§ 2.1 Real Numbers and Number Lines Real Numbers and Number LinesReal Numbers and Number Lines §...

Segment Measure and Coordinate GraphingSegment Measure and Coordinate GraphingSegment Measure and Coordinate GraphingSegment Measure and Coordinate Graphing

§§ 2.1 2.1 Real Numbers and Number Lines

§§ 2.4 2.4 The Coordinate Plane

§§ 2.3 2.3 Congruent Segments

§§ 2.2 2.2 Segments and Properties of Real Numbers

§§ 2.5 2.5 Midpoints

1. Find the next three terms of the sequence 12, 17, 23, 30, … ,

2. Name the intersection of planes ABC and CDE in the figure. CD

��

38, 47, 57

A ray extends in only one direction and has an endpoint.

5 Minute-Check5 Minute-Check

A

B

D

E

F

C

3. How does a ray differ from a line?

A line extends in two directions.

4. Find the perimeter and area of a rectangle with length of 10 centimeters and width of 4 centimeters.

28 P cm240 A cm

Real Numbers and Number LinesReal Numbers and Number Lines

You will learn to find the distance between two points on anumber line.

1) Whole Numbers2) Natural Numbers3) Integers4) Rational Numbers5) Terminating Decimals6) Nonterminating Decimals7) Irrational Numbers8) Real Numbers9) Coordinate10 Origin11) Measure12) Absolute Value


Numbers that share common properties can be classified or grouped into sets.Different sets of numbers can be shown on number lines.

0 21 43 65 7 98 10

This figure shows the set of _____________ . whole numbers

The whole numbers include 0 and the natural, or counting numbers.

The arrow to the right indicate that the whole numbers continue _________.indefinitely


0 21 43-5 5-4 -2-3 -1

This figure shows the set of _______ . integers

positive integersnegative integers

The integers include zero, the positive integers, and the negative integers.The arrows indicate that the numbers go on forever in both directions.


A number line can also show ______________. rational numbers

0 21-1-2 4

3

3

82

313

81

5

5

3 11

8

A rational number is any number that can be written as a _______,where a and b are integers and b cannot equal ____.

a

bfraction

zero

The number line above shows some of the rational numbers between -2 and 2.

In fact, there are _______ many rational numbers between any two integers.infinitely


Rational numbers can also be represented by ________.decimals

30.375

8

20.666 . . .

3 0

07

Decimals may be __________ or _____________.terminating nonterminating

0.3750.49

terminating decimals.

0.666 . . .-0.12345 . . .

nonterminating decimals.

The three periods following the digits in the nonterminating decimals indicatethat there are infinitely many digits in the decimal.

The three periods following the digits in the nonterminating decimals indicatethat there are infinitely many digits in the decimal.


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.


Decimals that are nonterminating and do not repeat are called _______________.irrational numbers

6.028716 . . . and0.101001000 . . .

appear to be irrational numbers


The number line above shows some real numbers between -2 and 2.

____________ include both rational and irrational numbers.Real numbers

Postulate 2-1

Number Line

Postulate

Each real number corresponds to exactly one point on a numberline.

Each point on a number line corresponds to exactly one realnumber

0 21-1-2 3

81.8603 . . . 0.8 0.6 1.762 . . .0.25


The number that corresponds to a point on a number line is called the_________ of the point.coordinate

On the number line below, __ is the coordinate of point A.

The coordinate of point B is __

Point C has coordinate 0 and is called the _____.

-4

10

origin

x

11-5 -3 -1 1 3 5 7 9-6 -4 0 4 8-6 2 106-2

AB C


The distance between two points A and B on a number line is found byusing 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 distancebetween the points.

AB

measure

Postulate 2-3

Ruler

Postulate

Points on a line are paired with real numbers, and the measure ofthe distance between two points is the positive difference of the corresponding numbers.

measure = a – b

B A

ab


x

11-5 -3 -1 1 3 5 7 9-6 -4 0 4 82 106-2

AB

The measure of the distance between B and A is the positive difference10 – 2, or 8.

Another way to calculate the measure of the distance is by using____________.absolute value

10 2AB

8

8

2 10BA

8

8

1 2 43 5 76 8 9 10 11


x

-2 0 2-1-3 1

Use the number line below to find the following measures.

A B C F

BA5 8

3 3

3

3

1

CF ( 1) 2

3

3

1 2 3


x

-2 0 2-1-3 1

Use the number line below to find the following measures.

A D E F

DA1 8

3 3

7

3

7

3

EF 1

23

5

3

5

3

1 2 3


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?

313 154

159

159 miles

MH

Find the value or values of the variable that makes each equation true.

1.

2.

3.

4.

5.

6. Find the next three terms of the sequence. 6, 12, 24, . . .

3 63g

12 7 67x 22 32y

2 4 3 6 0z z

If 4 and 3, what is the value of the expression

2 5 3 ?

c d

d c

g = 21

x = 5

y = 4 or y = -- 4

z = -- 2

6

48, 96, 192


Segments and Properties of Real NumbersSegments 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


Given three collinear points on a line, one point is always _______ the othertwo points.

between

Definition

of

Betweenness

Point R is between points P and Q if and only if R, P, and Q arecollinear and _______________.

