1.2 Measuring and Constructing Segments Essential Question USING

How can you measure and construct a line
Measuring Line Segments Using
Nonstandard Units
12
Work with a partner.
11
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
H
INC
7
3
2
1
6
29
30
To be proficient in math,
you need to explain to
yourself the meaning of a
problem and look for entry
points to its solution.
6
5
4
5
USING
PROBLEM-SOLVING
STRATEGIES
8
7
4
9
a. Draw a line segment that has a length
of 6 inches.
b. Use a standard-sized paper clip to
measure the length of the line segment.
Explain how you measured the line
segment in “paper clips.”
3
10
2
G.2.B
G.5.B
Preparing for
G.5.C
Essential Question
segment?
1
TEXAS ESSENTIAL
KNOWLEDGE AND SKILLS
Measuring and Constructing Segments
CM
1.2
c. Write conversion factors from paper clips
ps
to inches and vice versa.
1 paper clip =
1 in. =
in.
paper clip
d. A straightedge is a tool that you can use to draw a straight line. An example of a
straightedge is a ruler. Use only a pencil, straightedge, paper clip, and paper to draw
another line segment that is 6 inches long. Explain your process.
Measuring Line Segments Using
Nonstandard Units
Work with a partner.
a. Fold a 3-inch by 5-inch index card
fo
ld
on one of its diagonals.
3 in.
b. Use the Pythagorean Theorem
to algebraically determine the
length of the diagonal in inches.
5 in.
Use a ruler to check your answer.
c. Measure the length and width of the index card in paper clips.
d. Use the Pythagorean Theorem to algebraically determine the length of the diagonal
in paper clips. Then check your answer by measuring the length of the diagonal in
paper clips. Does the Pythagorean Theorem work for any unit of measure? Justify
your answer.
Measuring Heights Using Nonstandard Units
Work with a partner. Consider a unit of length that is equal to the length of the
diagonal you found in Exploration 2. Call this length “1 diag.” How tall are you in
diags? Explain how you obtained your answer.
Communicate Your Answer
4. How can you measure and construct a line segment?
Section 1.2
HSTX_GEOM_PE_01.02.indd 11
Measuring and Constructing Segments
11
6/19/14 4:48 PM
1.2 Lesson
What You Will Learn
Use the Ruler Postulate.
Use the Segment Addition Postulate.
Core Vocabul
Vocabulary
larry
Use the Distance Formula.
postulate, p. 12
axiom, p. 12
coordinate, p. 12
distance, p. 12
between, p. 13
construction, p. 15
congruent segments, p. 15
Copy segments and compare segments for congruence.
Using the Ruler Postulate
In geometry, a rule that is accepted without proof is called a postulate or an axiom.
A rule that can be proved is called a theorem, as you will see later. Postulate 1.1 shows
how to find the distance between two points on a line.
Postulate
Postulate 1.1
names of points
Ruler Postulate
The points on a line can be matched one to
one with the real numbers. The real number
that corresponds to a point is the coordinate
of the point.
A
B
x1
x2
coordinates of points
The distance between points A and B, written
as AB, is the absolute value of the difference
of the coordinates of A and B.
A
AB
B
x2
x1
AB = x2 − x1
Using the Ruler Postulate
—.
Find the length of BE
B
−4
−3
−2
E
−1
0
1
2
3
4
5
6
SOLUTION
By the Ruler Postulate, the coordinate of B is −1 and the coordinate of E is 5.
BE = ∣ 5 − (−1) ∣
Definition of distance
= ∣5 + 1∣
Simplify.
= ∣6∣
Add.
=6
Find the absolute value.
— is 6.
So, the length of BE
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Find the length of the segment.
J
−6
—
1. MN
12
Chapter 1
HSTX_GEOM_PE_01.02.indd 12
−5
−4
—
2. LN
K
L
−3
−2
M
−1
0
—
3. KJ
1
N
2
3
4
—
4. JL
Basics of Geometry
6/19/14 4:49 PM
Using the Segment Addition Postulate
When three points are collinear, you can say that one point is between the other two.
A
D
B
E
F
C
Point B is between
points A and C.
Point E is not between
points D and F.
Postulate
Postulate 1.2
Segment Addition Postulate
If B is between A and C, then AB + BC = AC.
