Transcript Slide 1

Segment Measure and Coordinate Graphing

§ 2.1 Real Numbers and Number Lines

§ 2.2 Segments and Properties of Real Numbers

§ 2.3 Congruent Segments

§ 2.4 The Coordinate Plane

§ 2.5 Midpoints
5 Minute-Check
1. Find the next three terms of the sequence
12, 17, 23, 30, … , 38, 47, 57
2. Name the intersection of planes ABC and CDE
in the figure.
CD
D
3. How does a ray differ from a line?
A
A ray extends in only one direction and has an endpoint.
A line extends in two directions.
C
B
4. Find the perimeter and area of a rectangle with length of 10 centimeters
and width of 4 centimeters.
P  28 cm
A  40 cm2
E
F
Real Numbers and Number Lines
You will learn to find the distance between two points on a
number line.
1) Whole Numbers
2) Natural Numbers
3) Integers
4) Rational Numbers
5) Terminating Decimals
6) Nonterminating Decimals
7) Irrational Numbers
8) Real Numbers
9) Coordinate
10 Origin
11) Measure
12) Absolute Value
Real Numbers and Number Lines
Numbers that share common properties can be classified or grouped into sets.
Different sets of numbers can be shown on number lines.
This figure shows the set of _____________
whole numbers .
0
1
2
3
4
5
6
7
8
9
10
The whole numbers include 0 and the natural, or counting numbers.
indefinitely
The arrow to the right indicate that the whole numbers continue _________.
Real Numbers and Number Lines
This figure shows the set of _______
integers .
-5
-4
-3
negative integers
-2
-1
0
1
2
3
4
5
positive integers
The integers include zero, the positive integers, and the negative integers.
The arrows indicate that the numbers go on forever in both directions.
Real Numbers and Number Lines
A number line can also show ______________.
rational numbers
-2
5

3
11

8
-1
1 0

5
3
8
2
3
1
4
3
13
8
2
a
A rational number is any number that can be written as a _______,
fraction
b
where a and b are integers and b cannot equal ____.
zero
The number line above shows some of the rational numbers between -2 and 2.
infinitely many rational numbers between any two integers.
In fact, there are _______
Real Numbers and Number Lines
decimals
Rational numbers can also be represented by ________.
3
 0.375
8
2
 0.666 . . .
3
0
0
7
terminating or _____________.
nonterminating
Decimals may be __________
0.375
0.49
terminating decimals.
0.666 . . .
-0.12345 . . .
nonterminating decimals.
The three periods following the digits in the nonterminating decimals indicate
that there are infinitely many digits in the decimal.
Real Numbers and Number Lines
Some nonterminating decimals have a repeating pattern.
0.17171717 . . .
repeats the digits 1 and 7 to the right of the decimal point.
A bar over the repeating digits is used to indicate a repeating decimal.
0.171717 . . .  0.17
Each rational number can be expressed as a terminating decimal or a
nonterminating decimal with a repeating pattern.
Real Numbers and Number Lines
Decimals that are nonterminating and do not repeat
are called _______________.
irrational numbers
6.028716 . . .
and
0.101001000 . . .
appear to be irrational numbers
Real Numbers and Number Lines
Real numbers include both rational and irrational numbers.
____________
-2
1.8603 . . .
0
-1
0.8
0.25
3
8
1
0.6
2
1.762 . . .
The number line above shows some real numbers between -2 and 2.
Postulate 2-1
Number Line
Postulate
Each real number corresponds to exactly one point on a number
line.
Each point on a number line corresponds to exactly one real
number
Real Numbers and Number Lines
The number that corresponds to a point on a number line is called the
coordinate of the point.
_________
On the number line below, 10
__ is the coordinate of point A.
The coordinate of point B is __
-4
origin
Point C has coordinate 0 and is called the _____.
-6
-5
-4
A
C
B
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
x
11
Real Numbers and Number Lines
The distance between two points A and B on a number line is found by
using the Distance and Ruler Postulates.
Postulate 2-2
Distance
Postulate
For any two points on a line and a given unit of measure, there is a
unique positive real number called the measure of the distance
between the points.
A
B
measure
Postulate 2-3
Ruler
Postulate
Points on a line are paired with real numbers, and the measure of
the distance between two points is the positive difference of the
corresponding numbers.
A
B
b
measure = a – b
a
Real Numbers and Number Lines
The measure of the distance between B and A is the positive difference
10 – 2, or 8.
B
-6
-5
-4
-3
-2
-1
0
A
x
1
2
3
4
5
6
7
8
9
10
11
1
2
3
4
5
6
7
8
9
10
11
Another way to calculate the measure of the distance is by using
____________.
absolute value
AB  10  2
BA  2 10
8
 8
8
8
Real Numbers and Number Lines
Use the number line below to find the following measures.
A
-3
B
-2
1
C
-1
5  8
BA      
3  3
3

