Circles and Arcs. Objectives To find the measures of central angles and arcs To find the circumference and arc length

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1 10-6 ircles and rcs ommon ore tate tandards G-..1 Know precise definitions of... circle... G-..1 rove that all circles are similar. lso G-..2, G-..5 M 1, M 3, M 4, M 6, M 8 bjectives o find the measures of central angles and arcs o find the circumference and arc length Hm. Will the answer be more than 63 in. or less than 63 in.? he bicycle wheel shown at the right travels 63 in. in one complete rotation. If the wheel rotates only 120 about the center, how far does it travel? Justify your reasoning. Lesson L VocabularyV circle center diameter radius congruent circles central angle semicircle minor arc major arc adjacent arcs circumference pi concentric circles arc length MHEMIL IE In a plane, a circle is the set of all points equidistant from a given point called the center. You name a circle by its center. ircle ( } ) is shown below. diameter is a segment that contains the center of a circle and has both endpoints on the circle. radius is a segment that has one endpoint at the center and the other endpoint on the circle. ongruent circles have congruent radii. central angle is an angle whose vertex is the center of the circle. is the center of the circle. is a central angle. is a diameter. is a radius. Essential Understanding You can find the length of part of a circle s circumference by relating it to an angle in the circle. n arc is a part of a circle. ne type of arc, a semicircle, is half of a circle. minor arc is smaller than a semicircle. major arc is larger than a semicircle. You name a minor arc by its endpoints and a major arc or a semicircle by its endpoints and another point on the arc. is a is a minor arc. major arc. Lesson 10-6 ircles and rcs 649

2 roblem 1 Naming rcs How can you identify the minor arcs? ince a minor arc contains all the points in the interior of a central angle, start by identifying the central angles in the diagram. What are the minor arcs of? he minor arcs are, E,, and E. What are the semicircles of? he semicircles are E, E, E, and. What are the major arcs of that contain point? he major arcs that contain point are, E, E, and E. E Got It? 1. a. What are the minor arcs of }? b. What are the semicircles of }? c. What are the major arcs of } that contain point Q? Q Key oncept rc Measure rc Measure he measure of a minor arc is equal to the measure of its corresponding central angle. he measure of a major arc is the measure of the related minor arc subtracted from 360. Example Q 50 m = m = 50 m Q = m = 310 he measure of a semicircle is 180. djacent arcs are arcs of the same circle that have exactly one point in common. You can add the measures of adjacent arcs just as you can add the measures of adjacent angles. ostulate 10-2 rc ddition ostulate he measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. m = m + m 650 hapter 10 rea

3 roblem 2 Finding the Measures of rcs How can you find m? is formed by adjacent arcs and. Use the rc ddition ostulate. What is the measure of each arc in? m = m = 32 m = m + m m = = 90 is a semicircle Got It? m = 180 m = = What is the measure of each arc in }? a. m b. m c. m Q d. m Q 77 Q 28 he circumference of a circle is the distance around the circle. he number pi (p) is the ratio of the circumference of a circle to its diameter. heorem 10-9 ircumference of a ircle he circumference of a circle is p times the diameter. = pd or = 2pr d r he number p is irrational, so you cannot write it as a terminating or repeating decimal. o approximate p, you can use 3.14, 22 7, or the key on your calculator. Many properties of circles deal with ratios that stay the same no matter what size the circle is. his is because all circles are similar to each other. o see this, consider the circles at the right. here is a translation that maps circle so that it shares the same center with circle. h k here also exists a dilation with scale factor k that maps circle h to circle. translation followed by a dilation is a similarity transformation. ecause a similarity transformation maps circle to circle, the two circles are similar. oplanar circles that have the same center are called concentric circles. h k oncentric circles Lesson 10-6 ircles and rcs 651

