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Find the value of x****.
SOLUTION: The sum of the measures of the central angles of a circle with no interior points in common is 360. ANSWER: 170
SOLUTION: The sum of the measures of the central angles of a circle with no interior points in common is 360. ANSWER: 150
PRECISION and are diameters of
. Identify each arc as major arc , minor arc , or semicircle****. Then find its measure.
SOLUTION: Here, is the longest arc connecting the points I and J on Therefore, it is a major arc. is a major arc that shares the same endpoints as minor arc IJ. ANSWER: major arc; 270
SOLUTION: Here, is the shortest arc connecting the points I and H on Therefore, it is a minor arc. ANSWER: minor arc; 59
SOLUTION:
Here, is a diameter. Therefore, is a semicircle. The measure of a semicircle is 180, so ANSWER: semicircle; 180
- FITNESS The graph shows the results of a survey taken by high school students regarding what activities they participate in after school. a. Find. b. Find. c. Describe the type of arc that the category Sports represents. SOLUTION: a. Here, is a minor arc. The measure of the arc is equal to the measure of the central angle. Find the 9% of 360 to find the central angle. b. Here, is a minor arc. The measure of the arc is equal to the measure of the central angle. Find the 12% of 360 to find the central angle. c. The arc that represents the category Sports, , is the longest arc connecting the points A and D. Therefore, it is a major arc. ANSWER: a. 32. b. 43. c. major arc
SOLUTION:
Use the Arc Length equation to find the arc length, if
you know the radius and the arc measure, in
radians.
ANSWER:
43.5 m
SOLUTION: Use the Arc Length equation to find the arc length, if you know the arc measure and the angle measure, in radians. ANSWER: 3.14 cm Find the value of x****.
SOLUTION: The sum of the measures of the central angles of a circle with no interior points in common is 360°. ANSWER: 80
SOLUTION: The sum of the measures of the central angles of a circle with no interior points in common is 360°. ANSWER: 225
SOLUTION:
The sum of the measures of the central angles of a circle with no interior points in common is 360. ANSWER: 35
SOLUTION: The sum of the measures of the central angles of a circle with no interior points in common is 360. ANSWER: 40 and are diameters of. Identify each arc as a major arc, minor arc, or semicircle****. Then find its measure.
SOLUTION: Here, is the shortest arc connecting the points C and D on circle B. Therefore, it is a minor arc. ANSWER: minor arc; 55
SOLUTION: Here, is the shortest arc connecting the points A and C on circle B. Therefore, it is a minor arc. Since is a diameter, is a semicircle and has a measure of 180. Use angle addition to find the measure of. Therefore, the measure of is 125. ANSWER: minor arc; 125
20. m ( )
SOLUTION:
Here, is a diameter. Therefore, arc CFG is a semicircle and m (arc CFG ) = 180. ANSWER: semicircle; 180
- SHOPPING The graph shows the results of a survey in which teens were asked where the best place was to shop for clothes. a. What would be the arc measures associated with the mall and vintage stores categories? b. Describe the kinds of arcs associated with the category “Mall”and category “None of these”. c. Are there any congruent arcs in this graph? Explain. SOLUTION: a. The measure of the arc is equal to the measure of the central angle. The mall contributes 76% and the vintage stores contribute 4% of the total shopping. Find the 76% of 360 to find the central angle of the arc associated with the malls. Find the 4% of 360 to find the central angle of the arc associated with the vintage stores. b. The arc associated with the mall has a measure of 273.6. So, it is a major arc. The arc associated with none of these has a measure of 9% of 360 or 32.4. So, it is a minor arc. c. Yes; the arcs associated with the online and none of these categories have the same arc measure since each category accounts for the same percentage of the circle, 9%. ANSWER: a. 273.6, 14. b. major arc; minor arc c. Yes; the arcs associated with the online and none of these categories have the same arc measure since each category accounts for the same percentage of the circle, 9%.
