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Patterns and Inductive ReasoningPatterns and Inductive Reasoning
(For help, go the Skills Handbook, page 715.)
GEOMETRY LESSON 1-1GEOMETRY LESSON 1-1
Here is a list of the counting numbers: 1, 2, 3, 4, 5, . . .Some are even and some are odd.
1. Make a list of the positive even numbers.
2. Make a list of the positive odd numbers.
3. Copy and extend this list to show the first 10 perfect squares. 12 = 1, 22 = 4, 32 = 9, 42 = 16, . . .
4. Which do you think describes the square of any odd number? It is odd. It is even.
1-1
Patterns and Inductive ReasoningPatterns and Inductive Reasoning
1. Even numbers end in 0, 2, 4, 6, or 8: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, . . .
2. Odd numbers end in 1, 3, 5, 7, or 9: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, . . .
3. 12 = (1)(1) = 1; 22 = (2)(2) = 4; 32 = (3)(3) = 9; 42 = (4)(4) = 16; 52 = (5)(5) = 25; 62 = (6)(6) = 36; 72 = (7)(7) = 49; 82 = (8)(8) = 64; 92 = (9)(9) = 81; 102 = (10)(10) = 100
4. The odd squares in Exercise 3 are all odd, so the square of any odd number is odd.
GEOMETRY LESSON 1-1GEOMETRY LESSON 1-1
Solutions
1-1
Patterns and Inductive ReasoningPatterns and Inductive ReasoningGEOMETRY LESSON 1-1GEOMETRY LESSON 1-1
Each term is half the preceding term. So the next two terms are
48 ÷ 2 = 24 and 24 ÷ 2 = 12.
Find a pattern for the sequence. Use the pattern to
show the next two terms in the sequence.
384, 192, 96, 48, …
1-1
Patterns and Inductive ReasoningPatterns and Inductive Reasoning
Make a conjecture about the sum of the cubes of the first 25
counting numbers.
Find the first few sums. Notice that each sum is a perfect square and that the perfect squares form a pattern.
13 = 1 = 12 = 12
13 + 23 = 9 = 32 = (1 + 2)2
13 + 23 + 33 = 36 = 62 = (1 + 2 + 3)2
13 + 23 + 33 + 43 = 100 = 102 = (1 + 2 + 3 + 4)2
13 + 23 + 33 + 43 + 53 = 225 = 152 = (1 + 2 + 3 + 4 + 5)2
The sum of the first two cubes equals the square of the sum of the first two counting numbers.
GEOMETRY LESSON 1-1GEOMETRY LESSON 1-1
1-1
Patterns and Inductive ReasoningPatterns and Inductive ReasoningGEOMETRY LESSON 1-1GEOMETRY LESSON 1-1
This pattern continues for the fourth and fifth rows of the table.13 + 23 + 33 + 43 = 100 = 102 = (1 + 2 + 3 + 4)2
13 + 23 + 33 + 43 + 53 = 225 = 152 = (1 + 2 + 3 + 4 + 5)2
So a conjecture might be that the sum of the cubes of the first 25 counting numbers equals the square of the sum of the first 25 counting numbers, or (1 + 2 + 3 + … + 25)2.
The sum of the first three cubes equals the square of the sum of the first three counting numbers.
(continued)
1-1
Patterns and Inductive ReasoningPatterns and Inductive Reasoning
The first three odd prime numbers are 3, 5, and 7. Make and
test a conjecture about the fourth odd prime number.
The fourth prime number is 11.
One pattern of the sequence is that each term equals the preceding term plus 2.
So a possible conjecture is that the fourth prime number is 7 + 2 = 9.
However, because 3 X 3 = 9 and 9 is not a prime number, this conjecture is false.
GEOMETRY LESSON 1-1GEOMETRY LESSON 1-1
1-1
Patterns and Inductive ReasoningPatterns and Inductive Reasoning
The price of overnight shipping was $8.00 in 2000, $9.50 in
2001, and $11.00 in 2002. Make a conjecture about the price in 2003.
Write the data in a table. Find a pattern.
2000
$8.00
2001 2002
$9.50 $11.00
Each year the price increased by $1.50.
A possible conjecture is that the price in 2003 will increase by $1.50.
If so, the price in 2003 would be $11.00 + $1.50 = $12.50.
GEOMETRY LESSON 1-1GEOMETRY LESSON 1-1
1-1
Patterns and Inductive ReasoningPatterns and Inductive ReasoningGEOMETRY LESSON 1–1GEOMETRY LESSON 1–1
pages 6–9 Practice and Problem Solving
1. 80, 160
2. 33,333; 333,333
3. –3, 4
4. ,
5. 3, 0
6. 1,
7. N, T
8. J, J
9. 720, 5040
10. 64, 128
11. ,
1 16
1 32
1 36
1 49
12. ,
13. James, John
14. Elizabeth, Louisa
15. Andrew, Ulysses
16. Gemini, Cancer
17.
18.
15
16
19. The sum of the first 6 pos.
even numbers is
6•7, or 42.
20. The sum of the first 30 pos.
even numbers is
30•31, or 930.
21. The sum of the first 100
pos. even numbers is
100•101, or 10,100.
1-1
Patterns and Inductive ReasoningPatterns and Inductive ReasoningGEOMETRY LESSON 1–1GEOMETRY LESSON 1–1
22. The sum of the first
100 odd numbers is
1002, or 10,000.
23. 555,555,555
24. 123,454,321
25–28. Answers may vary.
Samples are given.
25. 8 + (–5 = 3) and 3 > 8
26. • > and • >
27. –6 – (–4) Ò –6 and
–6 – (–4) Ò –4
28. ÷ = and is
improper.
29. 758F
30. 40 push-ups;
answers may vary.
Sample: Not very
confident, Dino may
reach a limit to the
number of push-ups
he can do in his
allotted time for
exercises.
31. 31, 43
32. 10, 13
33. 0.0001, 0.00001
34. 201, 202
35. 63, 127
36. ,
37. J, S
38. CA, CO
39. B, C13
12
13
13
12
12/ /
/
12
13
32
32
3132
6364
1-1
Patterns and Inductive ReasoningPatterns and Inductive ReasoningGEOMETRY LESSON 1–1GEOMETRY LESSON 1–1
40. Answers may vary.
Sample: In Exercise
31, each number
increases by increasing
multiples of 2. In Exercise
33, to get the next term,
divide by 10.
41.
You would get a third line
between and parallel to
the first two lines.
42.
43.
44.
45.
46. 102 cm
47. Answers may vary. Samples are given.a. Women may soon outrun
men in running competitions.b. The conclusion was based
on continuing the trend shown in past records.
c. The conclusions are based on fairly recent records for women, and those rates of improvement may not continue. The conclusion about the marathon is most suspect because records date only from 1955.
1-1
Patterns and Inductive ReasoningPatterns and Inductive ReasoningGEOMETRY LESSON 1–1GEOMETRY LESSON 1–1
48. a.
b. about 12,000 radio stations in 2010
c. Answers may vary. Sample: Confident; the pattern has held for several decades.
49. Answers may vary. Sample: 1, 3, 9, 27, 81, . . .1, 3, 5, 7, 9, . . .
50. His conjecture is probably false because most people’s growth slows by 18 untilthey stop growing somewhere between 18 and 22 years.
51. a.
b. H and Ic. a circle
52. 21, 34, 5553. a. Leap years are years
that are divisible by 4.b. 2020, 2100, and 2400c. Leap years are years
divisible by 4, except the final year of a century which must be divisible by 400. So, 2100 will not be a leap year, but 2400 will be.
1-1
Patterns and Inductive ReasoningPatterns and Inductive ReasoningGEOMETRY LESSON 1–1GEOMETRY LESSON 1–1
54. Answers may vary.Sample:
100 + 99 + 98 + … + 3 + 2 + 1 1 + 2 + 3 + … + 98 + 99 + 100 101 + 101 + 101 + … + 101 + 101 + 101
The sum of the first 100 numbers is
, or 5050.
