Lesson 6.1 Tangent Popeties Investigation 1 Tangent Conjectue If you daw a tangent to a cicle, then Daw a adius to the point of tangency. What do you notice? pependicula Would this be tue fo all tangent lines? Yes Convese of the Tangent Theoem Daw a line pependicula to OT at point T, call it T. What type of line ist? tangent Would this wok fo any adius? Yes T O O T Wite the Tangent Conjectue in you notes. Wite the Convese of the Tangent Conjectue in you notes. Investigation 2 N Tangent Segments Conjectue Daw tangent segments to cicle E fom point N. What do you notice about these segments? They e conguent. Measue them. 4.5 cm 4.5 cm Wite the Tangent Segments Conjectue in you notes. E G Daw Kite NGE. Do you know any of the angles of this kite? What elationships can you make between the angles of this kite? Make sue you can justify you answes with popeties! m90 and mg 90 because tangents ae pependicula to the adii at the point of tangency. Sum of the angles of a quadilateal ae 360. So 360 90 90 meg mn and 180 meg m N lways? Yes S. Stiling Page 1 of 15
EXERCISES Lesson 6.1 Page 313-314 #1 5, 8 10. Show how you ae finding you answes! State the popeties you ae applying and show calculations!!. w = 180 130 = 50 50 O Tangent adius Quad sum 360 w = 360 90 90 130 = 50 Tangents fom a point outside a =. Isos. base angles = and sum = 180 x = (180 70)/2 = 55 60 Linea pai supplementay. Tangent adius sum = 180 y = 180 60 90 = 30 Tangent adius Quad. sum = 360 z = 360 180 75 = 105 13 6 13 6 6 13 6 13 Tangents fom a point outside a =. OR = O = P = PC = 13 TC = TD = DS = SR and TD = ½ of 12 = 6 Peim = 4 * 13 + 4 * 6 = 76 S. Stiling Page 2 of 15
t diamete Vaious lines. Tangents must be adii! X t Y Z 10. Daw an obtuse tiangle BC inscibed in the cicle given below. Is the longest side of tiangle BC longe o shote than the diamete? B Vaious tiangles. C Shote S. Stiling Page 3 of 15
Lesson 6.2 Chod Popeties Investigation 3 Chod Popeties If two chods in a cicle ae conguent, then Investigate the following: the cental angles associated with those chods the intecepted acs associated with those chods If B CD, then B D What if the chods ae not conguent? EF GH F 116 116 O 116 116 P H Wite you obsevations: mbo mdoc 116 equal cental angles mb mcd 116 equal intecepted acs C E G None of the measues ae equal. Wite the Chod Conjectues in you notes. EXERCISES Lesson 6.2 Pages 320 321 # 1 3, 5, 6, 8 11 Wite the popeties you ae using as you ae finding the missing measues. (You don t need to name them, you just need to state them.) 165 Cental angle = intecepted ac. x = 165 = chods cut = acs. Cicle s acs = 360. z = 360 276 = 84 128 70 = chods cut = acs and = Cental angles. w = 70 70 70 84 S. Stiling Page 4 of 15