- MODELING The table shows the distribution of Texas's endangered animals by species. a. If you were to construct a circle graph of this information, what would be the arc measures associated with the first two categories? b. Describe the kind of arcs associated with the first category and the last category. c. Are there any congruent arcs in this graph? Explain. SOLUTION: a. The measure of the arc is equal to the measure of the central angle. The “mammals” category contributes 25% and the “fish” contributes 28% in the distribution of endangered animal species. Find the 25% of 360 to find the central angle of the arc associated with the "mammals" category. Find the 28% of 360 to find the central angle of the arc associated with the "fish" category. b. The arc corresponding to "mammals" category measures 90, so it is a minor arc. However, the arc corresponding to the "fish" category measures 100.8, so it is a major arc. c. No; no categories share the same percentage of the circle. ANSWER: a. 90; 100. b. minor arc; major arc c. No; no categories share the same percentage of the circle.
ENTERTAINMENT Use the Ferris wheel shown to find each measure.
SOLUTION: The measure of the arc is equal to the measure of the central angle. We have, Therefore, ANSWER: 40
SOLUTION: The measure of the arc is equal to the measure of the central angle. We have, Therefore, ANSWER: 60
SOLUTION: Here, is a diameter. Therefore, is a semicircle and ANSWER: 180
SOLUTION:
Arc JFH is a major arc. Therefore, the measure of arc JFH is 300. ANSWER: 300
SOLUTION: Arc GHF is a major arc. Therefore, the measure of arc GHF is 320. ANSWER: 320
SOLUTION: Here, is a diameter. Therefore, is a semicircle and ANSWER: 180
SOLUTION: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. First find m ∠ JLK and then use the Arc Addition Postulate. Therefore, the measure of arc HK is 100. ANSWER: 100
- , if the diameter is 9 centimeters SOLUTION: Use the arc length equation with r = (9) or 4.
centimeters and m (arc QT ) = m ∠ QPT or 112.
Therefore, the length of arc QT is about 8. centimeters. ANSWER: 8.80 cm
- , if PS = 4 millimeters SOLUTION:
is a radius of and is a diameter. Use the
Arc Addition Postulate to find m (arc QR ).
Use the arc length equation with r = 4 millimeters and x = m (arc RQ ) or 68. Therefore, the length of arc QR is about 4. millimeters. ANSWER: 4.75 mm
- , if RT = 15 inches SOLUTION:
is a diameter, so r = (15) or 7.5 inches. Use
the arc length equation with x = m (arc RS ) or 130.
Therefore, the length of arc RS is about 17.02 inches. ANSWER: 17.02 in.
- , if RT = 11 feet SOLUTION:
Use the Arc Addition Postulate to find the m (arc
QR ).
Use the Arc Addition Postulate to find m (arc QRS ).
Use the arc length equation with r = ( RT ) or 5. feet and x = m (arc QRS ) or 198. Therefore, the length of arc QRS is about 19.01 feet. ANSWER: 19.01 ft
- , if PQ = 3 meters SOLUTION: Arc RTS is a major arc that shares the same endpoints as minor arc RS.
Use the arc length equation with r = PS or 3 meters
and x = m (arc RTS ) or 230.
Therefore, the length of arc RTS is about 12. meters. ANSWER: 12.04 m HISTORY The figure shows the stars in the Betsy Ross flag referenced at the beginning of the lesson.
- What is the measure of central angle A? Explain how you determined your answer. SOLUTION: There are 13 stars arranged in a circular way equidistant from each other. So, the measure of the central angle of the arc joining any two consecutive stars will be equal to ANSWER: stars between each star 45. If the diameter of the circle were doubled , what would be the effect on the arc length from the center of one star B to the next star C? SOLUTION: The measure of the arc between any two stars is about 27.7. Let be the arc length of the original circle and be the arc length for the circle when the diameter is doubled. Use the arc length equation with a radius of r and x = 27.7 to find.
Use the arc length equation with a radius of 2 r and x
= 27.7 to find.
The arc length for the second circle is twice the arc length for the first circle. Therefore, if the diameter of the circle is doubled, the arc length from the center of star B to the center of the next star C would double. ANSWER: The length of the arc would double.
SOLUTION:
The radius of circle B is 0.5 meters and the length of arc CD is 1.31 meters. Use the arc length equation and solve for x to find the measure of arc CD.
Therefore, the measure of arc CD is about 150o.
ANSWER:
- radius of SOLUTION: The measure of major arc JL is 340 and its arc length is 56.37 feet. Use the arc length equation to solve for the radius of circle K.