The sum of the first n numbers is .
55. a. 1, 3, 6, 10, 15, 21b. They are the same.c. The diagram shows the product of n
and n + 1 divided by 2 when n = 3. The result is 6.
100 • 1012
n(n+1)2
d.
56. B
57. I
58. [2] a. 25, 36, 49b. n2
[1] one part correct
1-1
Patterns and Inductive ReasoningPatterns and Inductive ReasoningGEOMETRY LESSON 1–1GEOMETRY LESSON 1–1
59. [4] a. The product of 11 and a three-digit number that
begins and ends in 1 is a four-digit number
that begins and ends in 1 and has middle digits that are each one
greater than the middle digit of the three-digit
number.(151)(11) = 1661(161)(11) = 1771
b. 1991
c. No; (191)(11) = 2101
[3] minor error in explanation
[2] incorrect description in part (a)
[1] correct products for (151)(11), (161)(11), and (181)(11)
60-67.
68. B
69. N
70. G
1-1
Patterns and Inductive ReasoningPatterns and Inductive ReasoningGEOMETRY LESSON 1-1GEOMETRY LESSON 1-1
Find a pattern for each sequence.
Use the pattern to show the next
two terms or figures.
1. 3, –6, 18, –72, 360
2.
Use the table and inductive reasoning. Make a conjecture about each value.
3. the sum of the first 10 counting numbers
4. the sum of the first 1000 counting numbers
Show that the conjecture is false by finding one
counterexample.
5. The sum of two prime numbers is an even number.
–2160; 15,120
55
500,500
Sample: 2+3=5, and 5 is not even
1-1
Points, Lines, and PlanesPoints, Lines, and Planes
(For help, go to the Skills Handbook, page 722.)
Solve each system of equations.
1. y = x + 5 2. y = 2x – 4 3. y = 2x
y = –x + 7 y = 4x – 10 y = –x + 15
4. Copy the diagram of the four points A, B, C,
and D. Draw as many different lines as you
can to connect pairs of points.
GEOMETRY LESSON 1-2GEOMETRY LESSON 1-2
1-2
Points, Lines, and PlanesPoints, Lines, and Planes
1. By substitution, x + 5 = –x + 7; adding x – 5 to both sides results in 2x = 2; dividing both sides by 2 results in x = 1. Since x = 1, y = (1) + 5 = 6. (x, y) = (1, 6)
2. By substitution, 2x – 4 = 4x – 10; adding –4x + 4 to both sides results in –2x = –6; dividing both sides by –2 results in x = 3. Since x = 3, y = 2(3) – 4 = 6 – 4 = 2. (x, y) = (3, 2)
3. By substitution, 2x = –x + 15; adding x to both sides results in 3x = 15; dividing both sides by 3 results in x = 5. Since x = 5, y = 2(5) = 10. (x, y) = (5, 10)
4. The 6 different lines are AB, AC, AD, BC, BD, and CD.
Solutions
GEOMETRY LESSON 1-2GEOMETRY LESSON 1-2
1-2
Points, Lines, and PlanesPoints, Lines, and Planes
Any other set of three points do not lie on a line, so no other set of three points is collinear.
For example, X, Y, and Z and X, W, and Z form triangles and are not collinear.
In the figure below, name three points that are
collinear and three points that are not collinear.
GEOMETRY LESSON 1-2GEOMETRY LESSON 1-2
Points Y, Z, and W lie on a line, so they are collinear.
1-2
Points, Lines, and PlanesPoints, Lines, and Planes
You can name a plane using any three or more points on that plane that are not collinear. Some possible names for the plane shown are the following:
plane RST
plane RSU
plane RTU
plane STU
plane RSTU
Name the plane shown in two different ways.
GEOMETRY LESSON 1-2GEOMETRY LESSON 1-2
1-2
Points, Lines, and PlanesPoints, Lines, and PlanesGEOMETRY LESSON 1-2GEOMETRY LESSON 1-2
As you look at the cube, the front face is on plane AEFB, the back face is on plane HGC, and the left face is on plane AED.
The back and left faces of the cube intersect at HD.
Planes HGC and AED intersect vertically at HD.
Use the diagram below. What is the intersection of plane HGC
and plane AED?
1-2
Points, Lines, and PlanesPoints, Lines, and Planes
Points X, Y, and Z are the vertices of one of the four triangular faces of the pyramid. To shade the plane, shade the interior of the triangle formed by X, Y, and Z.
Shade the plane that
contains X, Y, and Z.
GEOMETRY LESSON 1-2GEOMETRY LESSON 1-2
1-2
Points, Lines, and PlanesPoints, Lines, and PlanesGEOMETRY LESSON 1–2GEOMETRY LESSON 1–2
1. no
2. yes; line n
3. yes; line n
4. yes; line m
5. yes; line n
6. no
7. no
8. yes; line m
pages 13–16 Practice and Problem Solving
9. Answers may vary. Sample: AE, EC, GA
10. Answers may vary. Sample: BF, CD, DF
11. ABCD
12. EFHG
13. ABHF
14. EDCG
15. EFAD
16. BCGH
17. RS
18. VW
19. UV
20. XT
21. planes QUX and QUV
22. planes XTS and QTS
23. planes UXT and WXT
24. UVW and RVW
1-2
Points, Lines, and PlanesPoints, Lines, and PlanesGEOMETRY LESSON 1–2GEOMETRY LESSON 1–2
25.
26.
27.
28.
29.
30. S
31. X
32. R
33. Q
34. X
1-2
46. Postulate 1-1: Through any two points there is exactly one line.
47. Answer may vary.Sample:
48.
49. not possible
Points, Lines, and PlanesPoints, Lines, and PlanesGEOMETRY LESSON 1–2GEOMETRY LESSON 1–2
35. no
36. yes
37. no
38. coplanar
39. coplanar
40. noncoplanar
41. coplanar
42. noncoplanar
43. noncoplanar
44. Answers may vary. Sample: The plane of the ceiling and the plane of a wall intersect in a line.
45. Through any three noncollinear points there is exactly one plane. The ends of the legs of the tripod represent three noncollinear points, so they rest in one plane. Therefore, the tripod won’t wobble.
1-2
50.
51. not possible
52.
yes53.
yes
Points, Lines, and PlanesPoints, Lines, and PlanesGEOMETRY LESSON 1–2GEOMETRY LESSON 1–2
56.
no
57.
no
58.
yes
54.
no
55.
yes
1-2
Points, Lines, and PlanesPoints, Lines, and PlanesGEOMETRY LESSON 1–2GEOMETRY LESSON 1–2
68. Answers may vary. Sample:
Post. 1-3: If two planes intersect, then they intersect in exactly one line.
69. A, B, and D
70. Post. 1-1: Through any two points there is exactly one line.
59.
yes
60. always
61. never
62. always
63. always
64. sometimes
65. never
66. a. 1b. 1c. 1d. 1e. A line and a point not on the line are always coplanar.
67.
Post. 1-4: Through three noncollinear points there is exactly one plane.
1-2
71. Post. 1-3: If two planes intersect, then they intersect in exactly one line.
72. The end of one leg might not be coplanar with the ends of the other three legs. (Post. 1-4)
73.
yes
Points, Lines, and PlanesPoints, Lines, and PlanesGEOMETRY LESSON 1–2GEOMETRY LESSON 1–2
76.
no
77.
yes
74.
yes
75.
no
1-2
Points, Lines, and PlanesPoints, Lines, and PlanesGEOMETRY LESSON 1–2GEOMETRY LESSON 1–2
81. a. Since the plane is flat, the line would have to curve so as to contain the 2 points and not lie in the plane; but lines are straight.
b. One plane; Points A, B, and C are
noncollinear. By Post. 1-4, they
are coplanar.Then, by part
(a), AB and BC are coplanar.
82. 1
78.
no
79. Infinitely many; explanations may vary. Sample: Infinitely many planes can intersect in one line.
80.
