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Lesson 1-5
Segments and
Their Measures
1
Objectives
• Measure segments.
• Add segment lengths.
Key Vocabulary
•
•
•
•
•
•
Coordinate
Distance
Length
Between
Congruent
Congruent Segments
Postulates
• Postulate 5: Segment Addition Postulate
The Number Line (a ruler)
To every point on a line there
corresponds
a unique real number.
To every real number there
corresponds a unique point on the line.
Definition: Coordinate – The line number
at corresponds to a point on a line.
Points On A Line
Names of points
• 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.
• In the diagram, x1 and x2 are
coordinates. The small
numbers are subscripts. The
coordinates are read as
“x sub 1” and
“x sub 2.”
A
B
x1
x2
Coordinates of points
A
x1
AB
B
x2
Distance
Names of points
A
B
x1
x2
Coordinates of points
• Definition: Distance – is a numerical
description of how far apart objects are. It is
the absolute value of the difference
between the coordinates of A and B.
• The distance is always positive.
• AB is also called the length of segment AB.
The Ruler
Points on a line can be paired with the real numbers in such a way
that:
• Any two chosen points can be paired with coordinates on a ruler.
• The distance between any two points on a number line is the
absolute value of the difference of the real numbers corresponding
to the points.
Formula: Take the absolute value of the difference
of the two coordinates a and b: │a – b │
The Ruler
A
B
x1
x2
AB = |x2 – x1|
• The length AB can be found by |x2-x1|.
• **The symbol for the length of AB is
AB.**
Example: Find AB.
A
B
Point A is at 2.5 and B is at 5.
So, AB = |5 - 2.5| = 2.5
Note:
• It doesn’t matter how you place the ruler.
For example, if the ruler in The Example is
placed so that A is aligned with 1, then B
aligns with 3.5. The difference in the
coordinates is the same.
A
B
Ruler: Example
Find the distance between P and K.
G
H
I
J
K
L
M
N
O
P
Q
R
-5
Note:
S
5
The coordinates are the numbers on the ruler or number line!
The capital letters are the names of the points.
Therefore, the coordinates of points P and K are 3 and -2 respectively.
Substituting the coordinates in the formula │a – b │
PK = | 3 - -2 | = 5
Remember: Distance is
always positive
Example 1
Measure the total length of the shark’s tooth to the
1
nearest 8 inch. Then measure the length of the exposed
part.
SOLUTION
Use a ruler to measure in inches.
1.
Align the zero mark of the ruler
with one end of the shark’s tooth.
2. Find the length of the shark’s tooth, AC.
AC = 2
1
1
–0 =2
4
4
3.
Find the length of the exposed part, BC.
3
7
1
= 1
BC = 2 –
8
8
4
Example 2
A. Find the length of AB using the ruler.
The ruler is marked in millimeters.
Point B is closer to the 42 mm mark.
Answer: AB is 42 millimeters long.
Example 2
B. Find the length of AB using the ruler.
Each centimeter is divided into fourths.
Point B is closer to the 4.5 cm mark.
Answer: AB is 4.5 centimeters long.
Your Turn:
A. 2 mm
B. 1.8 mm
C. 18 mm
D. 20 mm
Your Turn
B.
A. 1 cm
B. 2 cm
C. 2.5 cm
D. 3 cm
Example 3
A.
Each inch is divided into sixteenths. Point E is closer
to the 3-inch mark.
Example 3
B.
Your Turn:
A.
A.
B.
C.
D.
Your Turn:
B.
A.
B.
C.
D.
Betweenness
Definition:
X is between A and B if AX + XB = AB.
Betweenness refers to collinear points only.
X
A
X
B
A
B
AX + XB = AB
AX + XB > AB
X is between A and B
X is not between A and B
Because the 3 points are not collinear.
Is Alex between Ty and Josh?
Yes!
Ty
Alex
Josh
How about now?
No, but
why not?
In order for a point to be between 2 others, all 3
points MUST BE collinear!!
Between
A
B
• When three points lie
on a line, you can say
that one of them is
between the other
two. This concept
applies to collinear
points only.
Point B is
between points
A and C.
C
Postulate 5: 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
AB
C
BC
Segment Addition
If C is between A and B, then AC + CB = AB.
Example: If AC = x , CB = 2x and AB = 12, then, find x, AC
and CB.