Therefore, the radius of circle K is about 9.50 feet.
ANSWER:
9.50 ft ALGEBRA In , m ∠ HCG = 2 x and m ∠ HCD = 6 x + 28. Find each measure.
SOLUTION: Here, ∠ HCG and ∠ HCD form a linear pair. So, the sum of their measures is 180.
So, m ∠ HCG = 2(19) or 38 and m ∠ HCD = 6(19) +
28 or 142.
is a diameter of circle C, so arc HFE is a
semicircle and has a measure of 180.
Therefore, the measure of arc EF is 52. ANSWER: 52
SOLUTION: Here, ∠ HCG and ∠ HCD form a linear pair. So, the sum of their measures is 180.
So, m ∠ HCG = 2(19) or 38 and m ∠ HCD = 6(19) +
28 or 142.
The measure of an arc is equal to the measure of its
related central angle.
Therefore, m (arc HD ) = m ∠ HCD or 142.
ANSWER:
SOLUTION:
Here, ∠ HCG and ∠ HCD form a linear pair. So, the sum of their measures is 180.
So, m ∠ HCG = 2(19) or 38 and m ∠ HCD = 6(19) +
28 or 142.
Use the Arc Addition Postulate to find the measure
of arc HGF.
Therefore, the measure of arc HGF is 128. ANSWER: 128
- RIDES A pirate ship ride follows a semicircular path , as shown in the diagram. a. What is? b. If CD = 62 feet , what is the length of? Round to the nearest hundredth. SOLUTION: a. From the figure, is 22 + 22 = 44º less than the semi circle centered at C. Therefore, b. The length of an arc is given by the formula, where x is the central angle of the arc and r is the radius of the circle. Here, x = 136 and r = 62. Use the formula. ANSWER: a. 136 b. 147.17 ft
Use the arc length equation with r = 13 and x =
m (arc JL ) or 67.4.
Therefore, the length of arc JL is about 15.29 units.
e. Use the arc length equation with r = 13 and x = m (arc JK ) or 44.8.
Therefore, the length of arc JK is about 10.16 units.
ANSWER:
a. 67. b. 22. c. 44. d. 15.29 units e. 10.16 units
ARC LENGTH AND RADIAN MEASURE In
this problem, you will use concentric circles to show
that the length of the arc intercepted by a central
angle of a circle is dependent on the circle’s radius.
a. Compare the measures of arc and arc.
Then compare the lengths of arc and arc.
What do these two comparisons suggest?
b. Use similarity transformations (dilations) to
explain why the length of an arc intercepted by a
central angle of a circle is proportional to the circle’s
radius r. That is, explain why we can say that for this
diagram,.
c. Write expressions for the lengths of arcs and
. Use these expressions to identify the constant of
proportionality k in = kr.
d. The expression that you wrote for k in part c
converts the degree measure of an angle to the
radian measure of an angle. Use it to find the
radian measure of an angle measuring 90°.
SOLUTION:
a. By definition, the measure of an arc is equal to the measure of its related central angle. So, m (arc ) = x and m (arc ) = x. By the transitive property, m (arc ) = m (arc ). Using the arc length equation, = and =. Since , then <. These comparisons suggest that arc measure is not affected by the size of the circle, but arc length is affected.
b. Since all circles are similar, the larger circle is a
dilation of the smaller by some factor k , so
or. Likewise, the arc intercepted on the
larger circle is a dilation of the arc intercepted on the
smaller circle, so or. Then
substitute for k.
c. = or and = or ; Therefore, for = kr,
d. For an angle measuring 90, x = 90. Then, k =
or.
ANSWER:
a. m (arc ) = m (arc ); < ; these comparisons suggest that arc measure is not affected by the size of the circle, but arc length is affected. b. Since all circles are similar, the larger circle is a dilation of the smaller by some factor k , so or. Likewise, the arc intercepted on the larger circle is a dilation of the arc intercepted on the smaller circle, so or. Thus or. c. and ; d.