By Post. 1-1, points D and B determine a line and points A and D determine a line. The distress signal is on both lines and, by Post. 1-2, there can be only one distress signal.
1-2
Points, Lines, and PlanesPoints, Lines, and PlanesGEOMETRY LESSON 1–2GEOMETRY LESSON 1–2
94. 25, -5
95. 34
96. 44
83.
84. 1
85. A
86. I
87. B
88. H
89. [2] a. ABD, ABC, ACD, BCD
b. AD, BD, CD[1] one part correct
90.
91. I, K
92. 42, 56
93. 1024, 4096
14
1-2
Points, Lines, and PlanesPoints, Lines, and PlanesGEOMETRY LESSON 1-2GEOMETRY LESSON 1-2
1. Name three collinear points.
2. Name two different planes that contain points C and G.
3. Name the intersection of plane AED and plane HEG.
4. How many planes contain the points A, F, and H?
5. Show that this conjecture is false by finding one counterexample: Two planes always intersect in exactly one line.
Use the diagram at right.
D, J, and H
planes BCGF and CGHD
HE
1
Sample: Planes AEHD and BFGC never intersect.
1-2
Segments, Rays, Parallel Lines and PlanesSegments, Rays, Parallel Lines and Planes
(For help, go to Lesson 1-2.)
Judging by appearances, will the lines intersect?
1. 2. 3.
Name the plane represented by each surface of the box.
4. the bottom 5. the top
6. the front 7. the back
8. the left side 9. the right side
GEOMETRY LESSON 1-3GEOMETRY LESSON 1-3
1-3
Segments, Rays, Parallel Lines and PlanesSegments, Rays, Parallel Lines and Planes
1. no 2. yes 3. no
4. NMR 5. PQL 6. NKL
7. PQR 8. PKN 9. LQR
GEOMETRY LESSON 1-3GEOMETRY LESSON 1-3
Solutions
1-3
Name the segments and rays in the figure.
The labeled points in the figure are A, B, and C.
A segment is a part of a line consisting of two endpoints and all points between them. A segment is named by its two endpoints. So the segments are BA (or AB) and BC (or CB).
A ray is a part of a line consisting of one endpoint and all the points of the line on one side of that endpoint. A ray is named by its endpoint first, followed by any other point on the ray. So the rays areBA and BC.
Segments, Rays, Parallel Lines and PlanesSegments, Rays, Parallel Lines and PlanesGEOMETRY LESSON 1-3GEOMETRY LESSON 1-3
1-3
Segments, Rays, Parallel Lines and PlanesSegments, Rays, Parallel Lines and PlanesGEOMETRY LESSON 1-3GEOMETRY LESSON 1-3
Use the figure below. Name all segments that
are parallel to AE. Name all segments that are skew to AE.
Parallel segments lie in the same plane, and the lines that contain them do not intersect. The three segments in the figure above that are parallel to AE are BF, CG, and DH.
Skew lines are lines that do not lie in the same plane. The four lines in the figure that do not lie in the same plane as are AE, BC, CD, and GH.
1-3
Segments, Rays, Parallel Lines and PlanesSegments, Rays, Parallel Lines and Planes
Planes are parallel if they do not intersect. If the walls of your classroom are vertical, opposite walls are parts of parallel planes. If the ceiling and floor of the classroom are level, they are parts of parallel planes.
Identify a pair of parallel planes in your classroom.
GEOMETRY LESSON 1-3GEOMETRY LESSON 1-3
1-3
Segments, Rays, Parallel Lines and PlanesSegments, Rays, Parallel Lines and PlanesGEOMETRY LESSON 1–3GEOMETRY LESSON 1–3
12. BC
13. BE, CF
14. DE, EF, BE
15. AD, AB, AC
16. BC, EF
17. ABC || DEF
1.
2.
3.
4.
5. RS, RT, RW, ST, SW, TW
6. RS, ST, TW, WT, TS, SR
7. a. TS or TR, TWb. SR, ST
8. 4; RY, SY, TY, WY
9. Answers may vary.Sample: 2; YS or YR, YT or YW
10. Answers may vary.Check students’ work.
11. DF
Pages 19-22 Practice and Problem Solving
1-3
25. true
26. False; they are skew.
27. true
28. False; they intersect above CG.
29. true
30. False; they intersect above pt. A.
Segments, Rays, Parallel Lines and PlanesSegments, Rays, Parallel Lines and PlanesGEOMETRY LESSON 1–3GEOMETRY LESSON 1–3
31. False; they are ||.
32. False; they are ||.
33. Yes; both name the segment with endpoints X and Y.
34. No; the two rays have different endpoints.
35. Yes; both are the line through pts. X and Y.
18. BE || AD
19. CF, DE
20. DEF, BC
21. FG
22. Answers may vary. Sample: CD, AB
23. BG, DH, CL
24. AF
Pages 18-20 Answers may vary. Samples are given
1-3
Pages 19-22 Exercises
36.
37. always
38. never
39. always
40. always
Segments, Rays, Parallel Lines and PlanesSegments, Rays, Parallel Lines and PlanesGEOMETRY LESSON 1–3GEOMETRY LESSON 1–3
41. never
42. sometimes
43. always
44. sometimes
45. always
46. sometimes
47. sometimes
48. Answers may vary. Sample: (0, 0); check students’ graphs.
49. a. Answers may vary. Sample: northeast
and southwestb. Answers may vary. Sample: northwest
and southeast, east and west
50. Two lines can be parallel, skew, or intersecting in one point. Sample: train tracks–parallel; vapor trail of a northbound jet and an eastbound jet at different altitudes– skew; streets that cross–intersecting
1-3
Segments, Rays, Parallel Lines and PlanesSegments, Rays, Parallel Lines and PlanesGEOMETRY LESSON 1–3GEOMETRY LESSON 1–3
55. a. The lines of intersection
are parallel.
b. Examples may vary. Sample: The floor and ceiling are parallel. A wall
intersects both. The lines of intersection
are parallel.
56. Answers may vary. Sample: The diamond structure makes it tough, strong, hard, and durable. The graphite structure makes it soft and slippery.
57. a.
one segment; EF
b.
3 segments; EF, EG, FG
51. Answers may vary. Sample: Skew lines cannot be contained in one plane. Therefore, they have “escaped” a plane.
52. ST || UV
53. Answers may vary.Sample: XY and ZWintersect in R.
54. Planes ABC and DCBFintersect in BC.
1-3
Segments, Rays, Parallel Lines and PlanesSegments, Rays, Parallel Lines and PlanesGEOMETRY LESSON 1–3GEOMETRY LESSON 1–3
58. No; two different planes cannot intersect in more than one line.
59. yes; plane P, for example
60. Answers may vary.Sample: VR, QR, SR
61. QR
62. Yes; no; yes; explanations may vary.
63. D
64. H
65. B
66. F
67. B
68. C
69. D
57. c.
Answers may vary. Sample: For each “new” point, the number of new segments equals the number of “old” points.d. 45 segments
e. n(n - 1)2
1-3
79.
80.
81.
82. 1.4, 1.48
83. –22, –29
84. FG, GH
85. P, S
86. No; whenever you subtract a negative number, the answer is greater than the given number. Also, if you subtract 0, the answer stays the same.
Segments, Rays, Parallel Lines and PlanesSegments, Rays, Parallel Lines and PlanesGEOMETRY LESSON 1–3GEOMETRY LESSON 1–3
71–78. Answers may vary. Samples are
given.
71. EF
72. A
73. C
74. AEF and HEF
75. ABH
76. EHG
77. FG
78. B
70. [2] a. Alike: They do not intersect.
Different: Parallel lines are coplanar
and skew lines lie in different
planes.
b. No; of the 8 other lines shown, 4 intersect
JM and 4 are skew
to JM.
[1] one likeness, one difference
1-3
Segments, Rays, Parallel Lines and PlanesSegments, Rays, Parallel Lines and PlanesGEOMETRY LESSON 1-3GEOMETRY LESSON 1-3
Use the figure below for Exercises 1-3.