AC
If AB + BC = AC, then B is between A and C.
A
B
AB
C
BC
Using the Segment Addition Postulate
a. Find DF.
D
23
b. Find GH.
E
F
35
36
F
21
G
H
SOLUTION
a. Use the Segment Addition Postulate to write an equation. Then solve the equation
to find DF.
DF = DE + EF
Segment Addition Postulate
DF = 23 + 35
Substitute 23 for DE and 35 for EF.
DF = 58
Add.
b. Use the Segment Addition Postulate to write an equation. Then solve the
equation to find GH.
FH = FG + GH
Segment Addition Postulate
36 = 21 + GH
Substitute 36 for FH and 21 for FG.
15 = GH
Subtract 21 from each side.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Use the diagram at the right.
23
5. Use the Segment Addition Postulate to find XZ.
6. In the diagram, WY = 30. Can you use the
X
50
Y
Z
W
Segment Addition Postulate to find the distance
between points W and Z ? Explain your reasoning.
144
J 37 K
L
7. Use the diagram at the left to find KL.
Section 1.2
HSTX_GEOM_PE_01.02.indd 13
Measuring and Constructing Segments
13
6/19/14 4:49 PM
Using the Distance Formula
You can use the Distance Formula to find the distance between two points in a
coordinate plane.
Core Concept
The Distance Formula
READING
The red mark at the corner
of the triangle that makes
a right angle indicates a
right triangle.
y
If A(x1, y1) and B(x2, y2) are points in a
coordinate plane, then the distance between
A and B is
——
AB = √(x2 – x1)2 + (y2 – y1)2 .
B(x2, y2)
y2 − y1
A(x1, y1)
x2 − x1 C(x2, y1)
x
The Distance Formula is related to the Pythagorean Theorem, which you will see
again when you work with right triangles.
Distance Formula
(AB)2
= (x2 −
x1)2
Pythagorean Theorem
+ (y2 −
y1)2
B(x2, y2)
y
c 2 = a2 + b 2
c
b
y2 − y1
A(x1, y1)
x2 − x1
a
C(x2, y1)
x
Using the Distance Formula
Your school is 4 miles east and 1 mile south of your apartment. A recycling center,
where your class is going on a field trip, is 2 miles east and 3 miles north of your
apartment. Estimate the distance between the recycling center and your school.
SOLUTION
You can model the situation using a coordinate plane with your apartment at the origin
(0, 0). The coordinates of the recycling center and the school are R(2, 3) and S(4, −1),
respectively. Use the Distance Formula. Let (x1, y1) = (2, 3) and (x2, y2) = (4, −1).
——
RS = √ (x2 − x1)2 + (y2 − y1)2
——
= √(4 − 2)2 + (−1 − 3)2
—
READING
4
R(2, 3)
2
Subtract.
= √4 + 16
Evaluate powers.
—
N
Substitute.
= √22 + (−4)2
—
The symbol ≈ means “is
approximately equal to.”
Distance Formula
= √ 20
Add.
≈ 4.5
Use a calculator.
W
2
−2
E
S(4, −1)
S
So, the distance between the recycling center and your school is about 4.5 miles.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
8. In Example 3, a park is 3 miles east and 4 miles south of your apartment. Find the
distance between the park and your school.
14
Chapter 1
HSTX_GEOM_PE_01.02.indd 14
Basics of Geometry
6/19/14 4:49 PM
Constructing and Comparing Congruent Segments
A construction is a geometric drawing that uses a limited set of tools, usually a
compass and straightedge.
Copying a Segment
Use a compass and straightedge to construct a line segment
—
that has the same length as AB.
A
B
SOLUTION
Step 1
Step 2
A
B
Step 3
A
C
B
C
Draw a segment Use a straightedge
—.
to draw a segment longer than AB
Label point C on the new segment.
Measure length Set your
—.
compass at the length of AB
A
B
C
D
Copy length Place the compass at
C. Mark point D on the new segment.
— has the same length as AB
—.
So, CD
Core Concept
Congruent Segments
READING
In the diagram, the
red tick marks indicate
—
AB ≅ —
CD . When there
is more than one pair of
congruent segments, use
multiple tick marks.
