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10-2 Measuring Angles and Arcs
Pre-Calculus
Standard
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Find the value of x.
1.
SOLUTION:
The sum of the measures of the central angles of a circle with no interior points in common is 360.
ANSWER:
170
2.
SOLUTION:
The sum of the measures of the central angles of a circle with no interior points in common is 360.
ANSWER:
150
are diameters of
3.
SOLUTION:
is the longest arc connecting the points Therefore, it is a major arc.
CCSS PRECISION and are diameters of . Identify each arc as a major arc, minor arc, or semicircle. Then find its measure.
3.
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
4.
SOLUTION:
Here, is the shortest arc connecting the points I and H on Therefore, it is a minor arc.
ANSWER:
minor arc; 59
5.
SOLUTION:
is a diameter. Therefore, is a
ANSWER:
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10 - 2 Measuring Angles and Arcs
ANSWER:
5.
SOLUTION:
Here, is a diameter. Therefore, is a semicircle. The measure of a semicircle is 180, so
ANSWER:
semicircle; 180
- RESTAURANTS The graph shows the results of a
SOLUTION:
is a minor arc.
is a minor arc.
ANSWER:
ANSWER:
- RESTAURANTS The graph shows the results of a survey taken by diners relating what is most important about the restaurants where they eat.
a. Find.
b. Find. c. Describe the type of arc that the category Great Food represents.
SOLUTION: a. Here, is a minor arc. The measure of the arc is equal to the measure of the central angle. Find the 22% 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 8% of 360 to find the central angle.
c. The arc that represents the category Great Food , is the longest arc connecting the points C and D. Therefore, it is a major arc.
ANSWER: a. 79. b. 28. c. major arc
7.
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10 - 2 Measuring Angles and Arcs
Find the value of x.
12.
SOLUTION:
The sum of the measures of the central angles of a circle with no interior points in common is 360.
ANSWER:
80
13.
SOLUTION:
The sum of the measures of the central angles of a circle with no interior points in common is 360.
ANSWER:
225
14.
SOLUTION:
circle with no interior points in common is 360.
ANSWER:
ANSWER:
14.
SOLUTION:
The sum of the measures of the central angles of a circle with no interior points in common is 360.
ANSWER:
35
15.
SOLUTION:
The sum of the measures of the central angles of a circle with no interior points in common is 360.
ANSWER:
40
16.
SOLUTION:
is the shortest arc connecting the points eSolutions Manual - Powered by Cognero Therefore, it is a minor arc. Page 4
10 - 2 Measuring Angles and Arcs
ANSWER:
and are diameters of. Identify each arc as a major arc, minor arc, or semicircle. Then find its measure.
16.
SOLUTION:
Here, is the shortest arc connecting the points C and D on Therefore, it is a minor arc.
ANSWER:
minor arc; 55
17.
SOLUTION:
Here, is the shortest arc connecting the points A and C on Therefore, it is a minor arc.
Since is a diameter, arc ACD is a semicircle and has a measure of 180. Use angle addition to find the measure of arc AC.
Therefore, the measure of arc AC is 125.
ANSWER: minor arc; 125
18. m(arc CFG)
SOLUTION:
Here, is a diameter. Therefore, arc CFG is a semicircle and m(arc CFG) = 180.
ANSWER: semicircle; 180
SOLUTION:
is the longest arc connecting the points Therefore, it is a major arc.
is a diameter. Therefore, arc is a
ANSWER:
19.
SOLUTION:
Here, is the longest arc connecting the points C and D on Therefore, it is a major arc. Arc CGD is a major arc that shares the same endpoints as minor arc CD.
Therefore, the measure of arc CGD is 305.
ANSWER:
major arc; 305
20.
SOLUTION:
Here, is the longest arc connecting the points G and F on Therefore, it is a major arc. Arc GCF shares the same endpoints as minor arc GF.
Therefore, the measure of arc GCF is 325.
ANSWER: major arc; 325
21.
SOLUTION:
Here, is a diameter. Therefore, is a semicircle. The measure of a semicircle is 180, so m(arc ACD) = 180.
ANSWER: semicircle; 180
22.
SOLUTION:
is the shortest arc connecting the points Therefore, it is a minor arc.
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10 - 2 Measuring Angles and Arcs
- CCSS MODELING The table shows the results of a survey in which Americans were asked how long food could be on the floor and still be safe to eat. 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 “not so safe” category contributes 78% and the “three-second rule” contributes 10% in the supporters in the survey. Find the 78% of 360 to find the central angle of the arc associated with the not so safe category.
Find the 10% of 360 to find the central angle of the arc associated with the three-second rule category.
b. The arc corresponding to not safe to eat category measures 280, so it is a major arc. Similarly, the arc corresponding to the ten-second rule measures 4% of 360 or 14, so it is a minor arc. c. No; no categories share the same percentage of the circle.