1. Name the segments that form the triangle. 2. Name the rays that have point T as their endpoint.
3. Explain how you can tell that no lines in the figure are parallel or skew.
Use the figure below for Exercises 4 and 5.
4. Name a pair of parallel planes.
5. Name a line that is skew to XW.
TO, TP, TR, TS
The three pairs of lines intersect, So they cannot be parallel or skew.
AC or BD
RS, TR, ST plane BCD || plane XWQ
1-3
Measuring Segments and AnglesMeasuring Segments and Angles
(For help, go to the Skills Handbook pages 719 and 720.)
Simplify each absolute value expression.
1. |–6| 2. |3.5| 3. |7 – 10|
4. |–4 – 2| 5. |–2 – (–4)| 6. |–3 + 12|
Solve each equation.
7. x + 2x – 6 = 6
8. 3x + 9 + 5x = 81
9. w – 2 = –4 + 7w
1-4
GEOMETRY LESSON 1-4GEOMETRY LESSON 1-4
Measuring Segments and AnglesMeasuring Segments and Angles
1. The number of units from 0 to –6 on the number line is 6.
2. The number of units from 0 to 3.5 on the number line is 3.5.
3. |7 – 10| = |–3|, and the number of units from 0 to –3 on the number line is 3.
4. |–4 – 2| = |–6|, and the number of units from 0 to –6 on the number line is 6.
5. |–2 – (–4)| = |–2 + 4| = |2|, and the number of units from 0 to 2 on the
number line is 2.
6. |–3 + 12| = |9|, and the number of units from 0 to 9 on the number line is 9.
7. Combine like terms: 3x – 6 = 6; add 6: 3x = 12; divide by 3: x = 4
8. Combine like terms: 8x + 9 = 81; subtract 9: 8x = 72; divide by 8: x = 9
9. Add –7w + 2: –6w = –2; divide by –6: w = 13
GEOMETRY LESSON 1-4GEOMETRY LESSON 1-4
Solutions
1-4
Measuring Segments and AnglesMeasuring Segments and Angles
Use the Ruler Postulate to find the length of each segment.
XY = | –5 – (–1)| = | –4| = 4
ZY = | 2 – (–1)| = |3| = 3
ZW = | 2 – 6| = |–4| = 4
Find which two of the segments XY, ZY, and ZW
are congruent.
GEOMETRY LESSON 1-4GEOMETRY LESSON 1-4
Because XY = ZW, XY ZW.
1-4
Measuring Segments and AnglesMeasuring Segments and AnglesGEOMETRY LESSON 1-4GEOMETRY LESSON 1-4
Use the Segment Addition Postulate to write an equation.
AN + NB = AB Segment Addition Postulate(2x – 6) + (x + 7) = 25 Substitute.
3x + 1 = 25 Simplify the left side. 3x = 24 Subtract 1 from each side. x = 8 Divide each side by 3.
AN = 10 and NB = 15, which checks because the sum of the segment lengths equals 25.
If AB = 25, find the value of x. Then find AN and NB.
AN = 2x – 6 = 2(8) – 6 = 10 NB = x + 7 = (8) + 7 = 15
Substitute 8 for x.
1-4
Measuring Segments and AnglesMeasuring Segments and AnglesGEOMETRY LESSON 1-4GEOMETRY LESSON 1-4
Use the definition of midpoint to write an equation.
RM = MT Definition of midpoint5x + 9 = 8x – 36 Substitute.
5x + 45 = 8x Add 36 to each side. 45 = 3x Subtract 5x from each side. 15 = x Divide each side by 3.
RM and MT are each 84, which is half of 168, the length of RT.
M is the midpoint of RT. Find RM, MT, and RT.
RM = 5x + 9 = 5(15) + 9 = 84MT = 8x – 36 = 8(15) – 36 = 84
Substitute 15 for x.
RT = RM + MT = 168
1-4
Measuring Segments and AnglesMeasuring Segments and AnglesGEOMETRY LESSON 1-4GEOMETRY LESSON 1-4
Name the angle below in four ways.
The name can be the vertex of the angle: G.
Finally, the name can be a point on one side, the vertex, and a point on the other side of the angle: AGC, CGA.
The name can be the number between the sides of the angle: 3.
1-4
Measuring Segments and AnglesMeasuring Segments and Angles
Because 0 < 80 < 90, 2 is acute.
m 2 = 80
Use a protractor to measure each angle.m 1 = 110
Because 90 < 110 < 180, 1 is obtuse.
Find the measure of each angle. Classify each as acute, right,
obtuse, or straight.
GEOMETRY LESSON 1-4GEOMETRY LESSON 1-4
1-4
Measuring Segments and AnglesMeasuring Segments and Angles
Use the Angle Addition Postulate to solve.
m 1 + m 2 = m ABC Angle Addition Postulate.
42 + m 2 = 88 Substitute 42 for m 1 and 88 for m ABC.
m 2 = 46 Subtract 42 from each side.
Suppose that m 1 = 42 and m ABC =88. Find m 2.
GEOMETRY LESSON 1-4GEOMETRY LESSON 1-4
1-4
Measuring Segments and AnglesMeasuring Segments and AnglesGEOMETRY LESSON 1–4GEOMETRY LESSON 1–4
15. 130
16. XYZ, ZYX, Y
17. MCP, PCM, C or 1
18. ABC, CBA
19. CBD, DBC
9. 25
10. a. 13b. RS = 40, ST = 24
11. a. 7b. RS = 60, ST = 36, RT = 96
12. a. 9b. 9; 18
13. 33
14. 34
1. 9; 9; yes
2. 9; 6; no
3. 11; 13; no
4. 7; 6; no
5. XY = ZW
6. ZX = WY
7. YZ < XW
8. 24
pages 29–33 Practice and Problem Solving
1-4
20-23. Drawings may vary.
20.
21.
22.
23.
Measuring Segments and AnglesMeasuring Segments and AnglesGEOMETRY LESSON 1–4GEOMETRY LESSON 1–4
33. –2.5, 2.5
34. –3.5, 3.5
35. –6, –1, 1, 6
36. a. 78 mib. Answers may vary. Sample: measuring
with a ruler
37–41. Check students’ work.
24. 60; acute
25. 90; right
26. 135; obtuse
27. 34
28. 70
29. Q
30. 6
31. –4
32. 1
1-4
Measuring Segments and AnglesMeasuring Segments and AnglesGEOMETRY LESSON 1–4GEOMETRY LESSON 1–4
60. 150
61. 30
62. 100
63. 40
64. 80
65. 125
66. 125
49. Answers may vary. Sample: (15, 0), (–9, 0), (3, 12), (3, –12)
50–54. Check students’ work.
55. about 42°
56–58. Answers may vary. Samples are given.
56. 3:00, 9:00
57. 5:00, 7:00
58. 6:00, 12:32
59. 180
42. true; AB = 2, CD = 2
43. false; BD = 9, CD = 2
44. false; AC = 9, BD = 9, AD = 11, and 9 + 9 ≠ 11
45. true; AC = 9, CD = 2, AD = 11, and 9 + 2 = 11
46. 2, 12
47. 115
48. 65
1-4
Measuring Segments and AnglesMeasuring Segments and AnglesGEOMETRY LESSON 1–4GEOMETRY LESSON 1–4
75. 12; m AOC = 82,m AOB = 32,m BOC = 50
76. 8; m AOB = 30,m BOC = 50,m COD = 30
77. 18; m AOB = 28,m BOC = 52,m AOD = 108
78. 7; m AOB = 28,m BOC = 49,m AOD = 111
79. 30
71. y = 15; AC = 24, DC = 12
72. ED = 10, DB = 10, EB = 20
73. a. Answers may vary. Sample: The two rays come together at a sharp point.b. Answers may vary. Sample: Molly had an acute pain in her knee.
74. 45, 75, and 165, or 135, 105, and 15
67–68. Answers may vary. Samples are
given
67. QVM and VPN
68. MNP and MVN
69. MQV and PNQ
70. a. 19.5b. 43; 137c. Answers may
vary. Sample: The sum of the
measures should be 180.
