NYS COMMON CORE MATHEMATICS CURRICULUM

Transcription

1 Exit Ticket Sample Solutions 1. Find the arc length of PPPPPP. AAAAAA llllllllllll(pppp ) = ( ) AAAAAA llllllllllll(pppp ) = CCCCCCCCCCCCCCCCCCCCCCCCCC(cccccccccccc OO) = The arc length of PPPPPP is ( )cm or cm. 2. Find the area of sector PPPPPP. AAAAAAAA(ssssssssssss PPPPPP) = (ππ(1111)22 ) AAAAAAAA(ssssssssssss PPPPPP) = The area of sector PPPPPP is cccc 22. Problem Set Sample Solutions 1. PP and QQ are points on the circle of radius 55 cm and the measure of arc PPPP is Find, to one decimal place each of the following: a. The length of arc PPPP AAAAAA llllllllllll(pppp ) = 7777 ( ) AAAAAA llllllllllll(pppp ) = 2222 The arc length of PPPP is 2222 cm or approximately cm. b. Find the ratio of the arc length to the radius of the circle. ππ = radians Date: 10/22/14 113

2 c. The length of chord PPPP The length of PPPP is twice the value of xx in OOOOOO. xx = 55 ssssss 3333 PPPP = 2222 = 1111 ssssss 3333 Chord PPPP has a length of 1111 ssssss 3333 cccc or approximately cccc. d. The distance of the chord PPPP from the center of the circle. The distance of chord PPPP from the center of the circle is labeled as yy in OOOOOO. yy = 55 cccccc 3333 The distance of chord PPPP from the center of the circle is 55 cccccc 3333 cccc or approximately 44 cccc. e. The perimeter of sector PPPPPP. PPPPPPPPPPPPPPPPPP(ssssssssssss) = PPPPPPPPPPPPPPPPPP(ssssssssssss) = The perimeter of sector PPPPPP is ( )cccc or approximately cccc. f. The area of the wedge between the chord PPPP and the arc PPPP AAAAAAAA(wwwwwwwwww) = AAAAAAAA(ssssssssssss) AAAAAAAA( PPPPPP) AAAAAAAA( PPPPPP) = 11 (1111 ssssss 3333)(55 cccccc 3333) 22 AAAAAAAA(ssssssssssss PPPPPP) = 7777 (ππ(55)22 ) AAAAAAAA(wwwwwwwwww) = 7777 (ππ(55)22 ) 11 (1111 ssssss 3333)(55 cccccc 3333) 22 The area of sector PPPPPP is approximately cccc 22. g. The perimeter of this wedge PPPPPPPPPPPPPPPPPP(wwwwwwwwww) = ssssss 3333 The perimeter of the wedge is approximately cccc. 2. What is the radius of a circle if the length of a 4444 arc is 9999? 9999 = 4444 (222222) rr = 3333 The radius of the circle is 3333 units. Date: 10/22/14 114

3 3. Arcs AAAA and CCCC both have an angle measure of 3333, but their arc lengths are not the same. OOOO = 44 and BBBB = 22. a. What are the arc lengths of arcs AAAA and CCCC? AAAAAA llllllllllll AAAA = 3333 ( ) AAAAAA llllllllllll AAAA = ππ The arc length of AAAA is 22 ππ units. 33 AAAAAA llllllllllll CCCC = 3333 ( ) AAAAAA llllllllllll CCCC = ππ The arc length of CCCC is ππ units. b. What is the ratio of the arc length to the radius for all of these arcs? Explain = ππ radians, the angle is constant, so the ratio of arc length to radius will be the angle measure, ππ c. What are the areas of the sectors AAAAAA and CCCCCC? AAAAAAAA(ssssssssssss AAAAAA) = 3333 (ππ(44)22 ) AAAAAAAA(ssssssssssss AAAAAA) = ππ The area of the sector AAAAAA is ππ uuuuuuuuuu22. AAAAAAAA(ssssssssssss CCCCDD) = 3333 (ππ(66)22 ) The area of the sector AAAAAA is 3333 uuuuuuuuuu In the circles shown, find the value of xx. The circles shown have central angles that are equal in measure. a. b. xx = radians xx = 3333 Date: 10/22/14 115

4 c. d. xx = xx = The concentric circles all have center AA. The measure of the central angle is The arc lengths are given. a. Find the radius of each circle. Radius of inner circle: ππ 22 = rr, rr = Radius of middle circle: = rr, rr = Radius of outer circle: = rr, rr = b. Determine the ratio of the arc length to the radius of each circle, and interpret its meaning. ππ is the ratio of the arc length to the radius of each 44 circle. It is the measure of the central angle in radians. 6. In the figure, if PPPP = 1111 cm, find the length of arc QQQQ? Since 66 is 11 of 9999, then the arc length of QQQQ is 11 of 1111 cm; the arc length of QQQQ is cm. Date: 10/22/14 116

5 7. Find, to one decimal place, the areas of the shaded regions. a. SSSSSSSSSSSS AAAAAAAA = AAAAAAAA oooo ssssssssssss AAAAAAAA oooo TTTTTTTTTTTTTTTT (or 33 (AAAAAAAA oooo cccccccccccc) + AAAAAAAA oooo tttttttttttttttt) 44 SSSSSSSSSSSS AAAAAAAA = 9999 (ππ(55)22 ) (55)(55) SSSSSSSSSSSS AAAAAAAA = The shaded area is approximately uuuuuuuu 22. b. The following circle has a radius of 22. SSSSSSSSSSSS AAAAAAAA = 33 (AAAAAAAA oooo cccccccccccc) + AAAAAAAA oooo tttttttttttttttt 44 Note: The triangle is a triangle with legs of length 22 (the legs are comprised by the radii, like the triangle in the previous question). SSSSSSSSSSSS AAAAAAAA = (ππ(22)22 ) (22)(22) SSSSSSSSSSSS AAAAAAAA = The shaded area is approximately. 44 uuuuuuuuuu 22. Date: 10/22/14 117

6 c. SSSSSSSSSSSS AAAAAAAA = (AAAAAAAA oooo 22 ssssssssssssss) + (AAAAAAAA oooo 22 tttttttttttttttttt) SSSSSSSSSSSS AAAAAAAA = ππ SSSSSSSSSSSS AAAAAAAA = ππ The shaded area is approximately uuuuuuuuuu 22. Date: 10/22/14 118