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1 2
Chapter 1,
continued
37. A;
220
2
Slope 5 }
5 2}3
023
2
The y-intercept, b, is 2, so the function f(x) 5 2}3x 1 2
best describes the graph.
38. G;
Points that lie in Quadrant II have negative x-coordinates
and positive y-coordinates. So, the coordinates (21, 5)
represent a point that lies in Quadrant II.
1.3 Guided Practice (pp. 16–18)
}
1. ####$
MN is a segment bisector of PQ.
PQ 5 2 1 1 }8 2 5 3 }4
7
3
5
8
5
8
1
1
UW 5 1}4 in.
5. EF 5 FG 5 13 cm
EF 1 FG 5 EG
13 1 13 5 EG
1
6. AB 5 BC 5 }(AC)
2
1
1
7. PQ 5 QR 5 }(PR)
2
QR 5 }
9 } 5 4 }4 in.
21 22
1
1
BC 5 }2 (19) 5 9 }2 cm
1
1
7
3
3
11. x 1 5 5 2x
UV 5 2 1 2 }8 2 5 5}4 m
18
x5}
7
3
RQ 5 }2 1 4}4 2 5 2 }8 in.
1
10. UV 5 2(UT)
7x 5 18
1
1
9. RQ 5 } (PQ)
2
LM 5 }2 (137) 5 68 }2 mm
7x 2 7 5 11
3
55x
AM 5 x 1 5
1 2
41
18
5 }
275}
7
7
5 5 1 5 5 10
12.
PQ 5 2 1 }
5}
7
72
82
7x 5 8x 2 6
2x 5 26
2
41x
4. } 5 21
2
4 1 x 5 22
4 1 y 5 24
x 5 26
y 5 28
The coordinates of endpoint V are (26, 28).
5. It does not matter which ordered pair you substitute for
(x1, y1) or which you substitute for (x2, y2) because the
distance between the two points is the same no matter
which you start with.
Î(x2 2 x1)
2x 1 7 5 5
2x 5 22
EM 5 7x 5 7(6) 5 42
41y
} 5 22
2
}}
2
13. 6x 1 7 5 4x 1 5
x56
117 218
3. M }, } 5 (4, 5)
2
2
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
} 1 } 5 UW
RT 5 10 }4 in.
5x 2 7 5 11 2 2x
6. B; AB 5
UV 1 VW 5 UW
1
1
5 }8 1 5 }8 5 RT
1
8. LM 5 MN 5 } (LN)
2
}
2. Line * is a segment bisector of PQ.
1
RS 1 ST 5 RT
EG 5 26 cm
Lesson 1.3
41
5
4. UV 5 VW 5 } in.
8
1
3. RS 5 ST 5 5 } in.
8
Mixed Review for TAKS
JM 5 6x 1 7
5 6(21) 1 7 5 1
14. 6x 2 11 5 10x 2 51
211 5 4x 2 51
40 5 4x
10 5 x
PR 5 6x 2 11 1 10x 2 51
5 6(10) 2 11 1 10(10) 2 51
5 60 2 11 1 100 2 51 5 98
1 ( y2 2 y2)
2
}}}
5 Ï(1 2 (23)) 1 (24 2 2)
2
x 5 21
2
}
5 Ï16 1 36
}
15. x 1 15 5 4x 2 45
60 5 3x
20 5 x
5 Ï52 ø 7.2
}
The approximate length of AB is 7.2 units.
1.3 Exercises (pp. 19 – 22)
SU 5 x 1 15 1 4x 2 45
5 20 1 15 1 4(20) 2 45 5 70
16. 2x 1 35 5 5x 2 22
35 5 3x 2 22
Skill Practice
}
57 5 3x
B(4, 26), you can use the distance formula.
19 5 x
1. To find the length of AB, with endpoints A(27, 5) and
2. To bisect a segment means to intersect a segment at its
midpoint. You cannot bisect a line because it continues
forever in both directions and, therefore, has no midpoint.