1-4
86. [2] a.
b. An obtuse measures between 90 and 180 degrees; the least and greatest whole number values are 91 and 179 degrees. Part of
ABC is 12°. So the least and greatest measures
for DBC are 79 and 167.
[1] one part correct
Measuring Segments and AnglesMeasuring Segments and AnglesGEOMETRY LESSON 1–4GEOMETRY LESSON 1–4
87. never
88. never
89. always
90. never
91. always
92. always
93. always
94. never
95. 25, 30
96. 3125; 15,625
97. 30, 34
80. a–c. Check students’ work.
81. Angle Add. Post.
82. C
83. F
84. D
85. H
1-4
Measuring Segments and AnglesMeasuring Segments and AnglesGEOMETRY LESSON 1-4GEOMETRY LESSON 1-4
Use the figure below for Exercises 4–6.
4. Name 2 two different ways.
5. Measure and classify 1, 2, and BAC.
6. Which postulate relates the measures of 1, 2, and BAC?
14Angle Addition Postulate
DAB, BAD
Use the figure below for Exercises 1-3.
1. If XT = 12 and XZ = 21, then TZ = 7.
2. If XZ = 3x, XT = x + 3, and TZ = 13, find XZ.
3. Suppose that T is the midpoint of XZ. If XT = 2x + 11 and XZ = 5x + 8, find the value of x.
9
24 90°, right; 30°, acute; 120°, obtuse
1-4
Basic ConstructionBasic Construction
In Exercises 1-6, sketch each figure.
1. CD 2. GH 3. AB
4. line m 5. acute ABC 6. XY || ST
7. DE = 20. Point C is the midpoint of DE. Find CE.
8. Use a protractor to draw a 60° angle.
9. Use a protractor to draw a 120° angle.
(For help, go to Lesson 1-3 and 1-4.)
GEOMETRY LESSON 1-5GEOMETRY LESSON 1-5
1-5
Basic ConstructionBasic Construction
1. The figure is a segment whose endpoints are C and D.
2. The figure is a ray whose endpoint is G.
3. The figure is a line going through the points A and B.
4. 5. The figure is an angle whose
measure is between 0° and 90°.
6. The figure is two segments in a plane whose corresponding
lines do not intersect.
7. Since C is a midpoint, CD = CE; also, CD + CE = 20;
substituting results in CE + CE = 20, or 2CE = 20, so CE = 10.
8. 9.
GEOMETRY LESSON 1-5GEOMETRY LESSON 1-5
Solutions
1-5
Basic ConstructionBasic ConstructionGEOMETRY LESSON 1-5GEOMETRY LESSON 1-5
Step 2: Open the compass to the length of KM.
Construct TW congruent to KM.
Step 1: Draw a ray with endpoint T.
Step 3: With the same compass setting, put the compass point on point T. Draw an arc that intersects the ray. Label the point of intersection W.
TW KM
1-5
Basic ConstructionBasic Construction
~
Step 3: With the same compass setting, put the compass point on point Y. Draw an arc that intersects the ray. Label the point of intersection Z.
Construct Y so that Y = G.
Step 1: Draw a ray with endpoint Y.
Step 2: With the compass point on point G, draw an arc that intersects both sides of G. Label the points of intersection E and F.
GEOMETRY LESSON 1-5GEOMETRY LESSON 1-5
1-5
Basic ConstructionBasic ConstructionGEOMETRY LESSON 1-5GEOMETRY LESSON 1-5
(continued)
Step 4: Open the compass to the length EF. Keeping the same compass setting, put the compass point on Z. Draw an arc that intersects the arc you drew in Step 3. Label the point of intersection X.
Step 5: Draw to complete Y.
Y G
1-5
Basic ConstructionBasic ConstructionGEOMETRY LESSON 1-5GEOMETRY LESSON 1-5
Step 1: Put the compass point on
point A and draw a short arc. Make
sure that the opening is less than AB.12
Start with AB.
Step 2: With the same compass setting, put the compass point on point B and draw a short arc.
Without two points of intersection, no line can be drawn, so the perpendicular bisector cannot be drawn.
Use a compass opening less than AB. Explain why the
construction of the perpendicular bisector of AB is not possible.
12
1-5
Basic ConstructionBasic ConstructionGEOMETRY LESSON 1-5GEOMETRY LESSON 1-5
–3x = –48 Subtract 4x from each side. x = 16 Divide each side by –3.
m AWR = m BWR Definition of angle bisector x = 4x – 48 Substitute x for m AWR and
4x – 48 for m BWR.
m AWB = m AWR + m BWR Angle Addition Postulatem AWB = 16 + 16 = 32 Substitute 16 for m AWR and
for m BWR.
Draw and label a figure to illustrate the problem
WR bisects AWB. m AWR = x and m BWR = 4x – 48. Find m AWB.
m AWR = 16 m BWR = 4(16) – 48 = 16 Substitute 16 for x.
1-5
Basic ConstructionBasic Construction
Step 1: Put the compass point on vertex M. Draw an arc that intersects both sides of M. Label the points of intersection B and C.
Step 2: Put the compass point on point B. Draw an arc in the interior of M.
Construct MX, the bisector of M.
GEOMETRY LESSON 1-5GEOMETRY LESSON 1-5
1-5
Basic ConstructionBasic Construction
Step 4: Draw MX. MX is the angle bisector of M.
(continued)
Step 3: Put the compass point on point C. Using the same compass setting, draw an arc in the interior of M. Make sure that the arcs intersect. Label the point where the two arcs intersect X.
GEOMETRY LESSON 1-5GEOMETRY LESSON 1-5
1-5
6.
7.
8.
Basic ConstructionBasic ConstructionGEOMETRY LESSON 1–5GEOMETRY LESSON 1–5
9. a. 11; 30b. 30c. 60
10. 5; 50
11. 15; 48
12. 11; 56
13.
1.
2.
3.
4.
5.
1-5
Basic ConstructionBasic ConstructionGEOMETRY LESSON 1–5GEOMETRY LESSON 1–5
16. Find a segment on XY so that you can construct YZ as its bisector.
17. Find a segment on SQ so that you can construct SP as its bisector. Then bisect PSQ.
18. a. CBD; 41b. 82c. 49; 49
19. a-b.
14.
15.
1-5
Basic ConstructionBasic ConstructionGEOMETRY LESSON 1–5GEOMETRY LESSON 1–5
21. Explanations may vary. Samples are given.a. One midpt.; a midpt.
divides a segment into two segments. If there were more than one
midpt. the segments wouldn’t be .
b. Infinitely many; there’s only 1 midpt. but there exist infinitely many lines through the midpt. A
segment has exactly one bisecting line because there can be only one line to a segment at its midpt.
c. There are an infinite number of lines in space that are to a
segment at its midpt. The
lines are coplanar.
20. Locate points A and B on a line. Then construct a at A and B as in Exercise 16.Construct AD and BCso that AB = AD = BC.
1-5
27.
28. a.
They appear to meet at one
pt.
Basic ConstructionBasic ConstructionGEOMETRY LESSON 1–5GEOMETRY LESSON 1–5
25. They are both correct. If you mult. each side of Lani’s eq. by 2, the result is Denyse’s eq.
26. Open the compass to more than half the measure of the segment. Swing large arcs from the endpts. to intersect above and below the segment. Draw a line through the two pts. where the arcs intersect. The pt. where the line and segment intersect is the midpt. of the segment.
22.
23.
24.
1-5
33. a.
b. They are all 60°.c. Answers may vary. Sample: Mark a pt., A. Swing a long arc from A. From a pt. P on the arc, swing
another arc the same size that intersects the
arc at a second pt., Q. Draw PAQ.
To construct a 30° , bisect the 60° .
Basic ConstructionBasic ConstructionGEOMETRY LESSON 1–5GEOMETRY LESSON 1–5
30.
31. impossible; the short segments are not long enough to form a .