68 34 112 34 Cental angle = intecepted ac. mc 68 Radii = so COB isos. & base = sum = 180 (180 112)/2 = 34 mb 34 65 65 65 115 115 Linea pai supplementay. moi 65 Cental angle = intecepted ac. w = 115 = chods cut = acs and = cental angles. x = 115 and y = 65 82 110 120 82 48 120 mc 130 so mb 130 48 82 Cental angle = intecepted ac. x = 48, y = 82, w = 110 Cicle s acs add to 360. 360 48 82 110 = 120 z = 120 96 42 96 42 96 Cicle s acs add to 360. = chods cut = acs and = cental angles. mft 360 72 288 288 3 = 96 = x y = 96 Radii = so FOE isos. & 96 base =, sum = 180 (180 96)/2 = 42 = z 96 66 66 66 48 66 66 66 Cental angle = intecepted ac., so coesponding angles =. x = 66 Since adii of a cicle =, OB isos. & base angles =. sum = 180, so 180 66 66 = 48 = y moc 180 114 66 and z = 66. Radius = 18 so the diamete = 36. The diamete would have to be the longest chod of the cicle, so the chod can t be geate than 36. S. Stiling Page 5 of 15
Lesson 6.3 cs and ngles Investigation 4 The Big Question: What is the measue of an inscibed angle? What is the measue of mb? Daw an inscibed angle, What is m XB? 38 XB. What is the measue of mcd? Daw an inscibed angle, What is m CYD? 71 CYD. 76 X 142 38 O 38 76 76 B C Y P 142 142 71 71 D What is the elationship between an inscibed angle and its intecepted ac? Wite the Inscibed ngle Conjectue in you notes. inscibed angle = ½ intecepted ac Investigation 5: Inscibed Quadilateals Use you notes and daw a cyclic quadilateal in P. Remembe each angle must be an inscibed angle and each side must be a chod. Label you quadilateal BCD. P Measue all of the angles of you quadilateal. e thee any elationships between the angles? The opposite angles ae supplementay in a cyclic quadilateal. Wite the Cyclic Quadilateal Conjectue in you notes. Ty to daw a cyclic paallelogam in cicle O. What type of paallelogam can be inscibed in a cicle? O Only ectangles (and squaes) can be inscibed in a cicle.. Wite the Cyclic Paallelogam Conjectue in you notes. S. Stiling Page 6 of 15
Investigation 6 Given B CF ED. Examine the measues of the acs. What could you conclude about the intecepted acs? 22 F B 22 C mf mbc mef mdc. 54 22 22 P 54 54 54 Wite the Paallel Lines (Secants) Intecepted cs Conjectue in you notes. E D EXERCISES Lesson 6.3 Pages 327 328 # 1 14 Wite the popeties you ae using as you ae finding the missing measues. (You don t need to name them, you just need to state them.) 65 Inscibed angle = ½ intecepted ac. 120 30 60 Inscibed angle = ½ intecepted ac. Semi cicle measues 180 180 120 = 60 60 2 = 30 70 140 40 42 84 Inscibed angle = ½ intecepted ac. 95 * 2 = 190 c = 190 120 = 70 Inscibed angle = ½ intecepted ac & Semi cicle = 180 20 * 2 = 40 d = 180 40 = 140 180 96 = 84 e = 84 2 = 42 50 100 Radius tangent. sum = 180 180 90 40 = 50 Cental angle = intecepted ac. h = 50 90 105 x 150 Inscibed angle = ½ intecepted ac 75 * 2 = 150 Cicle s acs = 360 g = 360 150 110 = 100 x = (110 + 100)/2 = 105 Quad. sum = 360 f = 360 75 105 90 = 90 S. Stiling Page 7 of 15
50 130 130 Cental = ac & vetical s =. Radius tangent. Quad. sum = 360 w = 360 180 130 = 50 32 x Paallel secants cut = acs. Cicle = 360º 2x 64 360 2x 296 x 148 44 NDO is a semicicle. 180 136 = 44 Kite, so = chods make = acs so y = 44 142 76 142 Inscibed angle = ½ intecepted ac. 38 * 2 = 76 Cicle = 360 and = chods cut = acs k = (360 76)/2 = 142 60 60 60 60 Cicle = 360 & = chods cut = acs s = 360/6 = 60 Inscibed angle = ½ intecepted ac. = ½ (60 * 4) = 120 140 82 x Inscibed angle = ½ intecepted ac. 90 = ½ (98 + x) 180 = 98 + x, x = 82 m = 360 220 = 140 n = ½ (140 + 82) = 111 41 120 98 41 p Isos Δ, base s =. 180 98 q 41 2 Cental = intecepted ac. Paallel secants cut = acs. Cicle = 360º. 2 p 218 360 2 p 142 p 71 2b 2c 2a 2d 2e Inscibed angle = ½ intecepted ac. Cicle = 360º. 2a 2b 2c 2d 2e 360 2 a b c d e 360 a b c d e 180 Sum = 180º S. Stiling Page 8 of 15