3
1
F
2
0
3
1
CF  (1)  2
 3
3
2
x
Real Numbers and Number Lines
Use the number line below to find the following measures.
A
-3
E
D
-2
1
-1
1  8
DA      
3  3
2
0
3
F
1
2
1
EF   2
3
7

3
5

3
7

3

5
3
x
Real Numbers and Number Lines
Traveling on I-70, the Manhattan exit is at mile marker 313.
The Hays exit is mile marker 154.
What is the distance between these two towns?
MH  313 154
 159
 159 miles
Real Numbers and Number Lines
5 Minute-Check
Find the value or values of the variable that makes each
equation true.
1.
3g  63
2.
12 x  7  67
3.
2 y 2  32
4.
2 z  4  3z  6  0
5.
If c  4 and d  3, what is the value of the expression
2  5d  3c  ?
g = 21
x=5
y=4
or
y = -- 4
z = -- 2
6
6. Find the next three terms of the sequence. 6, 12, 24, . . . 48, 96, 192
Segments and Properties of Real Numbers
You will learn to apply the properties of real numbers to the
measure of segments.
1) Betweenness
2) Equation
3) Measurement
4) Unit of Measure
5) Precision
Segments and Properties of Real Numbers
Given three collinear points on a line, one point is always _______
between the other
two points.
Definition
of
Betweenness
Point R is between points P and Q if and only if R, P, and Q are
collinear and _______________.
PR + RQ = PQ
Q
R
P
PR
PQ
RQ
NOTE: If and only if (iff) means that both the statement and its converse are true.
Statements that include this phrase are called biconditionals.
Segments and Properties of Real Numbers
Segment measures are real numbers.
Let’s review some of the properties of real numbers relating to EQUALITY.
Properties of Equality for Real Numbers.
Reflexive Property
Symmetric Property
Transitive Property
For any number a,
a=a
For any numbers a and b,
if a = b, then b = a
For any numbers a, b, and c,
if a = b and b = c
then a = c
Segments and Properties of Real Numbers
Segment measures are real numbers.
Let’s review some of the properties of real numbers relating to EQUALITY.
Properties of Equality for Real Numbers.
Addition and
Subtraction
Properties
Multiplication and
Division
Properties
Substitution
Properties
For any numbers a, b, and c, if a = b,
then a + c = b + c
and a – c = b – c
For any numbers a, b, and c, if a = b,
then a * c = b * c
and
a÷c=b÷c
For any numbers a and b, if a = b,
then a may be replaced by b in any equation.
Segments and Properties of Real Numbers
If QS = 29 and QT = 52,
P
Q
find ST.
S
QS + ST = QT
QS + ST – QS = QT – QS
ST = QT – QS
ST = 52 – 29
= 23
T
Segments and Properties of Real Numbers
If PR = 27 and PT = 73,
P
Q
find RT.
R
S
PR + RT = PT
PR + RT – PR = PT – PR
RT = PT – PR
RT = 73 – 27
= 46
T
Segments and Properties of Real Numbers
5 Minute-Check
1. Points X, Y, and Z are collinear.
If XY = 32, XZ = 49, and YZ = 81, determine which point is
between the other two.
X
Y
Z
Refer to the figure below:
A
Suppose AC = 49 and AB = 14.
B
2. Find BC.
C
BC = AC - AB
BC = 49 - 14
BC = 35
3. Suppose D is 5 units to the right of C. What is AD?
AD =
AC + 5
= 54
Congruent Segments
You will learn to identify congruent segments and find the
midpoints of segments.
In geometry, two segments with the same length are called
congruent segments
________
_________
Definition of
Congruent
Segments
Two segments are congruent if and only if
they
have the same length
________________________
Congruent Segments
In the figures at the right, AB is
congruent to BC, and PQ is
A
congruent to RS.
The symbol  is used to
represent congruence.
AB  BC, and PQ  RS.
R
B
C
Congruent Segments
Use the number line to determine if the statement is True or False.
Explain you reasoning.
RS  TY
R
-6
-5
S
-4
-3
-2
-1
T
0
1
2
Because RS = 4 and TY = 5,
3
4
RS  TY
So, RS is not congruent to TY,
and the statement is false.
5
6
Y
7
8
9
10
11
x
Congruent Segments
Since congruence is related to the equality of segment measures, there are