B
2x
A x C
Step 1: Draw a figure
12
Step 2: Label fig. with given info.
AC + CB = AB
Step 3: Write an equation
x + 2x = 12
Step 4: Solve and
find all the answers
3x = 12
x = 4
x = 4
AC = 4
CB = 8
Example 4
Find XZ. Assume that the figure is not drawn to
scale.
___
XZ is the measure of XZ. Point Y is between X and
Z. XZ can be found by adding XY and YZ.
Add.
Your Turn:
Find BD. Assume that the figure is not
drawn to scale.
50.4 mm
A. 16.8 mm
B. 57.4 mm
16.8 mm
B
C. 67.2 mm
D. 84 mm
C
D
Example 5
Find LM. Assume that the figure is not drawn
to scale.
Point M is between L and N.
LM + MN = LN
Betweenness of points
LM + 2.6 = 4
Substitution
LM + 2.6 – 2.6 = 4 – 2.6
LM = 1.4
Subtract 2.6 from
each side.
Simplify.
Your Turn:
Find TU. Assume that the figure is not
drawn to scale.
3 in
A.
in.
T
B.
in.
C.
in.
D.
in.
U
V
Example 6
Use the map to find the distance
from Athens to Albany.
SOLUTION
Because the three cities lie on a line, you can use the Segment
Addition Postulate.
AM = 80 miles
Use map.
MB = 90 miles
Use map.
AB = AM + MB
= 80 + 90
Segment Addition Postulate
Substitute.
= 170
Add.
ANSWER The distance from Athens to Albany is 170
miles.
Example 7
Use the diagram to find EF.
SOLUTION
DF = DE + EF
16 = 10 + EF
16 – 10 = 10 + EF – 10
6 = EF
Segment Addition Postulate
Substitute 16 for DF and 10 for DE.
Subtract 10 from each side.
Simplify.
Your Turn:
Find the length.
1. Find AC.
ANSWER
20
ANSWER
8
2. Find ST.
Example 8
ALGEBRA Find the value of x and ST if T is between
S and U, ST = 7x, SU = 45, and TU = 5x – 3.
Draw a figure to represent
this situation.
ST + TU = SU
7x + 5x – 3 = 45
7x + 5x – 3 + 3 = 45 + 3
12x = 48
Betweenness of points
Substitute known values.
Add 3 to each side.
Simplify.
Example 8
x =4
Simplify.
Now find ST.
ST = 7x
Given
= 7(4)
x=4
= 28
Multiply.
Answer: x = 4, ST = 28
Your Turn:
ALGEBRA Find the value of n and WX if W is
between X and Y, WX = 6n – 10, XY = 17, and
WY = 3n.
A. n = 3; WX = 8
B. n = 3; WX = 9
C. n = 9; WX = 27
D. n = 9; WX = 44
Definition of congruent
Congruent means having the same
measure.
≅
Congruent means same,
different from equal.
Congruent Segments
Definition: Segments with equal lengths. (congruent symbol:
A
If numbers are equal the objects are congruent.
C
D
AB: the segment AB ( an object )
AB: the distance from A to B ( a number )
Incorrect
notation:
)
B
Congruent segments can be marked with dashes.
Correct notation:

AB = CD
AB  CD
AB  CD
AB = CD
How to mark congruent
segments in a figure
A
E
AB  AE
D
B
C
How to mark congruent
segments in figures
A
F
D
B
G
C
BC  DF
E
AC  EG
Example 9
Are the segments shown in the
coordinate plane congruent?
SOLUTION
For a horizontal segment, subtract the x-coordinates.
DE = 1 – (–3) = 4 = 4
For a vertical segment, subtract the y-coordinates.
FG = –3 –1 = –4 = 4
ANSWER
DE and FG have the same length. So,
DE  FG.
Your Turn:
Plot the points in a coordinate plane. Then decide
whether AB and CD are congruent.
1. A(–2, 3), B(3, 3), C(–3, 4), D(–3, –1)
ANSWER
yes;
2. A(0, 5), B(0, –1), C(5, 0), D(–1, 0)
ANSWER
yes;
Joke Time
How do crazy people go through the
forest?
They take the psycho path.
What do you call a boomerang that
doesn't work?
A stick.
Assignment
• Section 1.5, pg. 31 – 33: #1 – 45 odd