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Chapter 9 – Circles
9.1
Answer Key
Parts of Circles
Answers
1.
diameter
2.
secant
3.
chord
4.
point of tangency
5.
common external tangent
6.
common internal tangent
7.
the center
8.
radius
9.
The diameter is the longest chord in any circle.
10.
ʘB: 4 units, ʘC: 5 units, D: 2 units, .ʘE: 2
11.
ʘD ≅ ʘE because they have the same radius length.
12.
They will have 2 common external tangents and no common internal tangents.
13.
CE = Radius of ʘE + Radius of ʘC = 5 + 2 = 7
14.
y=x−2
CK-12 Basic Geometry Concepts
1
Chapter 9 – Circles
9.2
Answer Key
Tangent Lines
Answers
1.
Yes
2.
No
3.
Yes
4.
𝑥 = 4√10
5.
𝑥 = 4√11
6.
x=9
7.
x=3
8.
x=5
9.
𝑥 = 8√2
10.
Yes, by AA. 𝑚∠𝐶𝐴𝐸 = 𝑚∠𝐷𝐵𝐸 = 90° and ∠𝐴𝐸𝐶 ≅ ∠𝐵𝐸𝐷 by vertical angles.
11.
CE = 25
12.
BE = 12
13.
DE = 15
14.
BC = 37, AD = 35
15.
32 units
16.
BH = BD and DF = FH. So, BDFH is a kite because adjacent sides are congruent.
17.
If the radius of the circle is 5, then the length of each side of the square is 10. The
perimeter would be 4 ∙ 10 = 40.
18.
No, because the opposite sides are congruent and at most a circle can touch three
sides. For a circle to be inscribe in any shape, it needs to touch all the sides of that
shape.
19.
Answers will vary.
20.
Answers will vary.
CK-12 Basic Geometry Concepts
2
Answer Key
Chapter 9 – Circles
21.
Statement
̅̅̅̅
̅̅̅̅
1. 𝐴𝐵 and 𝐶𝐵 with points of tangency at A
and C. ̅̅̅̅
𝐴𝐷 and ̅̅̅̅
𝐷𝐶 are radii.
̅̅̅̅ ≅ 𝐷𝐶
̅̅̅̅
2. 𝐴𝐷
3. ̅̅̅̅
𝐷𝐴 ⊥ ̅̅̅̅
𝐴𝐵 and ̅̅̅̅
𝐷𝐶 ⊥ ̅̅̅̅
𝐶𝐵
4. 𝑚∠𝐵𝐴𝐷 = 90° and 𝑚∠𝐵𝐶𝐷 = 90°
̅̅̅̅.
5. Draw 𝐵𝐷
6. ∆𝐴𝐷𝐵 and ∆𝐷𝐶𝐵 are right triangles
7. ̅̅̅̅
𝐷𝐵 ≅ ̅̅̅̅
𝐷𝐵
8. ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷
̅̅̅̅ ≅ 𝐶𝐵
̅̅̅̅
9. 𝐴𝐵
22.
23.
Reason
Given
All radii are congruent.
Tangent to a Circle Theorem
Definition of perpendicular lines
Connecting two existing points
Definition of right triangles (Step 4)
Reflexive PoC
HL
CPCTC
a)
ABCD is a kite.
b)
The line that connects the center and the external point B bisects ABC.
̅̅̅̅ ≅ 𝐵𝑇
̅̅̅̅ by the Two Tangents Theorem and the transitive property.
̅̅̅̅ ≅ 𝐶𝑇
𝐴𝑇
CK-12 Basic Geometry Concepts
3
Chapter 9 – Circles
9.3
Answer Key
Arcs in Circles
Answers
1.
minor arc
2.
major arc
3.
semicircle
4.
major arc
5.
minor arc
6.
semicircle
7.
Yes, CD  DE because their corresponding central angles are congruent.
8.
mCD  66
9.
mCAE  228
10.
100, 260
11.
175, 185
12.
38, 322
13.
109, 251
14.
124, 236
15.
34, 326
16.
Yes, they are in the same circle with equal central angles.
17.
Yes, the central angles are vertical angles and they are equal, which means the arcs
equal.
18.
No, we don’t know the measure of the corresponding central angles.
19.
90
20.
49
21.
82
22.
16
CK-12 Basic Geometry Concepts
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Chapter 9 – Circles
23.
188
24.
172
25.
196
26.
270
27.
x = 54
28.
x = 47
29.
x = 25
CK-12 Basic Geometry Concepts
Answer Key
5
Chapter 9 – Circles
9.4
Answer Key
Chords in Circles
Answers
1.
̅̅̅̅ ≅ 𝐷𝐹
̅̅̅̅
𝐴𝐶
2.
AC ≅ DF
3.
DJ ≅ JF
4.
̅̅̅̅
𝐷𝐸 ≅ ̅̅̅
𝐸𝐽
5.
AGH ≅ HGC
6.
DGF ≅ AGC
7.