- ERROR ANALYSIS Brody says that and are congruent since their central angles have the same measure. Selena says they are not congruent. Is either of them correct? Explain your reasoning. SOLUTION: Brody has incorrectly applied Theorem 10.1. The arcs are congruent if and only if their central angles are congruent and the arcs and angles are in the same circle or congruent circles. The circles containing arc WX and arc YZ are not congruent because they do not have congruent radii. The arcs will have the same degree measure but will have different arc lengths. So, the arcs are not congruent. Therefore, Selena is correct. ANSWER: Selena; the circles are not congruent because they do not have congruent radii. So, the arcs are not congruent. CONSTRUCT ARGUMENTS Determine whether each statement is sometimes , always , or never true. Explain your reasoning.
- The measure of a minor arc is less than 180. SOLUTION: By definition, an arc that measures less than 180 is a minor arc. Therefore, the statement is always true. ANSWER: Always; by definition, an arc that measures less than 180 is a minor arc.
- If a central angle is obtuse, its corresponding arc is a major arc. SOLUTION: Obtuse angles intersect arcs between and. So, the corresponding arc will measure less than . Therefore, the statement is never true. ANSWER: Never; obtuse angles intersect arcs between and .
- The sum of the measures of adjacent arcs of a circle depends on the measure of the radius. SOLUTION: Postulate 10.1 says that the measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. The measure of each arc would equal the measure of its related central angle. The radius of the circle does not depend on the radius of the circle. Therefore, the statement is never true. ANSWER: Never; the sum of the measures of adjacent arcs depends on the measures of the arcs.
63. CHALLENGE The time shown on an analog clock
is 8:10. What is the measure of the angle formed by the hands of the clock? SOLUTION: At 8:00, the minute hand of the clock will point at 12 and the hour hand at 8. At 8:10, the minute hand will point at 2 and the hour hand will have moved of the way between 8 and 9. The sum of the measures of the central angles of a circle with no interior points in common is 360. The numbers on an analog clock divide it into 12 equal arcs and the central angle related to each arc between consecutive numbers has a measure of or 30. To find the measure of the angle formed by the hands, find the sum of the angles each hand makes with 12. At 8:10, the minute hand is at 2 which is two arcs of 30 from 12 or a measure of 60.
When the hour hand is on 8, the angle between the
hour hand and 12 is equal to four arcs of 30 or 120.
At 8:10, the hour hand has moved of the way
from 8 to 9. Since the arc between 8 and 9
measures 30, the angle between the hand and 8 is
(30) or 5. The angle between the hour hand and 12
is 120 – 5 or 115.
The sum of the measures of the angles between the
hands and 12 is 60 + 115 or 175.
Therefore, the measure of the angle formed by the
hands of the clock at 8:10 is 175.
ANSWER:
- WRITING IN MATH Describe the three different types of arcs in a circle and the method for finding the measure of each one. SOLUTION: Minor arc, major arc, semicircle; the measure of a minor arc equals the measure of the corresponding central angle. The measure of a major arc equals 360 minus the measure of the minor arc with the same endpoints. The measure of a semicircle is 180 since it is an arc with endpoints that lied on a diameter. ANSWER: Sample answer: Minor arc, major arc, semicircle; the measure of a minor arc equals the measure of the corresponding central angle. The measure of a major arc equals 360 minus the measure of the minor arc with the same endpoints. The measure of a semicircle is 180.
- A model train runs on a circular track with a diameter of 4 meters, as shown. Which of the following is the best estimate of the distance the train travels as it moves from the station to the grain silo? A 12.6 m B 8.7 m C 4.4 m D 2.2 m SOLUTION: The circumference of the track is 4π m, or about 12.57 m. From the station to the grain silo is of the circumference. Then , so choice C is correct. ANSWER: C
- In ⊙ J , ∠ KJL ≅ ∠ LJM ≅ ∠ MJN. Sofia wants to calculate the length of. Which expression can Sofia use to find the required length? A B C D E SOLUTION: Since ∠ KJL ≅ ∠ LJM ≅ ∠ MJN, u se this relationship to determine the measure of each angle. KN is a diameter so, the sum of the measures of these three angles is 180. Solve for x, as shown in the figure below. Therefore, m ∠ KJL = m ∠ LJM = m ∠ MJN = 60_._ The length of would correspond to the measure of minor arc , which is 120. The circumference of the circle is. So, the length of =. The correct choice is D. ANSWER: D
