GUIDED PRACTICE
Vocabulary Check
✓
Concept Check
✓
3 APPLY
1. Sketch a circle. Then sketch and label a radius, a diameter, and a chord.
See margin.
2. How are chords and secants of circles alike? How are they different? See margin.
BASIC
¯
˘
3. XY is tangent to ›C at point P. What is m™CPX? Explain. See margin.
Day 1: pp. 599–600 Exs. 9–35
Day 2: pp. 600–602 Exs. 36–49,
54–56, 58–70 even
4. The diameter of a circle is 13 cm. What is the radius of the circle? 6.5 cm
Skill Check
✓
5. In the diagram at the right, AB = BD = 5 and
B
AVERAGE
¯
˘
AD = 7. Is BD tangent to ›C? Explain.
2. Both intersect a › in 2
points; a secant is a line,
a chord is a segment with
endpoints on the ›.
Day 1: pp. 599–600 Exs. 9–35
Day 2: pp. 600–602 Exs. 36–49,
54–56, 58–70 even
C
See margin.
D
A
¯
˘
ADVANCED
¯
˘
3. 90°; if a line is tangent to AB is tangent to ›C at A and DB is tangent to ›C at D. Find the value of x.
a ›, then it is fi to the
A
6. 4
7. 2
8. 5
A
B
2
A
radius drawn to the point
2x
of tangency.
C
x
B
C
x
C
4
D
B
D
STUDENT HELP
FINDING RADII The diameter of a circle is given. Find the radius.
Extra Practice
to help you master
skills is on p. 821.
9. d = 15 cm
10. d = 6.7 in.
11. d = 3 ft
12. d = 8 cm
3.35 in.
4 cm
7.5 cm
1.5 ft
FINDING DIAMETERS The radius of ›C is given. Find the diameter of ›C.
45
2
13. r = 26 in.
14. r = 62 ft
15. r = 8.7 in.
16. r = 4.4 cm
124 ft
8.8 cm
52 in.
17.4 in.
17. CONGRUENT CIRCLES Which two circles below are congruent? Explain
your reasoning.
radius is }} = 22.5,
22.5
which is the radius of
›C.
C
22
45
D
G
MATCHING TERMS Match the notation with the term that best describes it.
Æ
18. AB B
A. Center
STUDENT HELP
19. H E
B. Chord
HOMEWORK HELP
20. HF F
Example 1: Exs. 18–25,
42–45
Example 2: Exs. 26–31
Example 3: Exs. 32–35
Example 4: Exs. 36–39
Example 5: Exs. 40, 41
Example 6: Exs. 49–53
Example 7: Exs. 46–48
¯
˘
Æ
C. Diameter
21. CH D
D. Radius
22. C A
E. Point of tangency
Æ
23. HB C
¯
˘
24. AB H
¯
˘
25. DE G
A
COMMON ERROR
EXERCISES 18 AND 24
Students may confuse the notation
for chords and secants. Emphasize
that a secant is a line, while a chord
is a segment. A
苶B
苶 is a chord, while
fl°·
AB is a secant.
D
H
F. Common external tangent
H. Secant
HOMEWORK CHECK
To quickly check student understanding of key concepts, go over
the following exercises: Exs. 14,
26, 30, 36, 44, 46, 49, 55. See also
the Daily Homework Quiz:
• Blackline Master (Chapter 10
Resource Book, p. 26)
•
Transparency (p. 72)
B
C
G. Common internal tangent
BLOCK SCHEDULE
EXERCISE LEVELS
Level A: Easier
9–28
Level B: More Difficult
29–52, 54–56
Level C: Most Difficult
53, 57
PRACTICE AND APPLICATIONS
17. C and G; the diameter
of ›G is 45, so the
Day 1: pp. 599–600 Exs. 9–35
Day 2: pp. 600–602 Exs. 36–57,
58–70 even
pp. 599–602 Exs. 9–49, 54–56,
58–70 even
10
D
ASSIGNMENT GUIDE
E
G
F
10.1 Tangents to Circles
1, 5. See Additional Answers beginning on page AA1.
599
599
MATHEMATICAL REASONING
EXERCISE 39 The following
statement and its converse are
both true: A tangent is perpendicular to a radius drawn to the point of
tangency. Ask students to rewrite
the statement as a biconditional.