Possible Answers ̅̅̅̅
𝐴𝐺 , ̅̅̅̅
𝐻𝐺 , ̅̅̅̅
𝐶𝐺 , ̅̅̅̅
𝐹𝐺 , ̅̅̅
𝐽𝐺 , ̅̅̅̅
𝐷𝐺
8.
107
9.
8
10.
118
11.
133
12.
140
13.
120
14.
x = 64, y = 4
15.
x = 8, y = 10
16.
x ≈ 15.3, y ≈ 12.3
17.
x ≈ 18.36
18.
x = 9, y = 6
19.
x = 4.5
20.
x=3
21.
x=7
22.
x ≈ 6.63
23.
121.3
CK-12 Basic Geometry Concepts
6
Chapter 9 – Circles
24.
112.9
25.
̅̅̅̅
𝐵𝐹 ≅ ̅̅̅̅
𝐹𝐷 and BF  FD by Chord Theorem #3.
26.
̅̅̅̅
𝐶𝐴 ≅ ̅̅̅̅
𝐴𝐹 by Chord Theorem #4.
27.
̅̅̅̅ is a diameter by Chord Theorem #2.
𝑄𝑆
CK-12 Basic Geometry Concepts
Answer Key
7
Answer Key
Chapter 9 – Circles
9.5
Inscribed Angles in Circles
Answers
1.
semicircle
2.
congruent
3.
chords
4.
J
L
1
2
𝑚∠𝐽𝐾𝐿 = 𝑚∠𝐽𝑀𝐿
M
K
5.
51
6.
46
7.
x = 180, y = 21
8.
x = 60, y = 49
9.
x = 30, y = 60
10.
37
11.
42
12.
6
13.
10
CK-12 Basic Geometry Concepts
8
Answer Key
Chapter 9 – Circles
14.
Statement
1. Inscribed ABC and diameter ̅̅̅̅
𝐵𝐷
mABE = x and mCBE = y
2. x + y = mABC
3. ̅̅̅̅
𝐴𝐸 ≅ ̅̅̅̅
𝐸𝐵 and ̅̅̅̅
𝐸𝐵 ≅ ̅̅̅̅
𝐸𝐶
4. ∆𝐴𝐸𝐵 and ∆𝐸𝐵𝐶 are isosceles
5. mEAB = x and mECB = y
6. mAED = 2x and mCED = 2y
Reason
Given
8. mAD  mDC  mAC
Angle Addition Postulate
All radii are congruent
Definition of an isosceles triangle
Isosceles Triangle Theorem
Exterior Angle Theorem
The measure of an arc is the same as its
central angle.
Arc Addition Postulate
9. mAC = 2x + 2y
Substitution
10. mAC = 2(𝑥° + 𝑦°)
Distributive PoE
11. mAC = 2mABC
Subsitution
7. mAD = 2x and mDC = 2y
1
12. 𝑚∠𝐴𝐵𝐶 = 2 𝑚 AC
CK-12 Basic Geometry Concepts
Division PoE
9
Chapter 9 – Circles
9.6
Answer Key
Inscribed Quadrilaterals in Circles
Answers
1.
inscribed
2.
opposite, supplementary
3.
48
4.
120
5.
54
6.
45
7.
87
8.
27
9.
x = 200, y = 100
10.
x = 68, y = 99
11.
x = 93, y = 97
12.
x = 24
13.
x = 35
CK-12 Basic Geometry Concepts
10
Answer Key
Chapter 9 – Circles
9.7
Angles On and Inside a Circle
Answers
1.
x = 103
2.
x = 100
3.
x = 38
4.
x = 216
5.
x = 66
6.
x = 113
7.
x = 60, y = 40
8.
Statement
1. Intersecting chords ̅̅̅̅
𝐴𝐶 and ̅̅̅̅
𝐵𝐷.
̅̅̅̅
2. Draw 𝐵𝐶
A
D
Reason
Given
Construction
B
a°
C
3. 𝑚∠𝐷𝐵𝐶
1
̂
= 2 𝑚𝐷𝐶
1
̂
= 2 𝑚𝐴𝐵
4. 𝑚∠𝐴𝐶𝐵
5. 𝑚∠𝑎 = 𝑚∠𝐷𝐵𝐶 + 𝑚∠𝐴𝐶𝐵
1
̂ + 1 𝑚𝐴𝐵
̂
6. 𝑚∠𝑎 = 2 𝑚𝐷𝐶
2
9.
inside, intercepted
10.
on, half
CK-12 Basic Geometry Concepts
Inscribed Angle Theorem
Inscribed Angle Theorem
Exterior Angle Theorem
Substitution
11
Answer Key
Chapter 9 – Circles
9.8
Angles Outside a Circle
Answers
1.
x = 63, y = 243
2.
x = 42
3.
x = 150
4.
x = 70
5.
x = 152
6.
x = 180, y = z = 45
7.
x = 180, y = 60
8.
x = 27
9.
x = 5
10.
Statement
1. Intersecting secants ⃗⃗⃗⃗⃗
𝐴𝐵 and ⃗⃗⃗⃗⃗
𝐴𝐶 .