XZ 5 2x 1 35 1 5x 2 22
5 2(19) 1 35 1 5(19) 2 22 5 146
Geometry
Worked-Out Solution Key
5
Chapter 1,
continued
317 515
17. M }, } 5 M (5, 5)
2
2
2
29. R(4, 26), M(27, 8)
014 413
18. M }, } 5 M (2, 3.5)
2
2
1
2
24 1 6 4 1 4
19. M }, } 5 M (1, 4)
2
2
1
2
27 1 (23) 25 1 7
20. M }, } 5 M (25, 1)
2
2
1
2
41x
2
}58
4 1 x 5 214
26 1 y 5 16
x 5 218
y 5 22
The other endpoint is S (218, 22).
30. R(24, 26), M(3, 24)
28 1 11 27 1 5
21. M }, } 5 M (1.5, 21)
2
2
}53
24 1 x
2
} 5 24
23 1 (28) 3 1 6
22. M }, } 5 M (25.5, 4.5)
2
2
24 1 x 5 6
26 1 y 5 28
1
2
1
2
23. Substitute the given numbers and variables into the
midpoint formula.
1
x1 1 x2 y1 1 y2
M }
,}
2
2
x 5 10
31. PQ 5
2
}
32. QR 5
}
of the y-coordinates, x1 and x2 and y1 and y2 should have
been added, not subtracted.
}
5 Ï 25 1 4 5 Ï29 ø 5.4 units
33. ST 5
}}
Ï(x2 2 x1)2 1 ( y2 2 y1)2
}}
5 Ï(3 2 (21))2 1 (22 2 2)2
}
3 1 (21)
}
5 Ï 16 1 16 5 Ï32 ø 5.7 units
812
,}
1}
2 5 (5, 1)
2
2
34. D; MN 5
25. B(3, 0), M(0, 5)
y 5 10
26. B(5, 1), M(1, 4)
51x
}51
2
11y
}54
2
51x52
11y58
y57
The other endpoint is S(23, 7).
27. R(6, 22), M(5, 3)
}}
Ï(x2 2 x1)2 1 ( y2 2 y1)2
}}}
5 Ï(4 2 (23))2 1 (8 2 (29))2
01y
}55
2
The other endpoint is S (23, 10).
}
}
5 Ï 49 1 289 5 Ï338 ø 18.4 units.
35. The length of the segment is
{3 2 (24){ 5 {7{ 5 7 units.
x1 1 x2
24 1 3
x1 1 x2
26 1 2
1
The midpoint has the coordinate }
5}
5 2}2.
2
2
36. The length of the segment is
{2 2 (26){ 5 {8{ 5 8 units.
The midpoint has the coordinate }
5}
5 22.
2
2
37. The length of the segment is
}55
61x
2
22 1 y
}53
2
6 1 x 5 10
22 1 y 5 6
x54
y58
The other endpoint is S (4, 8).
28. R(27, 11), M(2, 1)
{25 2 (215){ 5 {40{ 5 40 units.
x1 1 x2
215 1 25
The midpoint has the coordinate }
5}
5 5.
2
2
38. The length of the segment is
{25 2 (220){ 5 {15{ 5 15 units.
The midpoint has the coordinate
11 1 y
2
}52
27 1 x
2
}51
27 1 x 5 4
11 1 y 5 2
y 5 29
The other endpoint is S (11, 29).
x1 1 x2
220 1 (25)
2
} 5 } 5 212.5.
2
39. The length of the segment is {1 2 (28){ 5 9 units.
The midpoint has the coordinate
x1 1 x2
28 1 1
2
} 5 } 5 23.5.
2
Geometry
Worked-Out Solution Key
Ï(x2 2 x1)2 1 ( y2 2 y1)2
}}
24. When calculating the averages of the x-coordinates and
x 5 11
}}
5 Ï(2 2 (23))2 1 (3 2 5)2
m n
1 2 22
x 5 23
}
5 Ï 16 1 4 5 Ï20 ø 4.5 units
M }, }
x 5 23
}}
Ï(x2 2 x1)2 1 ( y2 2 y1)2
}}
Simplify the expression.