properties of congruence that are similar to the corresponding properties
of equality.
These statements are called ________.
theorems
Theorems are statements that can be justified by using logical reasoning.
2–1
Congruence of segments is
reflexive.
AB  AB
2–2
Congruence of segments is
symmetric.
If AB  CD, then CD  AB
2–3
Congruence of segments is
transitive.
If AB  CD, and CD  EF
then AB  EF
Congruent Segments
There is a unique point on every segment called the _______.
midpoint
On the number line below, M is the midpoint of
ST.
What do you notice about SM and MT?
S
-6
-5
-4
-3
-2
-1
0
1
2
M
3
SM = MT
4
5
6
T
7
8
9
10
x
11
Congruent Segments
A point M is the midpoint of a segment
between S and T and SM = MT
Definition of
Midpoint
S
M
ST
if and only if M is
T
SM = MT
The midpoint of a segment separates the segment into two segments of
equal _____.
length
_____
congruent
So, by the definition of congruent segments, the two segments are _________.
Congruent Segments
In the figure, B is the midpoint of
AC .
Find the value of x.
B
A
5x - 6
C
2x
Check!
Since B is the midpoint:
Write the equation involving x:
Solve for x:
AB = BC
5x – 6 = 2x
5x – 2x – 6 = 2x – 2x
3x – 6 + 6 = 0 + 6
AB = 5x – 6
= 5(2) – 6
= 10 – 6
=4
3x = 6
x=2
BC = 2x
= 2(2)
=4
Congruent Segments
To bisect something means to separate it into ___
two congruent parts.
midpoint of a segment bisects the segment because it separates the
The ________
segment into two congruent segments.
A point, line, ray, or plane can also bisect a segment.
E
Point C bisects AB
DC bisects AB
D
A
C
EC bisects AB
B
G
Plane GCD bisects AB
Congruent Segments
5 Minute-Check
1.
DF is bisected at point E, and DF  8.
What do you know about the lengths of DE and EF ?
The lengths are the same, both are 4.
2.
In the figure below, R is the midpoint of QS.
Find the value of d.
R
Q
d+4
3.
S
d + 4 = 3d
4 = 2d
2=d
3d
True or False: If AB  CD, then CD  AB
True; segment congruence is symmetric.
4.
True or False: If AB  BC, then B is the midpoint of AC.
False; Points A, B, and C may not be collinear.
5.
If a box has 5 red marbles, 5 blue marbles, P( B)  P(G)  5  5  10
15 15 15
and 5 green marbles, what is the probability
of selecting either a blue or green marble?
The Coordinate Plane
You will learn to name and graph ordered pairs on a
coordinate plane.
In coordinate geometry, grid paper is used to locate points.
The plane of the grid is called the coordinate plane.
y
5
4
3
2
1
x
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
2
3
4
5
The Coordinate Plane
The horizontal number line
is called the ______.
x-axis
y
5
4
Quadrant II
(–, +)
The vertical number line
is called the ______.
y-axis
3
2
1
Quadrant I
(+, +)
O
x
The point of intersection of the two
origin
axes is called the _____.
-5
-4
-3
-2
-1
1
3
4
-1
Quadrant III-2
(–, –) -3
-4
The two axes separate the plane into
quadrants
four regions called _________.
2
-5
Quadrant IV
(+, –)
5
The Coordinate Plane
An ordered pair of real numbers, called coordinates of a point, locates a
point in the coordinate plane.
one point in the
Each ordered pair corresponds to EXACTLY ________
coordinate plane.
The point in the coordinate plane is called the graph of the ordered pair.
graphing the ordered pair.
Locating a point on the coordinate plane is called _______
Postulate 2 – 4
Completeness
Property for Points
in the Plane
Each point in a coordinate plane corresponds to exactly one
__________________________.
ordered pair of real numbers
Each ordered pair of real numbers corresponds to exactly one
point in the coordinate plane
__________________________.
The Coordinate Plane
Graphing an ordered pair, (point): (x, y)
Graph point A at (4, 3)
y
The first number, 4, is called the
x-coordinate
___________.
5
4
(4, 3)
3
It tells the number of units the point lies to