A tangent is perpendicular to a
radius if and only if the radius is
drawn to the point of tangency.
TEACHING TIPS
EXERCISE 40 Remind students
that they need to square a binomial
to solve this problem. Example 5 on
page 597 is a similar problem.
APPLICATION NOTE
EXERCISES 40–41 Golf is
played in many countries throughout the world. Professional golfers
come not only from the United
States, but also from Great Britain,
Spain, Japan, Mexico, and Fiji.
The Ryder Cup, an international
golf championship, is held every
other year.
36. No; 52 + 142 ≠ 152, so
by the Converse of the
Pythagorean Thm.,
¤ABC is not a right ¤,
Æ
Æ
so AB is not fi to AC .
¯˘
Then AB is not tangent
to ›C.
37. No; 52 + 152 ≠ 172, so
by the Converse of the
Pythagorean Thm.,
¤ABC is not a right ¤,
Æ
Æ
so AB is not fi to AC .
¯˘
Then AB is not tangent
to ›C.
38. Yes; BC = 12 + 8 = 20
and 122 + 162 = 202, so
by the Converse of the
Pythagorean Thm.,
¤ABC is a right ¤, and
¯˘
Æ
Æ
AB fi AC . Then AB is
tangent to ›C.
39. Yes; BD = 10 + 10 = 20
and 202 + 212 = 292, so
by the Converse of the
Pythagorean Thm.,
¤ABD is a right ¤, and
¯˘
Æ
Æ
AB fi BD . Then AB is
tangent to ›C.
29. 2 internal, 2 external
IDENTIFYING TANGENTS Tell whether the common tangent(s) are internal
or external.
external
internal
internal
26.
27.
28.
DRAWING TANGENTS Copy the diagram. Tell how many common tangents
the circles have. Then sketch the tangents. See margin.
29.
30.
31.
COORDINATE GEOMETRY Use the diagram at the right.
y
32. What are the center and radius of ›A?
(2, 2), 2
33. What are the center and radius of ›B?
(6, 2), 2
34. Describe the intersection of the two circles.
the single point (4, 2)
35. Describe all the common tangents of the
A
1
B
1
x
two circles.
the lines with equations y = 0, y = 4, and x = 4
¯
˘
DETERMINING TANGENCY Tell whether AB is tangent to ›C. Explain your
reasoning. 36–39. See margin.
36.
37.
A
5
5
C
15
FOCUS ON
PEOPLE
17
C
B
38.
39.
A
30. none
31. 2 external
12
D
C
29
C
10
16
B
8
B
15
A
14
A
21
B
GOLF In Exercises 40 and 41, use the following information.
RE
FE
L
AL I
TIGER WOODS At
age 15 Tiger Woods
became the youngest golfer
ever to win the U.S. Junior
Amateur Championship, and
at age 21 he became the
youngest Masters champion
ever.
600
600
A green on a golf course is in the shape of a circle. A golf
ball is 8 feet from the edge of the green and 28 feet from
a point of tangency on the green, as shown at the right.
Assume that the green is flat.
28
40. What is the radius of the green? 45 ft
41. How far is the golf ball from the cup at the center?
53 ft
Chapter 10 Circles
8
MEXCALTITLÁN The diagram shows the
layout of the streets on Mexcaltitlán Island.
42. Name two secants.
¯
˘
¯
˘
J
COMMON ERROR
EXERCISE 53 Students often
start an indirect proof with the
negation of one of the given statements. Emphasize that the given
statements are never changed.
They should start by supposing line
l is not tangent to 䉺Q.
A
F
E
B
C
D
45. If ¤LJK were drawn, one of its sides would be
¯
˘
tangent to the circle. Which side is it? JK
L
54.
¯
˘
¯
˘
xy USING ALGEBRA AB and AD are tangent to ›C. Find the value of x.
44. Yes; the diameter is the
46. 5
47.
B
longest chord of the ›. (Let
2x ⴙ 7
º1, 1
P be the center of the › and
consider ¤HPC. By the ¤
A
C
Inequality, HP + CP > HC.
5x ⴚ 8
But HP = r and CP = r, so
D
HP + CP = 2r = the
diameter of the ›.)
¯
˘
PROOF Write a proof.
49. PS is tangent to ›X at P, 49.
¯
˘
H
FA and EB
43. Name two chords. Æ Æ Æ Æ
any two of GD , HC , FA , EB
44. Is the diameter of the circle greater than HC?
Explain. See margin.