ANSWER: a. 280; 36 b. major arc; minor arc c. No; no categories share the same percentage of the circle.
ENTERTAINMENT Use the Ferris wheel
ENTERTAINMENT Use the Ferris wheel shown to find each measure.
26.
SOLUTION:
The measure of the arc is equal to the measure of the central angle. We have, Therefore,
ANSWER: 40
27.
SOLUTION:
The measure of the arc is equal to the measure of the central angle. We have, Therefore,
ANSWER: 60
28.
SOLUTION:
Here, is a diameter. Therefore, is a semicircle and
ANSWER: 180
29.
SOLUTION:
ANSWER:
30.
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10 - 2 Measuring Angles and Arcs
is a diameter. Therefore, is a
ANSWER:
29.
SOLUTION:
Arc JFH is a major arc.
Therefore, the measure of arc JFH is 300.
ANSWER: 300
30.
SOLUTION:
Arc GHF is a major arc.
Therefore, the measure of arc GHF is 320.
ANSWER: 320
31.
SOLUTION:
Here, is a diameter. Therefore, is a semicircle and
ANSWER: 180
32.
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
33.
SOLUTION:
ANSWER:
33.
SOLUTION:
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. is a diameter, so arc JF is a semicircle and has a measure of 180.
Therefore, the measure of arc JKG is 220.
ANSWER: 220
34.
SOLUTION:
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. and are diameters, so arc KG and arc FJ are semicircles with measures of 180. Find m∠GLH and then use the Arc Addition Postulate.
Therefore, the measure of arc KFH is 260.
ANSWER:
260
35.
SOLUTION:
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Use semicircle FJ to find m∠HLG and then use the Arc Addition Postulate.
Therefore, the measure of arc HGF is 120.
ANSWER:
120
eSolutions Manual - Powered by Cognero Use to find the length of each arc. Round toPage 8
10 - 2 Measuring Angles and Arcs
ANSWER:
- ,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 feet.
ANSWER: 19 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 m
HISTORY The figure shows the stars in the
42.
SOLUTION:
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
- If the diameter of the circle were doubled, what
SOLUTION:
be the arc length for the circle when
Therefore, if the diameter of the circle is doubled,
ANSWER:
44.
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- 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. 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 to find.
Use the arc length equation with a radius of 2r and
x = 27 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.
- FARMS The in Madera California is a
“ ”
SOLUTION:
ANSWER:
- FARMS The Pizza Farm in Madera, California, is a circle divided into eight equal slices, as shown at the right. Each “slice” is used for growing or grazing pizza ingredients. a. What is the total arc measure of the slices containing olives, tomatoes, and peppers? b. The circle is 125 feet in diameter. What is the arc length of one slice? Round to the nearest hundredth.
SOLUTION:
a. The circle is divided into eight equal slices. So, the measure of the central angle of each slice will be
Therefore, total arc measure of the slices
containing olives, tomatoes, and peppers will be 3(45) = 135. b. The length of an arc l is given by the formula,
where x is the central angle of the arc
l and r is the radius of the circle. The measure of the central angle of each slice will
be So, x = 45 and r 62 ft. Then,
ANSWER:
a. 135 b. 49 ft
CCSS REASONING Find each measure. Round
45.
SOLUTION:
.
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10 - 2 Measuring Angles and Arcs
49.
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:
142
50.
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 semi-circular
SOLUTION:
is 22 + 22 = 44º
ANSWER:
- RIDES A pirate ship ride follows a semi-circular 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 l is given by the formula,
where x is the central angle of the arc
l and r is the radius of the circle. Here, x = 136 and r = 62. Use the formula.
ANSWER:
a. 136 b. 147 ft
- PROOF Write a two-column proof of Theorem
SOLUTION:
(Definition of (Definition of
(Substitution)
ANSWER:
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10 - 2 Measuring Angles and Arcs
- PROOF Write a two-column proof of Theorem
- Given: Prove:
SOLUTION:
Proof: Statements (Reasons)
- (Given)
- (Definition of )
- , (Definition of arc measure)
- (Substitution)
- (Definition of arcs)
ANSWER: Proof: Statements (Reasons)
(Given)
(Definition of )
, (Definition of arc measure)
(Substitution)
(Definition of arcs)
COORDINATE GEOMETRY In the graph, point M is located at the origin. Find each measure in . Round each linear measure to the nearest hundredth and each arc measure to the nearest tenth degree.
a. b. c. d. length of
SOLUTION:
d. length of e. length of
SOLUTION:
a. The measure of arc JL equals the measure of the
related central angle JML. Construct a right triangle
by drawing and a perpendicular segment from
J to the x-axis. The legs of the right triangle will
have lengths of 5 and 12.