left or right of the origin.
the __________
2
(0, 0)
1
x
-5
The second number, 3, is called the
y-coordinate
___________.
It tells the number of units the point lies
_____________
above or below the origin.
What is the coordinate of the origin?
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
2
3
4
5
The Coordinate Plane
Graphing an ordered pair, (point): (x, y)
Graph point B at (2, –3)
y
The first number, 2, is called the
x-coordinate
___________.
5
4
3
It tells the number of units the point lies to
left or right of the origin.
the __________
2
1
x
-5
The second number, –3, is called the
y-coordinate
___________.
It tells the number of units the point lies
_____________
above or below the origin.
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
-5
(2, –3)
5
The Coordinate Plane
Name the points A, B, C, & D
y
5
4
Point A(x, y) = (3, 2)
3
B (–3, 2)
2
Point B(x, y) = (–3, 2)
Point C(x, y) = (–3, –2)
Point D(x, y) = (3, –2)
A (3, 2)
1
x
-5
-4
-3
-2
-1
1
2
3
4
5
-1
-2
C (–3, –2) -3
-4
-5
D (3, –2)
The Coordinate Plane
 On a piece of grid paper draw lines
representing the x-axis and the y-axis.
y
5
4
 Graph : Point A(x, y) = A(2, 4)
3
2
Point B(x, y) = B(2, 0)
Point C(x, y) = C(2, –3)
1
x
-5
-4
-3
-2
-1
1
2
3
-1
Point D(x, y) = D(2, –5)
-2
-3
Consider these questions:
-4
-5
x=2
Write
a general
about
pairs
thatpoints?
have the
1)
What
do you statement
notice about
the ordered
graphs of
these
same
x-coordinate.
They
lie on a vertical line.
They
lie on
vertical
line
that the
intersects
the x-axis
at thepoints?
x-coordinate.
2) What
doayou
notice
about
x-coordinates
of these
They are the same number.
4
5
The Coordinate Plane
On the same coordinate plane
y
5
4
 Graph : Point W(x, y) = W(–4, –4)
3
2
Point X(x, y) = X(–2, –4)
1
Point Y(x, y) = Y(0, –4)
x
-5
-4
-3
-2
-1
1
2
3
4
-1
Point Z(x, y) = Z(3, –4)
-2
-3
Consider these questions:
y=–4
-4
-5
Write
a general
about
pairs
thatpoints?
have the
1)
What
do you statement
notice about
the ordered
graphs of
these
same
y-coordinate.
They
lie on a horizontal line.
They
lie on
horizontal
line that
the y-axis
at the
y-coordinate.
2) What
doayou
notice about
theintersects
y-coordinates
of these
points?
They are the same number.
5
The Coordinate Plane
If a and b are real numbers,
a vertical line contains all points (x, y) such that _____
x=a
Theorem 2 – 4
and
y=b
a horizontal line contains all points (x, y) such that _____
y
Graph the lines: x = –3
y=2
5
4
(–3, 2)
3
2
1
Graph the point of intersection of these
lines.
x
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
2
3
4
5
The Coordinate Plane
5 Minute-Check
y
1) Name the coordinates of each point.
4
A = (2, 0)
B = (3, –2)
2
A
x
0
B
-2
-3
-2
-1
1
2
3
4
5
0
2) Graph point C at (0, –4).
y
x = –4
5
4
3) Graph x = –4.
3
(–4, 2)
y=2
2
1
x
-5
4) Graph y = 2.
-4
-3
-2
-1
1
2
3
-1
-2
-3
5) Graph and label the intersection of
x = –4
and
y=2
-4
-5
C(0, -4)
4
5
Midpoints
You will learn to find the coordinates of the midpoint
of a segment.
A
C
B
The midpoint of a line segment, AB , is the point C that ______
bisects the segment.
C
A
-7
-6
-5
-4
-3
-2
-1
B
0
1
C = [3 + (-5)] ÷ 2
= (-2) ÷ 2
= -1
2
3
4
5
6
7
Midpoints
On a number line, the coordinate of the midpoint of a segment
whose endpoints have coordinates a and b is
ab
2
Theorem 2 – 5
A
ab
2
B
Midpoints
Find the midpoint, C(x, y), of a segment on the coordinate plane.
Consider the x-coordinate:
y
x=1
10
It will be (midway)
between the lines
x = 1 and x = 9
x=9
9
8
A
y=7
7
6
C(x, y)
Consider the y-coordinate:
y
5
4
It will be (midway)
between the lines
y = 3 and y = 7
y=3
3
B
2
1
x
0
x
-1
-2
-2
-1
1
0
2
3
4
5
6
7
8
9
10
Midpoints
On a coordinate plane, the coordinates of the midpoint of a
segment whose endpoints have coordinates (x1, y1) and
(x2, y2) are
 x1  x2 y1  y2 
,