Mexcaltitlán Island,
Mexico
G
K
B
5x 2 ⴙ 9
A
48.
º2, 2
C
2x ⴙ 5
B
14
A
55. Square; B
苶D
苶 and A
苶D
苶 are tangent
to çC at A and B, respectively,
so åA and åB are right ås .
Then by the Interior Angles of a
Quadrilateral Thm., åD is also a
right å. Then CABD is a rectangle. Opp. sides of a ¥ are •, so
C苶A
苶• B
苶D
苶 and A
苶D
苶 • C苶B
苶. But C苶A
苶 and
C苶B
苶 are radii, so C苶A
苶 • C苶B
苶 and by
the Transitive Prop. of Cong., all
4 sides of CABD are •. CABD is
both a rectangle and a rhombus,
so it is a square by the Square
Corollary.
57. See Additional Answers beginning on page AA1.
B 3x 2 ⴙ 2x ⴚ 7
D
R
P
q
¯
˘
PS is tangent to ›Y at S,
GIVEN 䉴 PS is tangent to ›X at P.
X
¯
˘
¯
˘
RT is tangent to ›X at T,
PS is tangent to ›Y at S.
¯
˘
¯
˘
and RT is tangent to ›Y at
Æ
Æ
Æ
RT is tangent to ›X at T.
R. Then PQ £ TQ and QS £
¯
˘
Æ
QR . (2 tangent segments
RT is tangent to ›Y at R.
with the same ext. endpoint
Æ
Æ
PROVE 䉴 PS £ RT
are £.) By the def. of cong.,
PQ = TQ and QS = QR, so
PQ + QS = TQ + QR by
PROVING THEOREM 10.1 In Exercises 50–52,
the addition prop. of
you will use an indirect argument to prove
equality. Then by the
Segment Addition Post. and Theorem 10.1.
the Substitution Prop.,
GIVEN 䉴 l is tangent to ›Q at P.
Æ
Æ
PS = RT or PS £ RT .
Æ
52. QR cannot be both greater PROVE 䉴 l fi QP
T
Y
S
q
l
P R
Æ
than QP and less than QP, 50. Assume l and QP
are not perpendicular. Then the perpendicular segment
so the assumption that l
from Q to l intersects l at some other point R. Because l is a tangent, R
Æ
and QP are not fi must be
Æ
cannot be in the interior of ›Q. So, how does QR compare to QP? Write
false. Then l fi QP .
an inequality. QR > QP
53. Sample proof: Assume that l
Æ
Æ
is not tangent to P, that is, 51. QR
is the perpendicular segment from Q to l, so QR is the shortest segment
there is another point X on
from Q to l. Write another inequality comparing QR to QP. QR < QP
l that is also on ›Q. X is on
›Q so QX = QP. But the fi
52. Use your results from Exercises 50 and 51 to complete the indirect proof of
segment from Q to l is the
Theorem 10.1. See margin.
shortest such segment, so
QX > QP. QX cannot be both
PROVING THEOREM 10.2 Write an indirect proof of Theorem 10.2.
equal to and greater than 53.
QP. The assumption that
(Hint: The proof is like the one in Exercises 50–52.) See margin.
such a point X exists must
GIVEN 䉴 l is in the plane of ›Q.
be false. Then l is tangent
q
Æ
to P.
l fi radius QP at P.
PROVE 䉴 l is tangent to ›Q.
D
D
C
C
A
ADDITIONAL PRACTICE
AND RETEACHING
For Lesson 10.1:
l
P
10.1 Tangents to Circles
601
• Practice Levels A, B, and C
(Chapter 10 Resource Book,
p. 16)
• Reteaching with Practice
(Chapter 10 Resource Book,
p. 19)
•
See Lesson 10.1 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
601
Æ
4 ASSESS
Æ Æ Æ
55. What type of quadrilateral is CADB? Explain. See margin on page 601.
Transparency Available
The radius of 䉺C is given.
Find the diameter of 䉺C.
1. 13 ft 26 ft
2. 3.2 in. 6.4 in.
Test
Preparation
56. MULTI-STEP PROBLEM In the diagram, line j is tangent to ›C at P.
Æ
a. What is the slope of radius CP?
d.
y
★ Challenge
C
B
yes
A
苶B
苶 and A
苶D
苶 are tangent to 䉺C.
