Transcript 1-8-13

1/8/13
Unit 1 Basics of Geometry
Segments
and Rays
1
Segment
Definition: Part of a line that consists of two points called the
endpoints and all points between them.
How to sketch:
How to name:
A
B
AB or BA
The symbol AB is read as "segment AB".
AB (without a symbol) means the length of
the segment or the distance between points
A and B.
2
Distance on a number line :
Find the distance between P and K.
G
H
I
J
K
L
M
N
O
-5
Note:
P
Q
R
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
4
Between
Definition:
X is between A and B if AX + XB = AB.
X
A
X
AX + XB = AB
B
A
B
AX + XB > AB
5
The Segment Addition Postulate
Postulate: 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
Step 2: Label fig. with given info.
Step 3: Write an equation
Step 4: Solve and find all the answers
12
AC + CB = AB
x + 2x = 12
3x = 12
x = 4
x = 4
AC = 4
CB = 8
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Congruent Segments
Definition: Segments with equal lengths. (congruent symbol:

)
B
Congruent segments can be marked with dashes.
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 )
Correct notation:
AB = CD
AB  CD
Incorrect notation:
AB  CD
AB = CD
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Midpoint
Definition: A point that divides a segment into
two congruent segments
If DE  EF , then E is the midpoint of DF.
Formulas:
F
E
D
On a number line, the coordinate of the midpoint of a segment
whose endpoints have coordinates a and b is a  b .
2
In a coordinate plane, the coordinates of the midpoint of a
segment whose endpoints have coordinates ( x1 , y1 ) and ( x2 , y2 )
is
 x1  x2 y1  y2 
,


2 
 2
.
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Midpoint on Number Line - Example
Find the coordinate of the midpoint of the segment PK.
G
H
I
J
K
L
M
N
O
P
Q
-5
R
S
5
a  b 3  (2) 1

  0.5
2
2
2
Now find the midpoint on the number line.
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Segment Bisector
Definition: Any segment, line or plane that divides a segment into two
congruent parts is called segment bisector.
A
F
A
B
E
AB bisects DF.
B
D
E
AB bisects DF.
F
A
E
D
D
F
Plane M bisects DF.
B
AB bisects DF.
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Ray
Definition: RA : RA and all points Y such that
A is between R and Y.
A
How to sketch:
R
A
Y
R
How to name:
RA ( not AR )
RA or RY ( not RAY )
( the symbol RA is read as “ray RA” )
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Opposite Rays
Definition: If A is between X and Y, AX and AY are opposite rays.
( Opposite rays must have the same “endpoint” )
X
A
Y
opposite rays
not opposite rays
D
E
DE and ED are not opposite rays.
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Geometric Concepts
• Through any two points there is
exactly one line.
• A line contains at least two points.
• Through any three points, there is
exactly one plane.
• A plane contains at least three points.
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• If two planes intersect,
then the intersecting is a line.
• If two points lie in a plane,
then the line containing the two
points lie in the same plane.
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