2 
 2
y
( x1 , y1 )
Theorem 2 – 6
 x1  x2 y1  y2 
,


2 
 2
( x2 , y2 )
O
x
Midpoints
Find the midpoint, C(x, y), of a segment on the coordinate plane.
y
 x1  x2 y1  y2 
C 
,

2
2


x=1
10
x=9
9
8
A(1, 7)
73 
 1 9
C 
,

2
2


y=7
7
6
C(5, 5)
y
5
 10
C 
,
 2
C  5,5
10 
2 
4
y=3
3
B(9, 3)
2
1
x
0
x
-1
-2
-2
-1
1
0
2
3
4
5
6
7
8
9
10
Midpoints
Graph A(1, 1) and B(7, 9)
Draw AB
y
10
B(7, 9)
Estimate the midpoint
of AB.
9
8
7
Check your answer
using the midpoint formula.
6
5
 x1  x2 y1  y2 
C 
,

2
2


 1+7 1+9 
C 
,

2
2


 8 10 
C  , 
2 2 
C(4,
C 5)
4
3
2
1
A(1, 1)
x
0
-1
-2
-2
-1
1
0
2
3
4
5
6
7
8
9
10
Midpoints
Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.
x-coordinate of B
y-coordinate of B
y
x1  x2
3
2
y1  y2
5
2
10
9
B(-1, 8)
8
7
Replace x1 with 7
and y1 with 2
6
7  x2
3
2
2  y2
5
2
midpoint
4
3
A(7, 2)
2
1
2  y2  10
x
0
-1
-2
-2
Add or subtract to
isolate the variable
x2  1
C(3, 5)
5
Multiply each
side by 2
7  x2  6
B(x, y) is somewhere over there.
-1
1
0
y2  8
2
3
4
5
6
7
8
9
10
Midpoints
A
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
1
1
2
2
3
4
5
3
6
7
8
9
10
11