Line segments that have the same length are called congruent segments. You
— is equal to the length of CD
—,” or you can say “AB
— is
can say “the length of AB
—.” The symbol ≅ means “is congruent to.”
congruent to CD
A
B
Lengths are equal.
AB = CD
Segments are congruent.
— ≅ CD
—
AB
C
D
“is equal to”
“is congruent to”
Comparing Segments for Congruence
Plot J(−3, 4), K(1, 1), L(−1, −3), and M(3, 0) in a coordinate plane. Then determine
— and LM
— are congruent.
whether JK
y
J(−3, 4)
4
2
−4
−2
SOLUTION
K(1, 1)
M(3, 0)
2
−2
L(−1, −3)
4 x
Plot the points, as shown. Use the Distance Formula to find the length of
each segment.
JK =
——
—
—
√[1 − (−3)]2 + (1 − 4)2 = √42 + (−3)2 = √25 = 5
———
—
—
√[3 − (−1)]2 + [0 − (−3)]2 = √42 + 32 = √25 = 5
—
— have the same length. So, JK
— ≅ LM
—.
JK and LM
LM =
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
9. Plot A(−2, 4), B(2, 3), C(−2, −3), and D(1, −1) in a coordinate plane. Then
— and CD
— are congruent.
determine whether AB
Section 1.2
HSTX_GEOM_PE_01.02.indd 15
Measuring and Constructing Segments
15
6/19/14 4:49 PM
Exercises
1.2
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
—
1. WRITING Explain how XY and XY are different.
2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
7
3
B
C
A
Find AC + CB.
Find BC − AC.
Find AB.
Find CA + BC.
Monitoring Progress and Modeling with Mathematics
In Exercises 3 –10, find the length of the segment.
(See Example 1.)
A
−4
3.
B
−3
−2
−1
0
C
D
1
2
—
CD
4.
—
6. AD
—
EA
—
AB
10.
F
15
H
15.
37
H
G
13
F
16.
In Exercises 11–18, find FH. (See Example 2.)
11.
G
4
—
8. DB
9.
3
—
CE
—
7. AC
4
F
E
—
5. BC
14.
22
F
H
G
15
8
G
17.
14
F
42
H
12.
7
19
H
22
H
G
G
18.
F
13.
12
11
G
H
53
40
G
F
H
F
16
Chapter 1
HSTX_GEOM_PE_01.02.indd 16
Basics of Geometry
6/19/14 4:49 PM
32. USING STRUCTURE Determine whether the statements
In Exercises 19–26, find the distance between the
two points. (See Example 3.)
are true or false. Explain your reasoning.
19. A(13, 2) and B(7, 10)
C
B
20. C(−6, 5) and D(−3, 1)
A
21. E(3, 7) and F(6, 5)
D
E
H
F
22. G(−5, 4) and H(2, 6)
a. B is between A and C.
23. J(−8, 0) and K(1, 4)
b. C is between B and E.
24. L(7, −1) and M(−2, 4)
c. D is between A and H.
25. R(0, 1) and S(6, 3.5)
d. E is between C and F.
26. T(13, 1.6) and V(5.4, 3.7)
33. ERROR ANALYSIS Describe and correct the error in
—.
finding the length of AB
CONSTRUCTION In Exercises 27 and 28, use a compass
and straightedge to construct a copy of the segment.
A
B
1
27.
2
✗
28.
COMPARING SEGMENTS In Exercises 29 and 30, plot the
points in a coordinate plane. Then determine whether
— and CD
— are congruent. (See Example 4.)
AB
29. A(0, 2), B(−3, 8), C(−2, 2), D(0, −4)
30. A(1, 4), B(5, 1), C(−3, 1), D(1, 6)
B
A
C
Europe
601 mi
1282 mi
✗
b. The model airplane’s flight lasted nearly 38 hours.
Estimate the airplane’s average speed in
miles per hour.
Section 1.2
HSTX_GEOM_PE_01.02.indd 17
AB = (6 − 1)2 + [2 − (−4)]2
= 52 + 62
= 25 + 36
= 61
the position of three players during part of a water
polo match. Player A throws the ball to Player B,
who then throws the ball to Player C.
y
16
Player C
(24, 14)
12
Player B
8
0
a. Find the total distance the model airplane flew.