4 What do you need to find? You need to find the distance around the track, which is the circumference of a circle. roblem 3 Finding a istance Film 2-ft-wide circular track for a camera dolly is set up for a movie scene. he two rails of the track form concentric circles. he radius of the inner circle is 8 ft. How much farther does a wheel on the outer rail travel than a wheel on the inner rail of the track in one turn? uter edge 8 ft 2 ft Inner edge circumference of inner circle = 2pr Use the formula for the circumference of a circle. = 2p(8) ubstitute 8 for r. = 16p implify. he radius of the outer circle is the radius of the inner circle plus the width of the track. radius of the outer circle = = 10 circumference of outer circle = 2pr Use the formula for the circumference of a circle. = 2p(10) ubstitute 10 for r. = 20p implify. he difference in the two distances traveled is 20p - 16p, or 4p ft. 4p Use a calculator. wheel on the outer edge of the track travels about 13 ft farther than a wheel on the inner edge of the track. Got It? 3. a. car has a circular turning radius of 16.1 ft. he distance between the two front tires is 4.7 ft. How much farther does a tire on the outside of the turn travel than a tire on the inside? b. easoning uppose the radius of } is equal to the diameter of }. What is the ratio of the circumference of } to the circumference of }? Explain ft 4.7 ft 652 hapter 10 rea

5 he measure of an arc is in degrees, while the arc length is a fraction of the circumference. onsider the arcs shown at the right. ince the circles are concentric, there is a dilation that maps 1 to 2. he same dilation maps the slice of the small circle to the slice of the large circle. ince corresponding lengths of similar figures are proportional, 1 2 u u r 1 a 1 u r 2 a 2 r 1 r 2 = a 1 a 2 r 1 a 2 = r 2 a 1 a 1 = r 1 a 2 r 2 his means that the arc length a 1 is equal to the radius r 1 times some number. o for a given central angle, the length of the arc it intercepts depends only on the radius. n arc of 60 represents , or 1 6, of the circle. o its arc length is 1 6 of the circumference. his observation suggests the following theorem. heorem rc Length he length of an arc of a circle is the product of the ratio measure of the arc 360 and the circumference of the circle. m length of = 360 # 2pr = m 360 # pd r roblem 4 Finding rc Length What is the length of each arc shown in red? Leave your answer in terms of. How do you know which formula to use? It depends on whether the diameter is given or the radius is given. X 16 in. Y length of XY = mxy # mxy Use a formula 360 pd length of XY = # 360 2pr for arc length cm = # p(16) ubstitute. = # 2p(15) = 4p in. implify. = 20p cm X Y Got It? 4. What is the length of a semicircle with radius 1.3 m? Leave your answer in terms of p. Lesson 10-6 ircles and rcs 653

6 Lesson heck o you know HW? Use at the right to answer each question. For Exercises 5 and 6, leave your answers in terms of. 1. What is the name of a minor arc? 2. What is the name of a major arc? 3. What is the name of a semicircle? 4. What is m? 5. What is the circumference of }? 6 What is the length of? 81 9 cm 65 o you UNEN? 7. Vocabulary What is the difference between the measure of an arc and arc length? Explain. 8. Error nalysis Your class must find the length of. classmate submits the following solution. What is the error? m Length of = 2pr = 2p(4) = 9 p m MHEMIL IE 4 m 70 ractice and roblem-olving Exercises MHEMIL IE ractice Name the following in. F ee roblem the minor arcs 10. the major arcs 11. the semicircles E Find the measure of each arc in ee roblem 2. Find the circumference of each circle. Leave your answer in terms of m 3 ft 20 cm ee roblem in. 28. he camera dolly track in roblem 3 can be expanded so that the diameter of the outer circle is 70 ft. How much farther will a wheel on the outer rail travel during one turn around the track than a wheel on the inner rail? 654 hapter 10 rea