31x
2
y 5 22
5 Ï(5 2 1)2 1 (4 2 2)2
01m 01n
}50
26 1 y
2
The other endpoint is S(10, 22).
M 1}
,}
2
2 2
6
26 1 y
2
} 5 27
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
1
Chapter 1,
continued
40. The length of the segment is {22 2 (27){ 5 5 units.
The midpoint has the coordinate
x
27 1 (22)
} 5 } 5 24.5.
2
2
2 62 4
4 1 }4 2 5 1 }
4
x
}}
41. B; LF 5 Ï (3 2 (22)) 1 (1 2 2)
2
2
}}}
JR 5 Ï(2 2 1)2 1 (23 2 (21))2
}
5 Ï1 1 4 5 Ï 5 ø 2.24
The approximate difference in lengths is
5.10 2 2.24 5 2.86.
42. Substitute the coordinates of point P and of the midpoint
into the distance formula to find the length of the
segment from P to M.
}}
PM 5 Ï(x2 2 x1)2 1 ( y2 2 y1)2
}}
22x 5 224
x 5 12
x
12
1
5 1}2
JM 5 MK 5 }8 5 }
8
Problem Solving
1
48. MR 5 QM 5 18 } feet
2
QR 5 2(QM) 5 21 18 }2 2 5 37 feet
1
49. House
5 Ï(1 2 (22))2 1 (0 2 4)2
}
3x
x 5 3x 2 24
}
5 Ï25 1 1 5 Ï 26 ø 5.10
}
3x
2 1 }8 2 5 }
26
4
x1 1 x2
}
47. 2(JM) 5 JK
Library
School
}
5 Ï9 1 16 5 Ï 25 5 5
2.85 km
5.7 km
If the distance from point P to the midpoint is 5 units,
}
then the length of PQ is twice that, or 2(5) 5 10 units.
}}
43. AB 5 Ï (23 2 0)2 1 (8 2 2)2
}
5.7
2
} 5 2.85
}
5 Ï9 1 36 5 Ï 45 ø 6.71
}}}
CD 5 Ï(0 2 (22))2 1 (24 2 2)2
}
}
44. EF 5 Ï (5 2 1)2 1 (1 2 4)2
b. B(4, 2), C(2, 6)
}
}}
5 Ï16 1 9 5 Ï 25 5 5
BC 5 Ï (2 2 4)2 1 (6 2 2)2
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
}}
GH 5 Ï(1 2 (23))2 1 (6 2 1)2
}
5 Ï16 1 25 5 Ï41 ø 6.40
c. C(2, 6), D(5, 4)
}}
CD 5 Ï(5 2 2)2 1 (4 2 6)2
}}
45. JK 5 Ï (4 2 (24)) 1 (8 2 0)
2
}
}
5 Ï64 1 64 5 Ï128 ø 11.31
}}}
LM 5 Ï(3 2 (24))2 1 (27 2 2)2
}
}
5 Ï49 1 81 5 Ï130 ø 11.40
The segments are not congruent.
}
46. SP 5 PT, so P is the midpoint of ST.
S P T
21
0
x
1
SP 5 PT
x20512x
2x 5 1
1
x 5 }2
}
5 Ï9 1 4 5 Ï 13 ø 3.6 m
d. A(1, 1), D(5, 4)
}}
AD 5 Ï(5 2 1)2 1 (4 2 1)2
}
}
5 Ï16 1 9 5 Ï 25 5 5 m
e. B(4, 2), D(5, 4)
}}
BD 5 Ï(5 2 4)2 1 (4 2 2)2
}
2
}
5 Ï4 1 16 5 Ï 20 ø 4.5 m
}
The segments are not congruent.
}
}
5 Ï9 1 1 5 Ï10 ø 3.2 m
}}
2
}}
}
The segments are not congruent.
}
50. a. A(1, 1), B(4, 2)
AB 5 Ï(4 2 1)2 1 (2 2 1)2
5 Ï4 1 36 5 Ï 40 ø 6.32
}
The library is 2.85 kilometers from the house.
