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10-6: Circles and Arcs Goal: Be able to find the measures of central angles and arcs, and to find the circumference and arc length of circles. circle – the set of all points ______________ equidistant center from a given point, the __________. This is A (circle A) A center radius – a segment that has one __________ endpoint center and the other endpoint at the __________ circle on the _________. diameter – a segment that contains the center of a circle and has both __________ endpoints the circle. __________on C A B is a radius. A C , A D are also radii. A B CD D is a diameter. central angle – an angle whose _________ vertex center of the circle. is at the __________ C CAB A B is a central angle. B A D is a central angle. D There are ________ 360 degrees in a circle. The following information was determined from a survey determining how people really spend their time. How People Spend Their Time 7% 31% 18% Sleep Other Food Work Entertainment 20% Must Do 15% 9% Directions: Find the measure of each central angle to the nearest whole number. Find the corresponding percent of 360. 1.) Sleep .31(360 ) 1 1 2 4.) Work .20(360 ) 7 2 2.) Other .15(360 ) 5 4 5.) Entertainment .18(360 ) 6 5 3.) Food .09(360 ) 3 2 6.) Must Do .07 (360 ) 2 5 arc – part of a _________. circle C A E B D is a minor arc. AB 0 m AB 180 H F G FGH is a semicircle. m FG H 180 CDE is a major arc. 180 m C D E 360 ***The measure of a equals minor arc __________ the measure of its corresponding central angle. ______________ Identify the following in circle C. M Y C W D X X D , D Y , YM , M W , X W a.) two minor arcs _____________________________ D YW , D YX , YM X , YM D , M W D b.) two major arcs _____________________________ YM W , M W X , W X Y c.) two semicircles _____________________________ Find the measure of each arc. M 84° Y C 40° W 56° 84° D 56° 84° X 1 .) X D m XD m D C X 56 2 .) X Y m XY m XD m D Y 3 .) M W X so, m XY 56 40 96 M W X is a semicircle. So, m M W X 180 so, m D XM 56 180 236 4 .) D X M m DXM m DX m XW M 5 .) YM mYM 180 m XY 180 96 84 6.) m X C W m XCW m XW 84 circumference of a circle – the ___________ distance around the circle. r C d or C 2 r d Circles that lie in the same plane and have concentric circles the same center are ___________________. A car has a turning radius of 16.1 feet. The distance between the two front tires is 4.7 ft. In completing the (outer) turning circle, how much farther does a tire travel than a tire on the concentric inner circle? To find the radius of the inner circle, subtract 4.7 ft from the turning radius. circumference of outer circle: C 2 r 2 (16.1) 32.2 radius of the inner circle: 1 6 .1 4 .7 1 1 .4 circumference of inner circle: C 2 r 2 (11.4) 22.8 The difference of the two distances: 3 2 .2 2 2 .8 9 .4 2 9 .5 A tire on the turning circle travels about 29.5 ft farther than a tire on the inner circle. Arc Length – fraction of the_______________ circumference of a circle. C B length of B C r r CD ? 2 r 360 A Is AB m BC No. C 105 105 105 D B m A B m C D , but AB C D They are part of different circles with different radii. Congruent arcs – arcs that have the _________________ same measure in the same circle or ___________________. in congruent circles. and are __________________ Find the length of the arcs. C 1 .) B C 5 cm B m BC 60 2 r 360 60 2 (5) 360 5 m AD B 360 150 210 cm B 3 2.) A D B m ADB 150 A 2 r 360 210 360 18 cm 2 (18) 21 cm D