EXERCISES Lesson 6.5 Pages 337 340 # 1 13, 15, 19. On all poblems, show algebaic pocedues: wite the fomula, substitute in known infomation, then solve. On #1 6, leave you answes in tems of π. On #7 9, use the π appoximation on the calculato and ound final answes to 3 decimal places. Fo #10 15, see you book fo the poblem statement. 1. If C = 5π cm, find d. C d 5 d 5 d 2. If = 5 cm, find C. C 2 C 2 5 C 10 3. If C = 24 cm, find. C 2 24 2 24 2 2 2 12 4. If d = 5.5 cm, find C. C d C 5.5 5. If a cicle has a diamete of 12 cm, what is its cicumfeence? C d C 12 6. If a cicle has a cicumfeence of 46π, what is its diamete? C d 46 d 46 d 7. If d = 5 cm, find C. C d C 5 C 15.708 8. If = 4 cm, find C. C 2 C 2 4 C 8 C 25.133 9. If C = 44 m, find. C 2 44 2 44 2 2 2 22 7.003 10. bicycle tie with a 27 inch diamete, find C. C d C 27 C 84.823 in 11. Feis wheel with = 24 cm, find distance taveled by a seat in one evolution. C 2 C 2 24 C 48 C 150.796 S. Stiling Page 9 of 15
12. Cicle inscibed in a squae with peimete 24 cm, find C. 6 p 4s 24 4s 6 s C d C 6 C 18.850 13. Cicle with C = 16π inches is cicumscibed about a squae, find length of the diagonal. C d 16 d 6 16 P 16 d P 15. Find numbe of 1 inch tiles to put aound the edge of the pool. The cicula ends: C d C 18 C 56.549 Sides of the ectangle ae =. peim 56.549 2 30 116.549 ft 116.549 * 12 = 1398.6 one-inch tiles So need 1399 one-inch tiles 30 18 #19 Hint: Stat with the 42 degee angles! b = 90 c = 42 d = 70 e = 48 f = 132 g = 52 K 84 H 180 42 90 = 48 180 42 48 = 90 90 48 R 48 84/2 = 42 132 P 52 M 70 360 90 68 132 = 70 52 N (180 76)/2 = 52 S S. Stiling Page 10 of 15
Investigation 10: c Length So fa the measue of an ac = the measue of its cental angle (in degees). In the diagam, mb mcd 120 If you ae thinking in tems of tun o degees, it makes sense that if you ae. standing at point O you will tun 120 to get fom to B and you would tun the same amount of degees to tun fom C to D. D B OC = 12 cm But if you ae on the cicle itself, and if you ae taveling fom point to point B did you tavel the same distance fom point C to point D? O 120 4 cm 8 cm C NO! The distance fom C to D is longe than the distance fom to B. How can you explain this? The distance would be pat of the cicumfeence, but what pat? What pat (faction) of the cicle ae we talking about? 120 1 Faction = 360 3 O 4 cm and OC 12 cm If of the cicumfeence! 1 8 length of B = 2 4 8.378 cm 3 3 1 length of CD = 212 8 25.133 cm 3, how fa is it fom to B? How fa is it fom C to D? Think pat So if you ae looking at the length of the ac, and not the amount of tun (o degee of the ac), then it makes complete sense. S. Stiling Page 11 of 15
