sin sin Notes: Theorem 9.10 Lw of osines If hs sides of length,, nd, s shown, then the following re true. = + os = + os = + os Notes: opyright ig Ides Lerning, LL Geometry 75

36 Nme Dte 9.7 Notetking with Voulry (ontinued) Etr Prtie In Eerises 1 3, use lultor to find the trigonometri rtio. Round your nswer to four deiml ples. 1. sin 5. os tn 96 In Eerises 4 nd 5, find the re of the tringle. Round your nswer to the nerest tenth E E D G 19 4 F In Eerises 6-8, solve the tringle. Round deiml nswers to the nerest tenth Geometry opyright ig Ides Lerning, LL