AB = ∣ 1 + 4.5 ∣ = 5.5
35. MODELING WITH MATHEMATICS The figure shows
(18, 7)
Player A
4
Atlantic Ocean
5
finding the distance between A(6, 2) and B(1, −4).
Distance (m)
North
America
4
34. ERROR ANALYSIS Describe and correct the error in
31. MODELING WITH MATHEMATICS In 2003, a remote-
controlled model airplane became the first ever to fly
nonstop across the Atlantic Ocean. The map shows the
airplane’s position at three different points during its
flight. Point A represents Cape Spear, Newfoundland,
point B represents the approximate position after
1 day, and point C represents Mannin Bay, Ireland.
The airplane left from Cape Spear and landed in
Mannin Bay.
3
(8, 4)
0
4
8
12
16
20
24
28 x
Distance (m)
a. How far did Player A throw the ball? Player B?
b. How far would Player A have to throw the ball to
throw it directly to Player C?
Measuring and Constructing Segments
17
6/19/14 4:49 PM
40. MAKING AN ARGUMENT Your friend claims there
36. MODELING WITH MATHEMATICS Your school is
is an easier way to find the length of a segment than
the Distance Formula when the x-coordinates of the
endpoints are equal. He claims all you have to do
is subtract the y-coordinates. Do you agree with his
statement? Explain your reasoning.
20 blocks east and 12 blocks south of your house.
The mall is 10 blocks north and 7 blocks west of
your house. You plan on going to the mall right after
school. Find the distance between your school and the
mall assuming there is a road directly connecting the
school and the mall. One block is 0.1 mile.
41. REASONING You travel from City X to City Y. You
37. MATHEMATICAL CONNECTIONS Point S is between
know that the round-trip distance is 647 miles. City Z, a
city you pass on the way, is 27 miles from City X. Find
the distance from City Z to City Y. Justify your answer.
—. Use the information to write an
points R and T on RT
equation in terms of x. Then solve the equation and
find RS, ST, and RT.
a. RS = 2x + 10
ST = x − 4
RT = 21
b. RS = 3x − 16
ST = 4x − 8
RT = 60
c. RS = 2x − 8
ST = 11
RT = x + 10
d. RS = 4x − 9
ST = 19
RT = 8x − 14
42. HOW DO YOU SEE IT? The bar graph shows the
win-loss record for a lacrosse team over a period of
three years. Explain how you can apply the Ruler
Postulate (Post. 1.1) and the Segment Addition
Postulate (Post. 1.2) when interpreting a stacked
bar graph like the one shown.
Win-Loss Record
Wins
38. THOUGHT PROVOKING Is it possible to design a table
where no two legs have the same length? Assume
that the endpoints of the legs must all lie in the same
plane. Include a diagram as part of your answer.
Losses
Year 1
Year 2
Year 3
0
2
39. MODELING WITH MATHEMATICS You have to walk
4
6
8
10
12
Number of games
from Room 103 to Room 117.
103
107
113
43. MATHEMATICAL CONNECTIONS In the diagram,
— ≅ BC
—, AC
— ≅ CD
—, and AD = 12. Find the
AB
lengths of all segments in the diagram. Suppose you
choose one of the segments at random. What is the
probability that the measure of the segment is greater
than 3? Explain your reasoning.
117
86 ft
22 ft
101
105
109
111
115
119
121
D
a. How many feet do you travel from Room 103
to Room 117?
44. CRITICAL THINKING Is it possible to use the Segment
Addition Postulate (Post. 1.2) to show FB > CB or
that AC > DB? Explain your reasoning.
c. Why might it take you longer than the time in
part (b)?
A
Maintaining Mathematical Proficiency
D
F
C
B
Reviewing what you learned in previous grades and lessons
(Skills Review Handbook)
−4 + 6
2
−3 + 9
46. —
45. —
Solve the equation.
2
Chapter 1
HSTX_GEOM_PE_01.02.indd 18
−8 + (−4)
47. —
2
7+6
2
48. —
(Skills Review Handbook)
49. 5x + 7 = 9x − 17
18
B
A
b. You can walk 4.4 feet per second. How many
minutes will it take you to get to Room 117?
Simplify.
C
3+y
2
50. — = 6
−5 + x
2
51. — = −9
52. −6x − 13 = −x − 23
Basics of Geometry
6/19/14 4:49 PM