32. impossible; the short segments are not long enough to form a .
b.
c. The three bisectors of a
intersect in one pt.
29.
1-5
Basic ConstructionBasic ConstructionGEOMETRY LESSON 1–5GEOMETRY LESSON 1–5
Label the intersection K.
Open the compass to PQ.
With compass pt. on K,
swing an arc to intersect
the first arc. Label the
intersection R. Draw XR.
c. Point O is the center of the circle.
36. ; the line intersects.
37. D
38. F
39. [2] a. Draw XY. With the
compass pt. on B
swing an arc that
intersects BA and
BC. Label the
intersections
P and Q, respectively.
With the compass
point on X, swing a
arc intersecting XY.
34. a-c.
35. a-c.
1-5
41. 642. 1043. 444. 345.
46. 10047. 20 and 18048.
49. No; they do not have
the same endpt.50. Yes; they both
represent a segment with endpts. R and S.
Basic ConstructionBasic ConstructionGEOMETRY LESSON 1–5GEOMETRY LESSON 1–5
39. [2] b. With compass open to XK, put compass point on X and swing an arc intersecting XR. With compass on R and open to KR, swing
an arc to intersect the first arc. Label intersection T. Draw XT.
[1] one part correct40. [4] a. Construct its
bisector.b. Construct the
bisector. Then construct the bisector of two new
segments.
c. Draw AB. Do constructions as in
parts a and b. Open the compass to the
length of the shortest segment in part b.
With the pt. of the compass on B,
swing an arc in the opp. direction from A
intersecting AB at C. AC = 1.25 (AB).
[3] explanations are not thorough
[2] two explanations correct[1] part (a) correct
1-5
Basic ConstructionBasic ConstructionGEOMETRY LESSON 1-5GEOMETRY LESSON 1-5
For problems 1-4, check students’ work.
QN bisects DNB.
1. Construct AC so that AC NB.
2. Construct the perpendicular bisector of AC.
3. Construct RST so that RST QNB.
4. Construct the bisector of RST.
5. Find x.
6. Find m DNB. 88
Use the figure at right.
17
1-5
The Coordinate PlaneThe Coordinate Plane
(For help, go to the Skills Handbook pages 715 and 716.)
GEOMETRY LESSON 1-6GEOMETRY LESSON 1-6
1-6
Find the square root of each number to the nearest tenth. Use a calculator if necessary.
1. 25 2. 17 3. 123
Evaluate each expression for m = –3 and n = 7.
4. (m – n)2 5. (n – m)2 6. m2 + n2
Evaluate each expression for a = 6 and b = –8.
7. (a – b)2 8. 9.a2 + b2 a + b2
The Coordinate PlaneThe Coordinate PlaneGEOMETRY LESSON 1-6GEOMETRY LESSON 1-6
Solutions
1. 25 = 52 = 5 2. 17 4.1232 = 4.123
3. 123 11.092 = 11.09
4. (m – n)2 = (–3 –7)2
= (–10)2
= 100
5. (n – m)2 = –7 – (–3))2
= (7 + 3)2
=102 = 100
6. m2 + n2 = (–3)2 + (7)2
= 9 + 49= 58
7. (a – b)2 = (6 – (–8))2
= (6 + 8)2
=142 = 196
8. a2 + b2 = (6)2 + (–8)2
= 36 + 64= 100 = 10
9.
–22= = –1
a + b2
6 + (–8)2=
1-6
d = 82 + (–8)2 Simplify.
The Coordinate PlaneThe Coordinate Plane
Find the distance between R(–2, –6) and S(6, –2)
to the nearest tenth.
Let (x1, y1) be the point R(–2, –6) and (x2, y2) be the point S(6, –2).
To the nearest tenth, RS = 11.3.
128 11.3137085 Use a calculator.
GEOMETRY LESSON 1-6GEOMETRY LESSON 1-6
d = (x2 – x1)2 + (y2 – y1)2 Use the Distance Formula.
d = (6 – (–2))2 + (–2 – (–6))2 Substitute.
d = 64 + 64 = 128
1-6
The Coordinate PlaneThe Coordinate Plane
Oak has coordinates (–1, –2). Let (x1, y1) represent Oak. Symphony has coordinates (1, 2). Let (x2, y2) represent Symphony.
To the nearest tenth, the subway ride from Oak to Symphony is 4.5 miles.
20 4.472135955 Use a calculator.
GEOMETRY LESSON 1-6GEOMETRY LESSON 1-6
d = 22 + 42 Simplify.
d = (x2 – x1)2 + (y2 – y1)2 Use the Distance Formula.
How far is the subway ride from Oak to
Symphony? Round to the nearest tenth.
d = (1 – (–1))2 + (2 – (–2))2 Substitute.
d = 4 + 16 = 20
1-6
The Coordinate PlaneThe Coordinate PlaneGEOMETRY LESSON 1-6GEOMETRY LESSON 1-6
Use the Midpoint Formula. Let (x1, y1) be A(8, 9) and (x2, y2) be B(–6, –3).
The coordinates of midpoint M are (1, 3).
AB has endpoints (8, 9) and (–6, –3). Find the coordinates of
its midpoint M.
The midpoint has coordinates Midpoint Formula
( , )x1 + x2
2
y1 + y2
2
Substitute 8 for x1 and (–6) for x2. Simplify.
8 + (–6)2The x–coordinate is = = 1
22
Substitute 9 for y1 and (–3) for y2. Simplify.
9 + (–3)2The y–coordinate is = = 3
62
1-6
The Coordinate PlaneThe Coordinate PlaneGEOMETRY LESSON 1-6GEOMETRY LESSON 1-6
Find the x–coordinate of G. Find the y–coordinate of G.
4 + y2
25 =
1 + x2
2–1 = Use the Midpoint Formula.
The coordinates of G are (–3, 6).
The midpoint of DG is M(–1, 5). One endpoint is D(1, 4). Find
the coordinates of the other endpoint G.
–2 = 1 + x2 10 = 4 + y2Multiply each side by 2.
Use the Midpoint Formula. Let (x1, y1) be D(1, 4) and the midpoint
be (–1, 5). Solve for x2 and y2, the coordinates of G.( , )x1 + x2
2
y1 + y2
2
1-6
The Coordinate PlaneThe Coordinate PlaneGEOMETRY LESSON 1–6GEOMETRY LESSON 1–6
11. about 4.5 mi
12. about 3.2 mi
13. 6.4
14. 15.8
15. 15.8
16. 5
17. B, C, D, E, F
18. (4, 2)
19. (3, 1)
20. (3.5, 1)
21. (6, 1)
22. (–2.25, 2.1)
23. (3 , –3)
24. (10, –20)
25. (5, –1)
26. (0, –34)
27. (12, –24)
28. (9, –28)
29. (5.5, –13.5)
30. (8, 18)
31. (4, –11)
1. 6
2. 18
3. 8
4. 9
5. 23.3
6. 10
7. 25
8. 12.2
9. 12.0
10. 9 mi
pages 46–48 Practice and Problem Solving
78
1-6
The Coordinate PlaneThe Coordinate PlaneGEOMETRY LESSON 1–6GEOMETRY LESSON 1–6
41. IV
42.
The midpts. Are the same, (5, 4). The diagonals bisect each other.
32. 5.0; (4.5, 4)
33. 5.8; (1.5, 0.5)
34. 7.1; (–1.5, 0.5)
35. 5.4; (–2.5, 3)
36. 10; (1, –4)
37. 2.8; (–4, –4)
38. 6.7; (–2.5, –2)
39. 5.4; (3, 0.5)
40. 2.2; (3.5, 1)
43.
ST = (5 – 2)2 + (–3 – (–6))2 = 9 + 9 = 3 2 4.2TV = (6 – 5)2 + (–6 – (–3))2 = 1 + 9 = 10 3.2SW = (5 – 6)2 + (–9 – (–6))2 = 9 + 9 = 3 2 4.2
No, but ST = SW and TV = VW.