̅̅̅̅
2. Draw 𝐵𝐸 .
B
D
a°
Reason
Given
Construction
A
E
3.
C
1
̂
𝑚∠𝐵𝐸𝐶 = 2 𝑚𝐵𝐶
1
̂
𝑚∠𝐷𝐵𝐸 = 2 𝑚𝐷𝐸
4.
5. 𝑚∠𝑎 + 𝑚∠𝐷𝐵𝐸 = 𝑚∠𝐵𝐸𝐶
6. 𝑚∠𝑎 = 𝑚∠𝐵𝐸𝐶 − 𝑚∠𝐷𝐵𝐸
1
̂ − 1 𝑚𝐷𝐸
̂
7. 𝑚∠𝑎 = 2 𝑚𝐵𝐶
2
1
̂ − 𝑚𝐷𝐸
̂)
8. 𝑚∠𝑎 = (𝑚𝐵𝐶
2
CK-12 Basic Geometry Concepts
Inscribed Angle Theorem
Inscribed Angle Theorem
Exterior Angle Theorem
Subtraction PoE
Substitution
Distributive Property
12
Answer Key
Chapter 9 – Circles
9.9
Segments from Chords
Answers
1.
102, x = 5
2.
5, 10, x = 2
3.
x = 7.5
4.
x = 6√2
5.
x = 10
6.
x = 10
7.
x=9
8.
x=4
9.
10 inches
10.
Statement
̅̅̅̅ and ̅̅̅̅
1. Intersecting chords 𝐴𝐶
𝐵𝐸 with
segments a, b, c, and d.
2. ∠𝐴𝐸𝐷 ≅ ∠𝐵𝐶𝐷
∠𝐸𝐴𝐷 ≅ ∠𝐶𝐵𝐷
3. ∆𝐴𝐷𝐸~∆𝐵𝐷𝐶
𝑎
𝑑
4. 𝑐 = 𝑏
5. 𝑎𝑏 = 𝑐𝑑
CK-12 Basic Geometry Concepts
Reason
Given
Theorem 9-8
AA Similarity Postulate
Corresponding parts of similar triangles
are proportional
Cross multiplication
13
Answer Key
Chapter 9 – Circles
9.10
Segments from Secants
Answers
1.
3 + x, x = 3
2.
20, 8 + 7, x = 6
3.
x = 11
4.
x ≈ 17.14
5.
x=6
6.
Statement
̅̅̅̅ and 𝑅𝑇
̅̅̅̅ with segments a, b,
1. Secants 𝑃𝑅
c, and d.
2. ∠𝑅 ≅ ∠𝑅
3. ∠𝑄𝑃𝑆 ≅ ∠𝑆𝑇𝑄
4. ∆𝑅𝑃𝑆~∆𝑅𝑇𝑄
𝑎
𝑐
5.
=
𝑐+𝑑
𝑎+𝑏
6. 𝑎(𝑎 + 𝑏) = 𝑐(𝑐 + 𝑑)
7.
z=7
8.
x = 11
9.
m = 13.75
10.
n=3
11.
s=7
12.
x = 16
13.
x = 4. 6
14.
x ≈ 12.25
15.
x = 14
CK-12 Basic Geometry Concepts
Reason
given
Reflexive PoC
Theorem 9-8
AA Similarity Postulate
Corresponding parts of similar triangles
are proportional
Cross multiplication
14
Chapter 9 – Circles
9.11
Answer Key
Segments from Secants and Tangents
Answers
1.
x + 15, x = 5
2.
x=6
3.
x = 4√42
4.
x = 10
5.
The error is in the set up. It should be10 ∙ 10 = 𝑦 ∙ (15 + 𝑦). The correct answer is y = 5.
6.
2.2
7.
3
8.
12
9.
2. 6
10.
1
11.
3
12.
≈ 4.9
13.
5.5
14.
15. 36
15.
6
CK-12 Basic Geometry Concepts
15
Chapter 9 – Circles
9.12
Answer Key
Circles in the Coordinate Plane
Answers
1.
center: (-5, 3), radius = 4
2.
center: (0, -8), radius = 2
3.
center: (7, 10), radius = 2√5
4.
center: (-2, 0), radius = 2√2
5.
(𝑥 − 4)2 + (𝑦 + 2)2 = 16
6.
(𝑥 + 1)2 + (𝑦 − 2)2 = 7
7.
(𝑥 − 2)2 + (𝑦 − 2)2 = 4
8.
(𝑥 + 4)2 + (𝑦 + 3)2 = 25
9.
yes
10.
no
11.
yes
12.
yes
13.
(𝑥 − 2)2 + (𝑦 − 3)2 = 52
14.
(𝑥 − 10)2 + 𝑦 2 = 29
15.
(𝑥 + 3)2 + (𝑦 − 8)2 = 200
16.
(𝑥 − 6)2 + (𝑦 + 6)2 = 325
CK-12 Basic Geometry Concepts
16