1.2 Measuring and Constructing Segments - Schoolwires
1.2 Measuring and Constructing Segments
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CM
MAKING SENSE OF PROBLEMS
To be proficient in math, you need to explain to yourself the meaning of a problem and look for entry points to its solution.
Essential Question How can you measure and construct a
line segment?
Measuring Line Segments Using Nonstandard Units
Work with a partner.
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."
12 11 10 9 8 7 6 5 4 3 2 1
INCH
c. Write conversion factors from paper clips to inches and vice versa.
1 paper clip = in.
1 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
on one of its diagonals.
3 in.
fold
b. Use the Pythagorean Theorem
to algebraically determine the
length of the diagonal in inches.
Use a ruler to check your answer.
5 in.
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 Measuring and Constructing Segments
11
1.2 Lesson
Core Vocabulary
postulate, p. 12 axiom, p. 12 coordinate, p. 12 distance, p. 12 construction, p. 13 congruent segments, p. 13 between, p. 14
What You Will Learn
Use the Ruler Postulate. Copy segments and compare segments for congruence. Use the Segment Addition Postulate.
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 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.
names of points
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
x1
x2
AB = x2 - x1
Using the Ruler Postulate Measure the length of -- ST to the nearest tenth of a centimeter.
S
T
SOLUTION
Align one mark of a metric ruler with S. Then estimate the coordinate of T. For example, when you align S with 2, T appears to align with 5.4.
S
T
cm 1
2
3
4
5
6
ST = 5.4 - 2 = 3.4
Ruler Postulate
So, the length of S--T is about 3.4 centimeters.
Monitoring Progress
Help in English and Spanish at
Use a ruler to measure the length of the segment to the nearest --18 inch.
1. M
2.
N
P
Q
3.
4.
U
V
W
X
12
Chapter 1 Basics of Geometry
Step 1
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 that
haacsotmhepassasmaenldenstgrtahigahstAe--dBg.e
to
construct
a
line
segment
A
B
SOLUTION Step 2
Step 3
A
B
C
Draw a to draw
segment a segment
UlosnegaersttrhaainghA--tBed. ge
Label point C on the new segment.
READING
In the diagram, the
r--AeBd tic--CkDm.aWrkhseinndthiceartee
is more than one pair of congruent segments, use multiple tick marks.
A
B
A
B
C
McoemapsausrsealtetnhgethlenSgetthyoofuA--rB.
C
D
Copy length Place the compass at
CSo. ,MC--aDrkhpaositnhteDsaomnethleenngetwh assegA--mBe.nt.
Core Concept
Congruent Segments
Lccaoinnnegsrasuyeeg"nmttheteontlC--senDthg.at"htThohafevA--esBythmiesbesoaqlmuael
length to the
laernegcthalolefdC--cDo,n"gorruyeonut
means "is congruent to."
sceagnmsaeynt"sA--.BYoius
A
B
Lengths are equal. AB = CD
SegmenA--tsBareC--coDngruent.
C
D
"is equal to"
"is congruent to"
J(-3, 4)
y
K(2, 4)
L(1, 3)
2
-4 -2 -2
2
4x
M(1, -2)
Comparing Segments for Congruence
PwlhoettJh(e-r J--3K, 4a)n, dKL(--2M, 4a)r,eLc(o1n, g3r)u, eanntd. M(1, -2) in a coordinate plane. Then determine
SOLUTION
Plot the points, as shown. To find the length of a horizontal segment, find the absolute value of the difference of the x-coordinates of the endpoints.
JK = 2 - (-3) = 5
Ruler Postulate
To find the length of a vertical segment, find the absolute value of the difference of the y-coordinates of the endpoints.
LM = -2 - 3 = 5
Ruler Postulate
--JK and L--M have the same length. So, J--K L--M.
Monitoring Progress
Help in English and Spanish at
5.
PdelotetrAm(i-ne2,w4h)e, tBh(e3r,A--4B),
aCn(d0,C--2D),
and D(0, -2) are congruent.
in
a
coordinate
plane.
Then
Section 1.2 Measuring and Constructing Segments
13
Using the Segment Addition Postulate
When three points are collinear, you can say that one point is between the other two.
A B C
D
E
F
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. If AB + BC = AC, then B is between A and C.
AC
A
B
C
AB BC
a. Find DF.
Using the Segment Addition Postulate
D 23 E 35 F
b. Find GH.
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.
144 J 37 K
Monitoring Progress
Help in English and Spanish at
Use the diagram at the right.
6. Use the Segment Addition Postulate to find XZ.
7. In the diagram, WY = 30. Can you use the Segment Addition Postulate to find the distance between points W and Z ? Explain your reasoning.
23
50
X
Y
Z
W
8. Use the diagram at the left to find KL. L
14
Chapter 1 Basics of Geometry
Using the Segment Addition Postulate
The cities shown on the map lie approximately in a straight line. Find the distance from Tulsa, Oklahoma, to St. Louis, Missouri.
738 mi T Tulsa
377 mi L Lubbock
S St. Louis
SOLUTION
1. Understand the Problem You are given the distance from Lubbock to St. Louis and the distance from Lubbock to Tulsa. You need to find the distance from Tulsa to St. Louis.
2. Make a Plan Use the Segment Addition Postulate to find the distance from Tulsa to St. Louis.
3. Solve the Problem Use the Segment Addition Postulate to write an equation. Then solve the equation to find TS.
LS = LT + TS 738 = 377 + TS 361 = TS
Segment Addition Postulate Substitute 738 for LS and 377 for LT. Subtract 377 from each side.
So, the distance from Tulsa to St. Louis is about 361 miles.
4. Look Back Does the answer make sense in the context of the problem? The distance from Lubbock to St. Louis is 738 miles. By the Segment Addition Postulate, the distance from Lubbock to Tulsa plus the distance from Tulsa to St. Louis should equal 738 miles.
377 + 361 = 738
Monitoring Progress
Help in English and Spanish at
9. The cities shown on the map lie approximately in a straight line. Find the distance from Albuquerque, New Mexico, to Provo, Utah.
Provo P
680 mi Albuquerque A 231 mi
C Carlsbad
Section 1.2 Measuring and Constructing Segments
15
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