}
5 Ï1 1 4 5 Ï 5 ø 2.2 m
f. A(1, 1), C(2, 6)
}}
AC 5 Ï(2 2 1)2 1 (6 2 1)2
}
}
5 Ï1 1 25 5 Ï 26 ø 5.1 m
51. The objects at B and D are closest together. The objects
at A and C are farthest apart.
Geometry
Worked-Out Solution Key
7
Chapter 1,
continued
}}
52. AB 5 Ï (18 2 8)2 1 (7 2 4)2
}
Mixed Review for TAKS
}
5 Ï100 1 9 5 Ï109 ø 10.4
55. B;
x 2 1 4x 5 5
Player A threw the ball about 10.4 meters.
}}
x 2 1 4x 2 5 5 0
BC 5 Ï(24 2 18)2 1 (14 2 7)2
}
(x 1 5)(x 2 1) 5 0
}
5 Ï36 1 49 5 Ï85 ø 9.2
x1550
Player B threw the ball about 9.2 meters.
x 5 25
}}
AC 5 Ï(24 2 8)2 1 (14 2 4)2
}
53. a. P(10, 50), Q(10, 10), R(80, 10)
The other information needed to determine the number
of rolls of border Juan needs to purchase is the perimeter
of the room.
}}}
}
}}}
}
d 5 rt
}}}
}
r 5 6.5mi/h
PQ 5 Ï(10 2 10)2 1 (10 2 50)2 5 Ï1600 5 40
QR 5 Ï(80 2 10) 1 (10 2 10) 5 Ï4900 5 70
2
RP 5 Ï(80 2 10)2 1 (10 2 50)2 5 Ï6500 ø 80.62
40 1 70 1 81 5 191
The distance around the park is about 191 yards.
1
2
10 1 80 50 1 10
M }
,}
2
2
1
x51
56. F;
Player A would have thrown the ball about 18.9 meters to
player C.
x1 1 x2 y 1 y
1
2
b. M }
2 ,}
2
x2150
or
The solutions to the equation are 25 and 1.
}
5 Ï256 1 100 5 Ï356 ø 18.9
2
or
57. C;
t 5 24 min 5 0.4 h
d 5 6.5(0.4) 5 2.6
Sally would travel 2.6 miles in 24 minutes.
Quiz 1.1–1.3 (p. 22)
2
1. Sample answer:
l
k
M (45, 30)
C
}}}
QM 5 Ï(45 2 10) 1 (30 2 10)
2
2
}
P
}
A
5 Ï 1225 1 400 5 Ï 1625 ø 40.3
B
c. It takes him about 1.5 minutes.
1
PQ 5 40, QM 5 40, MR 5 }2 (80) 5 40, RQ 5 70
40 1 40 1 40 1 70 1 40 5 230
Divide the total distance, about 230 yards, by 150
yards per minute.
D
54.
2. DE 5 AE 2 AD
DE 5 26 2 15 5 11
15
AD
3. AB 5 } 5 } 5 5
3
3
4. AC 5 2AB 5 2(5) 5 10
5. BD 5 AC 5 10
6. CE 5 CD 1 DE 5 5 1 11 5 16
M
A
C
AB 5 2AM
1
B
7. BE 5 BD 1 DE 5 10 1 11 5 21
x1 1 x2 y1 1 y2
8. M }, }
2
2
1
2
22 1 2 21 1 3
M 1}
,}
5 M (0, 1)
2
2 2
CM 5 }2 CD
The coordinates of the midpoint are (0, 1). The distance
between R and S is about 5.7 units.
AB 5 4 + CM
RS 5 Ï(x2 2 x1)2 1 ( y2 2 y1) 2
2AM 5 4 1 }2 2CD
1
}}
}}}
5 Ï(2 2 (22))2 1 (3 2 (21))2
}
5 Ï 16 1 16 ø 5.66 units
2AM 5 2CD
AM 5 CD
AM and CD are equal.
Mixed Review for TEKS (p. 23)
1. B;
2.8
}
}
5 1.4 meters.
Because EG bisects FH, FG 5 }
2
8
Geometry
Worked-Out Solution Key
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
QM is about 40 yards.
