Ch 6 Worksheet L1 Shorten Key Lesson 6.1 Tangent Properties

Transcription

1 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 meg mn and 180 meg m N lways? Yes S. Stiling Page 1 of 15

2 EXERCISES Lesson 6.1 Page #1 5, Show how you ae finding you answes! State the popeties you ae applying and show calculations!!. w = = O Tangent adius Quad sum 360 w = = 50 Tangents fom a point outside a =. Isos. base angles = and sum = 180 x = (180 70)/2 = Linea pai supplementay. Tangent adius sum = 180 y = = 30 Tangent adius Quad. sum = 360 z = = Tangents fom a point outside a =. OR = O = P = PC = 13 TC = TD = DS = SR and TD = ½ of 12 = 6 Peim = 4 * * 6 = 76 S. Stiling Page 2 of 15

3 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

4 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 O 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 # 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 = = = chods cut = acs and = Cental angles. w = S. Stiling Page 4 of 15

5 Cental angle = intecepted ac. mc 68 Radii = so COB isos. & base = sum = 180 ( )/2 = 34 mb Linea pai supplementay. moi 65 Cental angle = intecepted ac. w = 115 = chods cut = acs and = cental angles. x = 115 and y = mc 130 so mb Cental angle = intecepted ac. x = 48, y = 82, w = 110 Cicle s acs add to = 120 z = Cicle s acs add to 360. = chods cut = acs and = cental angles. mft = 96 = x y = 96 Radii = so FOE isos. & 96 base =, sum = 180 (180 96)/2 = 42 = z Cental angle = intecepted ac., so coesponding angles =. x = 66 Since adii of a cicle =, OB isos. & base angles =. sum = 180, so = 48 = y moc 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

6 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 O B C Y P 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

7 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 P Wite the Paallel Lines (Secants) Intecepted cs Conjectue in you notes. E D EXERCISES Lesson 6.3 Pages # 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 Inscibed angle = ½ intecepted ac. Semi cicle measues = = Inscibed angle = ½ intecepted ac. 95 * 2 = 190 c = = 70 Inscibed angle = ½ intecepted ac & Semi cicle = * 2 = 40 d = = = 84 e = 84 2 = Radius tangent. sum = = 50 Cental angle = intecepted ac. h = x 150 Inscibed angle = ½ intecepted ac 75 * 2 = 150 Cicle s acs = 360 g = = 100 x = ( )/2 = 105 Quad. sum = 360 f = = 90 S. Stiling Page 7 of 15

8 Cental = ac & vetical s =. Radius tangent. Quad. sum = 360 w = = x Paallel secants cut = acs. Cicle = 360º 2x x 296 x NDO is a semicicle = 44 Kite, so = chods make = acs so y = Inscibed angle = ½ intecepted ac. 38 * 2 = 76 Cicle = 360 and = chods cut = acs k = (360 76)/2 = Cicle = 360 & = chods cut = acs s = 360/6 = 60 Inscibed angle = ½ intecepted ac. = ½ (60 * 4) = x Inscibed angle = ½ intecepted ac. 90 = ½ (98 + x) 180 = 98 + x, x = 82 m = = 140 n = ½ ( ) = p Isos Δ, base s = q 41 2 Cental = intecepted ac. Paallel secants cut = acs. Cicle = 360º. 2 p p 142 p 71 2b 2c 2a 2d 2e Inscibed angle = ½ intecepted ac. Cicle = 360º. 2a 2b 2c 2d 2e a b c d e 360 a b c d e 180 Sum = 180º S. Stiling Page 8 of 15

9 EXERCISES Lesson 6.5 Pages # 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 If C = 24 cm, find. C If d = 5.5 cm, find C. C d C If a cicle has a diamete of 12 cm, what is its cicumfeence? C d C 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 If = 4 cm, find C. C 2 C 2 4 C 8 C If C = 44 m, find. C bicycle tie with a 27 inch diamete, find C. C d C 27 C in 11. Feis wheel with = 24 cm, find distance taveled by a seat in one evolution. C 2 C 2 24 C 48 C S. Stiling Page 9 of 15

10 12. Cicle inscibed in a squae with peimete 24 cm, find C. 6 p 4s 24 4s 6 s C d C 6 C 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 Sides of the ectangle ae =. peim ft * 12 = one-inch tiles So need 1399 one-inch tiles #19 Hint: Stat with the 42 degee angles! b = 90 c = 42 d = 70 e = 48 f = 132 g = 52 K 84 H = = R 48 84/2 = P 52 M = N (180 76)/2 = 52 S S. Stiling Page 10 of 15

11 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 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? Faction = O 4 cm and OC 12 cm If of the cicumfeence! 1 8 length of B = cm length of CD = 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

12 EXERCISES Lesson 6.7 Pages 351 # 1 9, On all poblems, show algebaic pocedues: wite the fomula, substitute in known infomation, then solve. Leave you answes in tems of π!! length length length length length d d S. Stiling Page 12 of 15

13 Read the poblems fom the book pages Completes 4 laps in 6 minutes. Calculate aveage speed in metes pe minute. Round to two decimal places! Make a dawing. 100 metes Pd P P metes m m 24 min min 40 metes evolution in 20 seconds, what is the angula velocity? deg 20 sec sec Since all of the hoses otate 360º in one evolution, they all have the same angula velocity 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 metes 16 m 2.51 m 20 sec sec Hose #2: 1 ev 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

14 EXERCISES Chapte 6 Review Pages # 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 = = 55 b = 55 Inscibed angle = ½ intecepted ac. 110 * 2 = 220 a = = 65 Cicle = 360 = chods cut = acs. c = ( )/2 = 128 x Vetex inside so x x 62 Linea pai supp e Inscibed angle = ½ intecepted ac. 90 * 2 = 180 Cicle = 360 d = = Linea pai supp Vetex inside so f f 66 f C 2 C 2 20 C 40 C d 132 d d Equal chods cut = acs. length Equal chods cut = acs. ( )/2 = 70 length S. Stiling Page 14 of 15

15 x sum = 180. x = = 108 Inscibed angle = ½ intecepted ac. 108 * 2 = 216 but the angle intecepts a semicicle which = 180. x should = x =56 Paallel secants cut = acs, so x = 56. s acs add to 360º, but = 354, not Semi- = = 72 Inscibed angle = ½ intecepted ac = ltenate inteio s =, so lines. Cicle s acs = = 152 = chods cut = acs. So JI IM and JIM is isos. 70 Inscibed angle = ½ intecepted ac. mkim 140 & mki = 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