Find the value of x.
B
4.
5x ⫹ 5
Writing Explain how to find an equation for
j
C(4, 5)
P (8, 3)
2
a line tangent to ›C at a point other than P.
80
A
1
2
º }}
b. What is the slope of j? Explain. 2; perpendicular
lines have slopes that are opposite reciprocals.
c. Write an equation for j. y = 2x º 13
Tell whether A
苶B
苶 is tangent to 䉺C.
A
3.
82
Æ
54. Sketch ›C, CA, CB, BD, and AD. See margin on page 601.
DAILY HOMEWORK QUIZ
18
¯
˘
Æ
LOGICAL REASONING In ›C, radii CA and CB are perpendicular. BD
¯
˘
and AD are tangent to ›C.
2
x
See margin.
57. CIRCLES OF APOLLONIUS The Greek mathematician Apollonius
56. d. Let Q be the point onÆ
›C. Find the slope of CQ .
Take the opposite
reciprocal of this slope to
get the slope of the
tangent. Use this slope and
the coordinates for Q to
get the equation.
(c. 200 B.C.) proved that for any three circles with no common points or
common interiors, there are eight ways to draw a circle that is tangent to the
given three circles. The red, blue, and green circles are given. Two ways to
draw a circle that is tangent to the given three circles are shown below.
Sketch the other six ways. See margin on page 601.
C
EXTRA CHALLENGE
10(x ⫺ 3)
D
www.mcdougallittell.com
7
MIXED REVIEW
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 10.1 are available in
blackline format in the Chapter 10
Resource Book, p. 23 and at
www.mcdougallittell.com.
ADDITIONAL TEST
PREPARATION
1. OPEN ENDED Draw a pair of
circles with exactly 3 common
tangents. Show the tangents.
Sample answer:
Æ
3
59. Since slope of PS = }} = 58. TRIANGLE INEQUALITIES The lengths of two sides of a triangle are 4 and 10.
8
Use an inequality to describe the length of the third side. (Review 5.5)
Æ Æ Æ
slope of QR , PS ∞ QR .
Let x be the length of the third side; 6 < x < 14.
Æ
PARALLELOGRAMS Show that the vertices represent the vertices of a
=
Since slope of PQ
Æ
parallelogram. Use a different method for each proof. (Review 6.3)
º3
= slope of SR ,
Æ Æ
59–60. Sample answers are given.
PQ ∞ SR . Then PQRS is
59.
P(5,
0),
Q(2,
9),
R(º6,
6),
S(º3,
º3)
a ⁄ by def.
苶 = QR,
60. Since
PS
= 兹41
60. P(4, 3), Q(6, º8), R(10, º3), S(8, 8)
Æ
Æ
PS £ QR . Since slope
Æ
SOLVING PROPORTIONS Solve the proportion. (Review 8.1)
5
of PS = }} = slope of
4
Æ Æ
Æ
x
x
x
3
9
12
33
18
3
61. ᎏᎏ = ᎏᎏ 6 }5}
62. ᎏᎏ = ᎏᎏ 27
63. ᎏᎏ = ᎏᎏ 28
64. ᎏᎏ = ᎏᎏ 77
QR , PS ∞ QR . Since
11
6
7
5
2
3
x
42
one pair of opp. sides
3
2
5
9
10
8
4
3
2
of PQRS is both ∞ and £, 65. ᎏ
ᎏ = ᎏᎏ 2 }5}
66. ᎏᎏ = ᎏᎏ º8 67. ᎏᎏ = ᎏᎏ 9
68. ᎏᎏ = ᎏᎏ º9
x+2
xº3
xº1
2x
3
x
x
x
PQRS is a ⁄ .
69. m™A ≈ 23.2°, m™C ≈
SOLVING TRIANGLES Solve the right triangle. Round decimals to the
66.8°, AC ≈ 15.2
nearest tenth. (Review 9.6)
70. m™A = 47°, AB ≈ 14.7,
69. A
70. A
71.
C
BC ≈ 10.7
14
B
71. BC ≈ 11.5, m™A ≈ 55.2°,
m™B ≈ 34.8°
6
C
602
602
Chapter 10 Circles
10
8
43ⴗ
C
B
A
14
B