1-6
The Coordinate PlaneThe Coordinate PlaneGEOMETRY LESSON 1–6GEOMETRY LESSON 1–6
50. 1073 mi
51. 2693 mi
52. 328 mi
53–56. Answers may vary. Samples are
given.
53. (3, 6), (0, 4.5)
54. E (0, 0), (8, 4)
55. (1, 0), (–1, 4)
56. (0, 10), (5, 0)
44. 19.2 units; (–1.5, 0)
45. 10.8 units; (3, –4)
46. 5.4 units; (–1, 0.5)
47. Z; about 12 units
48. 165 units; The dist. TV is less than the dist. TU, so the airplane should fly from T to V to U for the shortest route.
49. 934 mi
57. exactly one pt., E (–5, 2)
58. exactly one pt., J (2, –2)
59. a–f. Answers may vary. Samples are given.
a. BC = AD
b. If two opp. sides of a quad. are both || and , then the other two opp. sides are .
1-6
The Coordinate PlaneThe Coordinate PlaneGEOMETRY LESSON 1–6GEOMETRY LESSON 1–6
f. If a pair of opp. sides of a
quad. are both || and , then
the segment joining the midpts. of the
other two sides has the same length as
each of the first pair of sides.
60. A (0, 0, 0)B (6, 0, 0)C (6, –3.5, 0)D (0, –3.5, 0)E (0, 0, 9)F (6, 0, 9)G (0, –3.5, 9)
c. The midpts. are the same.
d. If one pair of opp. sides of a quad. are both || and , then its diagonals bisect each other.
e. EF = AB
61.
62. 6.5 units
63. 11.7 units
64. B
65. I
1-6
66. A
67. C
68. A
69. [2] a. (–10, 8), (–1, 5), (8, 2)
b. Yes, R must
be (–10,
8) so that
RQ = 160.
[1] part (a) correct or plausible
explanation for part (b)
The Coordinate PlaneThe Coordinate PlaneGEOMETRY LESSON 1–6GEOMETRY LESSON 1–6
70.
71.
72.
73.
74. 10
75. 10
76. 48
77. TAP, PAT
78. 150
1-6
The Coordinate PlaneThe Coordinate PlaneGEOMETRY LESSON 1-6GEOMETRY LESSON 1-6
1. Find the distance between A and B to the nearest tenth.
2. Find BC to the nearest tenth.
3. Find the midpoint M of AC to the nearest tenth.
4. B is the midpoint of AD. Find the coordinates of endpoint D.
5. An airplane flies from Stanton to Mercury in a straight flight path. Mercury is 300 miles east and 400 miles south of Stanton. How many miles is the flight?
6. Toni rides 2 miles north, then 5 miles west, and then 14 miles south. At the end of her ride, how far is Toni from her starting point, measured in a straight line? 13 mi
A has coordinates (3, 8). B has coordinates (0, –4). C has coordinates (–5, –6).
12.4
5.4
(–1, 1)
(–3, –16)
500 mi
1-6
Perimeter, Circumference, and AreaPerimeter, Circumference, and Area
(For help, go to the Skills Handbook page 719 and Lesson 1-6.)
Simplify each absolute value.
1. |4 – 8| 2. |10 – (–5)| 3. |–2 – 6|
Find the distance between the points to the nearest tenth.
4. A(2, 3), B(5, 9) 5. K(–1, –3), L(0, 0)
6. W(4, –7), Z(10, –2) 7. C(–5, 2), D(–7, 6)
8. M(–1, –10), P(–12, –3) 9. Q(–8, –4), R(–3, –10)
GEOMETRY LESSON 1-7GEOMETRY LESSON 1-7
1-7
4. d = (x2 – x1)2 + (y2 – y1)2
d = (5 – 2)2 + (9 – 3)2
d = 32 + 62
d = 9 + 36 = 45
To the nearest tenth, AB = 6.7.
6. d = (x2 – x1)2 + (y2 – y1)2
d = (10 – 4)2 + ( – 2 –(– 7))2
d = 62 + 52
d = 36 + 25 = 61
To the nearest tenth, WZ = 7.8.
2. | 10 – (–5) | = | 10 + 5 | = | 15 | = 15
Perimeter, Circumference, and AreaPerimeter, Circumference, and AreaGEOMETRY LESSON 1-7GEOMETRY LESSON 1-7
1. | 4 – 8 | = | –4 | = 4
Solutions
3. | –2 – 6 | = | –8 | = 8
5. d = (x2 – x1)2 + (y2 – y1)2
d = (0 – (–1))2 + (0 – (–3))2
d = 12 + 32
d = 1 + 9 = 10
To the nearest tenth, KL = 3.2.
7. d = (x2 – x1)2 + (y2 – y1)2
d = (– 7 – (– 5))2 + (6 – 2)2
d = (–2)2 + 52
d = 4 + 16 = 20
To the nearest tenth, CD = 4.5.
1-7
Perimeter, Circumference, and AreaPerimeter, Circumference, and AreaGEOMETRY LESSON 1-7GEOMETRY LESSON 1-7
Solutions (continued)
8.
9.
d = (x2 – x1)2 + (y2 – y1)2
d = (–12 – (–1))2 + (–3 – (–10))2
d = (–11)2 + 72
d = 121 + 49 = 170
To the nearest tenth, MP = 13.0.
d = (x2 – x1)2 + (y2 – y1)2
d = (–3 – (–8))2 + (–10 – (–4))2
d = 52 + (–6)2
d = 25 + 36 = 61
To the nearest tenth, QR = 7.8.
1-7
Perimeter, Circumference, and AreaPerimeter, Circumference, and Area
Margaret’s garden is a square 12 ft on each side.
Margaret wants a path 1 ft wide around the entire garden.
What will the outside perimeter of the path be?
The perimeter is 56 ft.
P = 4s Formula for perimeter of a square
P = 4(14) = 56 Substitute 14 for s.
GEOMETRY LESSON 1-7GEOMETRY LESSON 1-7
Because the path is 1 ft wide, increase each side of the garden by 1 ft. s = 1 + 12 + 1 = 14
1-7
C = 2 (6.5) Substitute 6.5 for r.
The circumference of G is 13 , or about 40.8 cm..
Perimeter, Circumference, and AreaPerimeter, Circumference, and Area
C = 13 Exact answer.
C = 13 40.840704 Use a calculator.
GEOMETRY LESSON 1-7GEOMETRY LESSON 1-7
C = 2 r Formula for circumference of a circle.
G has a radius of 6.5 cm. Find the circumference of G in
terms of . Then find the circumference to the nearest tenth.
. .
1-7
Perimeter, Circumference, and AreaPerimeter, Circumference, and AreaGEOMETRY LESSON 1-7GEOMETRY LESSON 1-7
Quadrilateral ABCD has vertices
A(0, 0), B(9, 12), C(11, 12), and D(2, 0).
Find the perimeter.
Draw and label ABCD on a coordinate plane.
BC = |11 – 9| = |2| = 2 Ruler Postulate
DA = |2 – 0| = |2| = 2 Ruler Postulate
Find the length of each side. Add the lengths to find the perimeter.
AB = (9 – 0)2 + (12 – 0)2 = 92 + 122 Use the Distance Formula.
= 81 + 144 = 255 = 15
CD = (2 – 11)2 + (0 – 12)2 = (–9)2 + (–12)2 Use the Distance Formula.
= 81 + 144 = 255 = 15
1-7
Perimeter, Circumference, and AreaPerimeter, Circumference, and Area
(continued)
Perimeter = AB + BC + CD + DA
= 15 + 2 + 15 + 2
= 34
The perimeter of quadrilateral ABCD is 34 units.
GEOMETRY LESSON 1-7GEOMETRY LESSON 1-7
1-7
Perimeter, Circumference, and AreaPerimeter, Circumference, and Area
Write both dimensions using the same unit of measurement. Find the area of the rectangle using the formula A = bh.
36 in. = 3 ft Change inches to feet using 12 in. = 1 ft.
