Woksheets Chapte 6: Discoveing and Poving Cicle Popeties 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 line Would this wok fo any adius? Yes O T O T Investigation 2 Wite the Tangent Conjectue in you notes. Wite the Convese of the Tangent Conjectue in you notes. 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. Wite the Tangent Segments Conjectue in you notes. E x x 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? The intecepted ac? Make sue you can justify you answes with popeties! m = 90 and m G= 90 because tangents ae pependicula to the adii at the point of tangency. Sum of the angles of a quadilateal ae 360. Let m EG= x So 360 = 90 + 90 + x + m N and 180 = x + m N, always. lso since x = m EG= mg, since the cental angle = it s intecepted ac. S. Stiling Page 1 of 13
EXERCISES Lesson 6.1 Page 313-314 #1 5, 8, 9. Show how you ae finding you answes! State the popeties you ae applying when possible. 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 Linea pai supplementay. Tangent adius sum = 180 y = 180 60 90 = 30 Tangent adius Quad. sum = 360 z = 360 180 75 = 105 60 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 13
t 112 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 13
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: m BO = m DOC = 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 13
68 34 112 34 Cental angle = intecepted ac. mc = 68 Radii = so COB isos. & base = sum = 180 (180 112)/2 = 34 m B= 34 65 65 65 115 115 Linea pai supplementay. m OI = 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 =, Δ OB isos. & base angles =. sum = 180, so 180 66 66 = 48 = y m OC = 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 13
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? 76 76 Daw an inscibed angle, What is m XB? 38 XB. No matte whee you daw the inscibed angle it = 38 76 O 38 B X C 38 142 What is the measue of Daw an inscibed angle, What is m CYD? 71 mcd? 142 CYD. 71 142 P 71 Y D What is the elationship between an inscibed angle and its intecepted ac? inscibed angle = ½ ac Wite the Inscibed ngle Conjectue in you notes. EXERCISES Lesson 6.3 Pages 327 328 # 1 7, 9 11, 16 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 Inscibed angle = ½ intecepted ac. 95 * 2 = 190 c = 190 120 = 70 50 Radius tangent. sum = 180 180 90 40 = 50 Cental angle = intecepted ac. h = 50 S. Stiling Page 6 of 13
40 42 84 Inscibed angle = ½ intecepted ac & Semi cicle = 180 20 * 2 = 40 d = 180 40 = 140 180 96 = 84 e = 84 2 = 42 100 90 150 Inscibed angle = ½ intecepted ac 75 * 2 = 150 Cicle = 360 g = 360 150 110 = 100 (110 + 100)/2 = 105 Quad. sum = 360 f = 360 75 105 90 = 90 50 130 130 Cental = ac & vetical s =. Radius tangent. Quad. sum = 360 w = 360 180 130 = 50 44 NDO is a semicicle. 180 136 = 44 Kite, so = chods make = acs so y = 44 142 142 Inscibed angle = ½ intecepted ac. 38 * 2 = 76 Cicle = 360 & = 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 76 74 Inscibed angle = ½ intecepted ac. 37 * 2 = 74 But 35 ½ * 74 S. Stiling Page 7 of 13
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. 5π = dπ 5 = d 2. If = 5 cm, find C. 5 C = 10π 3. If C = 24 cm, find. 24 = 2π 24 2π = 2π 2π 12 = π 4. If d = 5.5 cm, find C. C = 5.5π 5. If a cicle has a diamete of 12 cm, what is its cicumfeence? C = 12π 6. If a cicle has a cicumfeence of 46π, what is its diamete? 46π = dπ 46 = d 7. If d = 5 cm, find C. C = 5π C 15.708 8. If = 4 cm, find C. 4 C = 8π C 25.133 9. If C = 44 m, find. 44 = 2π 44 2π = 2π 2π 22 = 7.003 π 10. bicycle tie with a 27 inch diamete, find C. C = 27π C 84.823 in 11. Feis wheel with = 24 cm, find distance taveled by a seat in one evolution. ( 24) C = 48π C 150.796 S. Stiling Page 8 of 13
12. Cicle inscibed in a squae with peimete 24 cm, find C. 6 p = 4s 24 = 4s 6 = s C = 6π C 18.850 13. Cicle with C = 16π inches is cicumscibed about a squae, find length of the diagonal. 16π = dπ 16 = d 6 16 P P 15. Find numbe of 1 inch tiles to put aound the edge of the pool. The cicula ends: C = 18π 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 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 N 52 (180 76)/2 = 52 S S. Stiling Page 9 of 13
Investigation 5: 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. But if you ae on the cicle itself, and if you ae taveling fom point to point B (on the cicle) did you tavel the same distance as you would fom point C to point D? D B O 120 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 of the cicle, 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! length of B = 1 iπ ( 4) 2 = 16 π 16.755 3 3 length of CD = 1 π ( 12) 2 = 48π 150.796 3, how fa is it fom to B? How fa is it fom C to D? Think pat cm i cm 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 10 of 13
EXERCISES Lesson 6.7 Pages 351 # 1 7 On all poblems, show algebaic pocedues: wite the fomula, substitute in known infomation, then solve. Leave you answes in tems of π. 80 2 π 3 360 2 = 6 π 9 4 = π 3 4.189 120 2 π 12 360 1 = 24 π 3 = 8π 25.133 210 210 2 π 12 360 7 = 24 π 12 = 14π 43.982 120 6π = 2π 360 3 3 2π 6π = 2π 2π 3 9 = 60 60 2 π 18 360 1 = 36 π 6 = 6π 18.850 80 80 2 π 9 360 2 = 18 π 9 = 4π 12.566 160 160 12π = dπ 360 9 9 4π 12π = d 4π 4π 9 27 = S. Stiling Page 11 of 13
EXERCISES Chapte 6 Review Pages 359 360 # 4 19 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 = 128 180 Inscibed angle = ½ intecepted ac. 90 * 2 = 180 Cicle = 360 d = 360 180 89 = 91 20 C = 40π C 125.664 132 = dπ 132 = d π d 42.017 100 108 Equal chods cut = acs. 100 2 π ( 27 ) 360 5 = 54 π 18 = 15π 47.124 sum = 180. 180 35 57 = 108 Inscibed angle = ½ intecepted ac. 108 * 2 = 216 but the angle intecepts a semicicle which = 180. It should = 90. 160 36 72 Equal chods cut = acs. (360 220)/2 = 70 70 2 π ( 36 ) 360 7 = 72π 36 = 14π 43.982 Semi- = 180. 180 108 = 72 Inscibed angle = ½ intecepted ac. 72 2 = 36 ltenate inteio s =, so lines. S. Stiling Page 12 of 13
152 Cicle = 360. 360 152 56 = 152 = chods cut = acs. So JI = IM and Δ JIM is isos. 70 Inscibed angle = ½ intecepted ac. mkim = 140 & mki = 140 70 = 70 = chods cut = acs. so Δ KIM is isos. B Need pependicula bisectos. C B Need angle bisectos. nd a adius dawn pependicula to a side. C S. Stiling Page 13 of 13