EXERCISES Lesson 6.7 Pages 351 # 1 9, 13 14 On all poblems, show algebaic pocedues: wite the fomula, substitute in known infomation, then solve. Leave you answes in tems of π!! length 80 2 3 360 2 6 9 4 3 4.189 length 120 2 12 360 1 24 3 8 25.133 210 length 210 2 12 360 7 24 12 14 43.982 120 6 2 360 3 3 2 6 2 2 3 9 60 length 60 2 18 360 1 36 6 6 18.850 80 length 80 2 9 360 2 18 9 4 12.566 160 160 12 d 360 9 9 4 12 d 4 4 9 27 72 72 72 40 2 360 5 40 5 2 2 1 2 5 100 S. Stiling Page 12 of 15
Read the poblems fom the book pages 352 353. 9. Completes 4 laps in 6 minutes. Calculate aveage speed in metes pe minute. Round to two decimal places! Make a dawing. 100 metes Pd 2 100 P 40 200 P 325.66 metes 325.66 m 13.57 m 24 min min 40 metes 13. 1 evolution in 20 seconds, what is the angula velocity? 360 18 deg 20 sec sec Since all of the hoses otate 360º in one evolution, they all have the same angula velocity. 14. 2 hoses complete 1 evolution in 20 seconds. The hoses ae 8 m and 6 m fom the cente. What ae the tangential velocities of the two hoses? Round to two decimal places! Hose #1: 1 ev 2 8 16 metes 16 m 2.51 m 20 sec sec Hose #2: 1 ev 2 6 12 metes 12 m 1.88 m 20 sec sec The hose on the outside is moving faste because he has to tavel futhe to make one evolution in the same amount of time (20 seconds). S. Stiling Page 13 of 15
EXERCISES Chapte 6 Review Pages 359 360 # 4 19, 21, 22 Wite the popeties you ae using as you ae finding the missing measues. (You don t need to name them. You just need to state them.) Mak diagams with the infomation as you go! The degee measue descibes the amount of tun, based on the cental angle. The ac length is pat of the cicumfeence. Measued in a unit of length, like inches. 90 b Tangent Radius Cental angle = intecepted ac. sum = 180 180 90 35 = 55 b = 55 Inscibed angle = ½ intecepted ac. 110 * 2 = 220 a = 220 155 = 65 Cicle = 360 = chods cut = acs. c = (360 104)/2 = 128 x Vetex inside so x 1 60 64 2 x 62 Linea pai supp e 180 62 118 92 180 Inscibed angle = ½ intecepted ac. 90 * 2 = 180 Cicle = 360 d = 360 180 89 = 91 92 Linea pai supp 180 88 92 Vetex inside so 92 1 118 f 2 184 118 f 66 f C 2 C 2 20 C 40 C 125.664 132 d 132 d d 42.017 100 Equal chods cut = acs. length 100 2 27 360 5 54 18 15 47.124 160 70 Equal chods cut = acs. (360 220)/2 = 70 length 70 2 36 360 7 72 36 14 43.982 S. Stiling Page 14 of 15
x sum = 180. x = 180 35 57 = 108 Inscibed angle = ½ intecepted ac. 108 * 2 = 216 but the angle intecepts a semicicle which = 180. x should = 90. 72 x =56 Paallel secants cut = acs, so x = 56. s acs add to 360º, but 84 + 56 + 56 + 158 = 354, not 360. 36 Semi- = 180. 180 108 = 72 Inscibed angle = ½ intecepted ac. 72 2 = 36 72 152 ltenate inteio s =, so lines. Cicle s acs = 360. 360 152 56 = 152 = chods cut = acs. So JI IM and JIM is isos. 70 Inscibed angle = ½ intecepted ac. mkim 140 & mki 140 70 70 = chods cut = acs. so KIM is isos. (Place answes below.) Need at least 2 angle bisectos and a adius dawn pependicula to a side (fo the adius). Point S is equidistant fom the tiangle s sides. Need at least 2 pependicula bisectos. Point V is equidistant fom the tiangle s vetices. B B C S V C S. Stiling Page 15 of 15