A = bh Formula for area of a rectangle.
A = (4)(3) Substitute 4 for b and 3 for h.
A = 12
You need 12 ft2 of fabric.
To make a project, you need a rectangular piece of fabric 36 in. wide and 4 ft long. How many square feet of fabric do you need?
GEOMETRY LESSON 1-7GEOMETRY LESSON 1-7
1-7
A = r2 Formula for area of a circle
A = (1.5)2 Substitute 1.5 for r.
A = 2.25
Perimeter, Circumference, and AreaPerimeter, Circumference, and Area
In B, r = 1.5 yd..
The area of B is 2.25 yd2..
GEOMETRY LESSON 1-7GEOMETRY LESSON 1-7
Find the area of B in terms of . .
1-7
Perimeter, Circumference, and AreaPerimeter, Circumference, and AreaGEOMETRY LESSON 1-7GEOMETRY LESSON 1-7
Find the area of the figure below.
Draw a horizontal line to separate the figure into three nonoverlapping figures: a rectangle and two squares.
1-7
Perimeter, Circumference, and AreaPerimeter, Circumference, and AreaGEOMETRY LESSON 1-7GEOMETRY LESSON 1-7
AR = bh Formula for area of a rectangleAR = (15)(5) Substitute 15 for b and 5 for h.AR = 75
AS = s2 Formula for area of a squareAS = (5)2 Substitute 5 for s.AS = 25
A = 75 + 25 + 25 Add the areas. A = 125
The area of the figure is 125 ft2.
Find each area. Then add the areas.
(continued)
1-7
1. 22 in.
2. 36 cm
3. 56 in.
4. 78 cm
5. 120 m
6. 48 in.
7. 38 ft
8. 15 cm
9. 10 ft
10. 3.7 in.
11. m
12. 56.5 in.
13. 22.9 m
14. 1.6 yd
15. 351.9 cm
Perimeter, Circumference, and AreaPerimeter, Circumference, and AreaGEOMETRY LESSON 1–7GEOMETRY LESSON 1–7
16.
14.6 units17.
25.1 units
pages 55–58 Practice and Problem Solving
12
1-7
29. in.2
30. 0.25 m2
31. 9.9225 ft2
32. 0.01 m2
33. 153.9 ft2
34. 54.1 m2
35. 452.4 cm2
36. 452.4 in.2
37. 310 m2
38. 19 yd2
20. 1 ft2 or 192 in.2
21. 4320 in.2 or 3 yd2
22. 1 ft2 of 162 in.2
23. 8000 cm2 or 0.8 m2
24. 5.7 m2 or 57,000 cm2
25. 120,000 cm2 or 12 m2
26. 6000 ft2 or 666 yd2
27. 400 m2
28. 64 ft2
Perimeter, Circumference, and AreaPerimeter, Circumference, and AreaGEOMETRY LESSON 1–7GEOMETRY LESSON 1–7
18.
16 units
19.
38 units
13
18
23
964
1-7
Perimeter, Circumference, and AreaPerimeter, Circumference, and AreaGEOMETRY LESSON 1–7GEOMETRY LESSON 1–7
39. 24 cm2
40. 80 in.2
41. a. 144 in.2
b. 1 ft2
c. 144; a square whose sides are 12 in. long and a square whose sides are 1 ft long are the same size.
42. a. 30 squaresb. 16; 9; 4; 1c. They are =.
Post 1-10
48. Answers may vary. Sample: For Exercise 46, you use feet because the bulletin board is too big for inches. You do not use yards because your estimated lengths in feet were not divisible by 3.
49. 16 cm
50. 96 cm2
51. 288 cm
43. 3289 m2
44–47. Answers may vary. Check students’
work. Samples are given.
44. 38 in.; 90 in.2
45. 39 in.; 93.5 in.2
46. 12 ft; 8 ft2
47. 8 ft; 3.75 ft2
1-7
Perimeter, Circumference, and AreaPerimeter, Circumference, and AreaGEOMETRY LESSON 1–7GEOMETRY LESSON 1–7
52. a. Yes; every square is a rectangle.
b. Answers may vary. Sample: No,
not all rectangles are squares.
c. A = ( ) or A =
53. 512 tiles
56. 38 units
57. 54 units2
58. 1,620,000 m2
59. 30 m
60. (4x – 2) units
61. Area; the wall is a surface.
62. Perimeter; weatherstripping must fit the edges of the door.
54.
perimeter = 10 unitsarea = 4 units2
55.
perimeter = 16 unitsarea = 15 units2
P4
P2
162
1-7
63. Perimeter; the fence must fit the perimeter of the garden.
64. Area; the floor is a surface.
65. 6.25 units2
Perimeter, Circumference, and AreaPerimeter, Circumference, and AreaGEOMETRY LESSON 1–7GEOMETRY LESSON 1–7
b.
c. 25 ft by 50 ft
66. a. base heightarea
1 98 98
2 96 192
3 94 282
::
24 521248
25 501250
26 481248
::
47 6 282
48 4 192
49 2 98
1-7
Perimeter, Circumference, and AreaPerimeter, Circumference, and AreaGEOMETRY LESSON 1–7GEOMETRY LESSON 1–7
67. a. 9b. 9c. 9d. 9
68. units2
69. units2
70. (9m2 – 24mn + 16n2)
units2
71. Answers may vary. Sample: one 8 in.-by-8 in. square + one 5 in.-by-5 in. square + two 4 in.-by-4 in. squares
72. 388.5 yd
73. 64
83. 9.2 units; (1, 6.5)
84. 6.7 units; (–2.5, –2)
85. 90
86. WI RI
87. 62 units
88. 18 units
89. 6 units
90. 33 units
74. 2336
75. 540
76. 216
77. 810
78. (15, 13)
79. 8.5 units; (5.5, 5)
80. 5.8 units; (1.5, 5.5)
81. 13.9 units; (3, 5.5)
82. 6.4 units; (–2, 3.5)
3a20
25a2
4
1-7
Perimeter, Circumference, and AreaPerimeter, Circumference, and AreaGEOMETRY LESSON 1-7GEOMETRY LESSON 1-7
256 in.2
81 cm2
296 in.
30 ft2
42 units
A rectangle is 9 ft long and 40 in. wide.
1. Find the perimeter in inches.
2. Find the area in square feet.
3. The diameter of a circle is 18 cm. Find the area in terms of .
4. Find the perimeter of a triangle whose vertices are X(–6, 2), Y(8, 2), and Z(3, 14).
5. Find the area of the figure below. All angles are right angles.
1-7
Tools of GeometryTools of GeometryGEOMETRY CHAPTER 1GEOMETRY CHAPTER 1
1. Div. each preceding
term by –2; , –
2. Add 2 to the preceding term; 10, 12
3. Rotate the U clockwise one-quarter turn. Alphabet is backwards;
8. B
9. a. 1b. infinitely manyc. 1d. 1
10. 29,054.0 ft2
11. never
12. sometimes
13. never
14. always
15. never
4. Answers may vary. Sample: 1, 2, 4, 8, 16, 32, . . .1, 2, 4, 7, 11, 16, . . .In the first seq. double each term. In the second seq., add consecutive counting numbers.
5. A, B, C
6. Answers may vary. Sample: A, B, C, D
7. Answers may vary. Sample: A, B, D, E
12
14
TEST
16. 10
17. a. (11, 19)b. MC = MD = 136
18. 19.1 units
19. 800 cm2 or 0.08 m2
20. 12.25 in.2
21. 63.62 cm2
22. 7
23. 9
Tools of GeometryTools of Geometry
31. 33 yd2
1. D
2. G
3. B
4. H
5. B
6. I
7. B
8. H
9. 61 in.
10. 756 in.2
11. 207 in.
12. 2 in.
24. Answers may vary. Sample: Some ways of naming an can help identify a side or vertex.
25.
26. Bisector27. VW28. 7 units29. AY30. E, AY
13
12
14
GEOMETRY CHAPTER 1GEOMETRY CHAPTER 1
TEST