Chapter 1,
continued
2. H; Use the Midpoint Formula.
x1 1 x2 y1 1 y2
M }
,}
2
2
1
1
5. D;
2
24 1 6 5 1 (25)
2
2
AB 5 18.7 km
BC 5 2AB 5 2(18.7) 5 37.4 km
2
AC 5 AB 1 BC 5 18.7 1 37.4 5 56.1 km
M }, } 5 M (1, 0)
AB 1 BC 1 CA 5 18.7 1 37.4 1 56.1 5 112.2 km
The coordinates of point E are (1, 0).
To find the coordinates of point D, substitute the
coordinates of point C into the midpoint formula, and
set each coordinate in the midpoint formula equal to the
corresponding coordinate from the midpoint E.
21x
}51
2
81y
}50
2
21x52
81y50
x50
Jill travels 112.2 kilometers.
6. G;
}
BC represents the intersection of planes ABC and BFE.
7. 23x 1 5 5 25x 2 4
5 5 2x 2 4
9 5 2x
4.5 5 x
y 5 28
PQ 5 23(4.5) 1 5 1 25(4.5) 2 4
The coordinates of point D are (0, 28).
5 103.5 1 5 1 112.5 2 4
3. A;
5 217
}
The length of PQ is 217 inches.
B(3, 1), C(3, 2), D(3, 5), E(7, 5), F(10, 5)
BD 5 {5 2 1{ 5 {4{ 5 4
8. d 5 rt
DF 5 {10 2 3{ 5 {7{ 5 7
r 5 2.4 km/h
Using the existing roads, a trip from B to F is
4 1 7 5 11 miles.
t 5 45 min 5 0.75 h
d 5 2.4(0.75) 5 1.8
BC 5 {2 2 1{ 5 {1{ 5 1
}}
}
5.4 2 1.8 5 3.6
}
CE 5 Ï(7 2 3)2 1 (5 2 2)2 5 Ï16 1 9 5 Ï25 5 5
The length to the end of the trail is 3.6 kilometers.
EF 5 {10 2 7{ 5 {3{ 5 3
Using the new road where possible, a trip from B to F is
1 1 5 1 3 5 9 miles, which is 11 2 9 5 2 miles shorter.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
4. H;
Lesson 1.4
1.4 Guided Practice (pp. 24–28)
1. ŽPQR
T (21, 3) y
ŽRQS
S
ŽPQS
(24, 1) 1
1
ŽPQS is a right angle.
x
2.
(3, 23)
Perimeter 5 QR 1 RS 1 ST 1 TQ
}}}
QR 5 Ï(0 2 3)2 1 [25 2 (23)]2
}
5 Ï9 1 4 5 Ï13 ø 3.6
}}}
RS 5 Ï(24 2 0)2 1 [1 2 (25)]2
}
}
5 Ï16 1 36 5 Ï52 ø 7.2
}}}
ST 5 Ï[21 2 (24)]2 1 (3 2 1)2
}
}
5 Ï9 1 4 5 Ï13 ø 3.6
}}}
TQ 5 Ï[3 2 (21)]2 1 (23 2 3)2
}
P
The rays form a straight angle.
R(0, 25)
}
R
}
5 Ï16 1 36 5 Ï52 ø 7.2
Perimeter ø 3.6 1 7.2 1 3.6 1 7.2 5 21.6
The perimeter of rectangle QRST is about 21.6 units.
3. mŽKLN 1 mŽNLM 5 1808
10x 2 5 1 4x 1 3 5 180
14x 2 2 5 180
14x 5 182
x 5 13
10(13) 2 5 5 125
mŽKLN 5 1258
14(13) 1 3 5 55
mŽNLM 5 558
4. mŽEFM 1 mŽHFG 5 908
2x 1 2 1 x 1 1 5 90
3x 1 3 5 90
3x 5 87
x 5 29
mŽEFM 5 2(29) 1 2 5 608
mŽMFG 5 29 1 1 5 308
Geometry
Worked-Out Solution Key
9
Fly UP