7 29. he wheel of a compact car has a 25-in. diameter. he wheel of a pickup truck has a 31-in. diameter. o the nearest inch, how much farther does the pickup truck wheel travel in one revolution than the compact car wheel? Find the length of each arc shown in red. Leave your answer in terms of cm ft 18 m ee roblem m 9 m in. pply 36. hink bout a lan Nina designed a semicircular arch made of wrought iron for the top of a mall entrance. he nine segments between the two concentric semicircles are each 3 ft long. What is the total length of wrought iron used to make this structure? ound your answer to the nearest foot. What do you know from the diagram? What formula should you use to find the amount of wrought iron used in the semicircular arches? 20 ft Find each indicated measure for. 37. m EF 38. mejh 40. m FG 41. mjeg 39. mfh 42. mhfj 43. ets hamster wheel has a 7-in. diameter. How many feet will a hamster travel in 100 revolutions of the wheel? E F G J H 70 EM 44. raffic Five streets come together at a traffic circle, as shown at the right. he diameter of the circle traveled by a car is 200 ft. If traffic travels counterclockwise, what is the approximate distance from East t. to Neponset t.? Neponset t. te ft 454 ft 244 ft 488 ft 45. Writing escribe two ways to find the arc length of a major arc if you are given the measure of the corresponding minor arc and the radius of the circle. Main t. East t. 40 Maple t. Lesson 10-6 ircles and rcs 655

8 46. ime Hands of a clock suggest an angle whose measure is continually changing. How many degrees does a minute hand move through during each time interval? a. 1 min b. 5 min c. 20 min lgebra Find the value of each variable c Q (4c 10) (x 40) (3x 20) (2x 60) Q 49. Landscape esign landscape architect is constructing a curved path through a rectangular yard. he curved path consists of two 90 arcs. He plans to edge the two sides of the path with plastic edging. What is the total length of plastic edging he will need? ound your answer to the nearest meter. 4 m 2 m 50. easoning uppose the radius of a circle is doubled. How does this affect the circumference of the circle? Explain arc of } has the same length as a 45 arc of }. What is the ratio of the radius of } to the radius of }? 2 m 4 m Find the length of each arc shown in red. Leave your answer in terms of ft in. 6 m hallenge 55. oordinate Geometry Find the length of a semicircle with endpoints (1, 3) and (4, 7). ound your answer to the nearest tenth. 56. In }, the length of is 6p cm and m is 120. What is the diameter of }? 57. he diagram below shows two 58. Given: } with } concentric circles. W. roof rove: m = m how that the length of is equal to the length of Q. V Q U W 656 hapter 10 rea

9 59. ports n athletic field is a 100 yd-by-40 yd rectangle, with a semicircle at each of the short sides. running track 10 yd wide surrounds the field. If the track is divided into eight lanes of equal width, what is the distance around the track along the inside edge of each lane? 10 yd 40 yd 100 yd tandardized est rep / 60. he radius of a circle is 12 cm. What is the length of a 60 arc? 3p cm 4p cm 5p cm 6p cm 61. What is the image of for a 135 clockwise rotation about the center Q of the regular octagon? W U V 62. Which of the following are the sides of a right triangle? U 6, 8, 12 8, 15, 17 9, 11, 23 5, 12, 15 Extended esponse 63. Quadrilateral has vertices (1, 1), (4, 1), (4, 6), and (1, 6). Quadrilateral V has vertices ( -3, 4), ( -3, -2), ( -13-2), and V( -13, 4). how that and V are similar rectangles. Mixed eview art of a regular dodecagon is shown at the right. 64. What is the measure of each numbered angle? 65. he radius is 19.3 mm. What is the apothem? 66. What is the perimeter and area of the dodecagon to the nearest millimeter or square millimeter? an you conclude that the figure is a parallelogram? Explain a r ee Lesson ee Lesson 6-3. Get eady! o prepare for Lesson 10-7, do Exercises 70 and What is the circumference of a circle with diameter 17 in.? 71. What is the length of a 90 arc in a circle with radius 6 cm? ee Lesson Lesson 10-6 ircles and rcs 657