Use a trigonometric ratio to find the angle of the triangle at M which equals the measure of central angle JML.
Therefore,
b. The measure of arc KL equals the measure of the
related central angle KML. Construct a right
triangle by drawing and a perpendicular
segment from K to the x-axis. The legs of the right
triangle will have lengths of 12 and 5.
Use a trigonometric ratio to find the angle of the triangle at M which equals the measure of central angle KML.
Therefore,
c. Use the Arc Addition Postulate to find m(arc
JK).
So, the measure of arc JK is 44.
d. Use the right triangle from part a and find the
length of which is a radius of circle M.
Use the arc length equation with r = 13 and x = m
(arc JL) or 67.
Therefore, the length of arc JL is about 15 units.
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10 - 2 Measuring Angles and Arcs
- 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. 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.
CCSS ARGUMENTS Determine whether each statement is sometimes, always, or never true. Explain your reasoning. 56. 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 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.
- CHALLENGE The measures of , , and are in the ratio 5:3:4. Find the measure of each arc.
SOLUTION:
If the measures of arc LM,arc MN ,and arc NL are in the ratio 5:3:4, then their measures are a multiple, x, of these numbers. So, m(arc LM) = 5x, m(arc MN) = 3x, and m(arc MN) = 4x. The arcs are adjacent and form the entire circle, so their sum is 360.
Therefore, m(arc LM) = 5(30) or 150, m(arc MN)
= 3(30) or 90, and m(arc NL) = 4(30) or 120.
ANSWER:
- OPEN ENDED Draw a circle and locate three
SOLUTION:
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10 - 2 Measuring Angles and Arcs
- OPEN ENDED Draw a circle and locate three points on the circle. Estimate the measures of the three nonoverlapping arcs that are formed. Then use a protractor to find the measure of each arc. Label your circle with the arc measures.
SOLUTION: Sample answer:
ANSWER:
Sample answer:
- CHALLENGE The time shown on an analog clock
SOLUTION:
- 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:
175
- WRITING IN MATH Describe the three different
SOLUTION:
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10 - 2 Measuring Angles and Arcs
.
ANSWER:
- SAT/ACT What is the area of the shaded region if r = 4?
A
B
C
D
E
SOLUTION:
A square with sides of 8 units can be drawn using the centers of the four circles as vertices. The shaded area is the difference between the areas of the square and the total of the areas of the four quarter sections of the circles formed by the square.
Find the area of the square and the quarter section of the circles.
Subtract the area of the four quarter circle sections from the square to find the area of the shaded region.
Therefore, the correct choice is A.
ANSWER: A
Refer to.
67.
ANSWER:
Refer to.
- Name the center of the circle.
SOLUTION: Since the circle is named circle J, it has a center at J.
ANSWER: J
- Identify a chord that is also a diameter.
SOLUTION: A diameter of a circle is a chord that passes through the center and is made up of collinear radii.
passes through the center, so it is a diameter.
ANSWER:
- If LN = 12, what is JM?
SOLUTION: Here, JM is a radius and LN is a diameter. The radius is half the diameter. Therefore,
ANSWER:
6.
Graph the image of each polygon with the given
70. – – –
SOLUTION:
) → (
1, 2) → ’
(2, 1) → ’
2) → ’
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10 - 2 Measuring Angles and Arcs
Graph the image of each polygon with the given vertices after a dilation centered at the origin with the given scale factor. 70. X(– 1 , 2), Y(2, 1), Z(– 1 , – 2); r = 3
SOLUTION: Multiply the x- and y-coordinates of each vertex by the scale factor k. That is, (x, y) → (k x, k y). Here multiply the x- and y-coordinates by the scale factor 3. X(-1, 2) → X’(-3, 6) Y(2, 1) → Y’(6, 3) Z(-1, -2) → Z’(-3, -6)
ANSWER:
- A(– 4 , 4), B(4, 4), C(4, – 4), D(– 4 , – 4); r = 0.
SOLUTION:
) → (
→ ’
→ ’
→ ’
→ ’
- A(– 4 , 4), B(4, 4), C(4, – 4), D(– 4 , – 4); r = 0.
SOLUTION: Multiply the x- and y-coordinates of each vertex by the scale factor k. That is, (x, y) → (k x, k y). Here multiply the x- and y- coordinates by the scale factor 0.
A(-4, 4) → A’(-1, 1)
B(4, 4) → B’(1, 1)
C(4, -4) → C’(1, -1)
D(-4, -4) → D’(-1, -1)
ANSWER:
- BASEBALL The diagram shows some dimensions is a
SOLUTION:
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10 - 2 Measuring Angles and Arcs
10-2 Measuring Angles and Arcs
Subject: Pre-Calculus
Standard
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