P QR

PR RQPQ

PR + RQ = PQ

NOTE: If and only if (iff) means that both the statement and its converse are true.Statements that include this phrase are called biconditionals.


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 For any number a, a = a

Symmetric PropertyFor any numbers a and b,

if a = b, then b = a

Transitive PropertyFor any numbers a, b, and c,

if a = b and b = c then a = c


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

For any numbers a, b, and c, if a = b,

then a + c = b + c and

Multiplication andDivision

Properties

For any numbers a, b, and c, if a = b,

then a * c = b * c and a ÷ c = b ÷ c

Substitution Properties

For any numbers a and b, if a = b,

then a may be replaced by b in any equation.

a – c = b – c


P Q S T

If QS = 29 and QT = 52, find ST.

QS + ST = QT

QS + ST – QS = QT – QS

ST = QT – QS

ST = 52 – 29 = 23

S


P Q T

If PR = 27 and PT = 73, find RT.

PR + RT = PT

PR + RT – PR = PT – PR

RT = PT – PR

RT = 73 – 27 = 46

R


1. Points X, Y, and Z are collinear. If XY = 32, XZ = 49, and YZ = 81, determine which point is between the other two.

Y ZX

Refer to the figure below: Suppose AC = 49 and AB = 14.

A CB

2. Find BC.

3. Suppose D is 5 units to the right of C. What is AD?

BC = AC - AB

BC = 49 - 14

BC = 35

AD = AC + 5 = 54

Congruent SegmentsCongruent Segments

In geometry, two segments with the same length are called ________ _________congruent segments

You will learn to identify congruent segments and find the midpoints of segments.

Definition of

Congruent

Segments

Two segments are congruent if and only if

________________________they have the same length


BA C

R

S

P

Q

In the figures at the right, AB is

congruent to BC, and PQ is

congruent to RS.

The symbol is used to

represent congruence.

AB BC, and PQ RS.


x

11-5 -3 -1 1 3 5 7 9-6 -4 0 4 82 106-2

R S T Y

Use the number line to determine if the statement is True or False.Explain you reasoning.

RS TY

So, RS is not congruent to TY,

and the statement is false.

Because RS = 4 and TY = 5, TYRS


Since congruence is related to the equality of segment measures, there areproperties of congruence that are similar to the corresponding propertiesof 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 , then AB CD CD AB

2 – 3 Congruence of segments is transitive.

If , and

then

AB CD CD EF AB EF


There is a unique point on every segment called the _______.midpoint

On the number line below, M is the midpoint of . ST

x

11-5 -3 -1 1 3 5 7 9-6 -4 0 4 82 106-2

S M T

What do you notice about SM and MT?

SM = MT


Definition ofMidpoint

A point M is the midpoint of a segment if and only if M is between S and T and SM = MT

ST

MS T

SM = MT

The midpoint of a segment separates the segment into two segments of_____ _____.equal length

So, by the definition of congruent segments, the two segments are _________.congruent


In the figure, B is the midpoint of .AC Find the value of x.

5x - 6 2x

CBA

Since B is the midpoint: AB = BC

Write the equation involving x: 5x – 6 = 2x

Solve for x: 5x – 2x – 6 = 2x – 2x

3x – 6 + 6 = 0 + 6

3x = 6

x = 2

AB = 5x – 6 = 5(2) – 6 = 10 – 6 = 4

BC = 2x = 2(2) = 4

Check!


To bisect something means to separate it into ___ congruent parts.two

The ________ of a segment bisects the segment because it separates thesegment into two congruent segments.

A point, line, ray, or plane can also bisect a segment.

Point C bisects AB

bisects ABDC��

bisects ABEC��

Plane GCD bisects AB

midpoint

B

A

E

G

CD


is bisected at point E, and DF 8.

What do you know about the lengths of DE and EF ?

DF

In the figure below, R is the midpoint of QS.

Find the value of d.

d + 4 3d

SRQ

1.

2.

True or False: If , then AB CD CD AB 3.

True or False: If , then is the midpoint of .AB BC B AC4.

The lengths are the same, both are 4.

d + 4 = 3d 4 = 2d

True; segment congruence is symmetric.

False; Points A, B, and C may not be collinear.

2 = d

5. If a box has 5 red marbles, 5 blue marbles, and 5 green marbles, what is the probability of selecting either a blue or green marble?

)()( GPBP 15

5

15

5

15

10

The Coordinate PlaneThe Coordinate Plane

You will learn to name and graph ordered pairs on acoordinate plane.

In coordinate geometry, grid paper is used to locate points.

The plane of the grid is called the coordinate plane.

y

x5-4 -2 1 3 5

5

-4

-2

1

3

5

-5 -1 4

-5

-1

4

-3

-5

2

2-5

-3


y

x5-4 -2 1 3 5

5

-4

-2

1

3

5

-5 -1 4

-5

-1

4

-3

-5

2

2-5

-3

The horizontal number lineis called the ______.x-axis

The vertical number lineis called the ______.y-axis

O

The point of intersection of the twoaxes is called the _____.origin

The two axes separate the plane intofour regions called _________.quadrants

Quadrant I(+, +)

Quadrant II

(–, +)

Quadrant III

(–, –)

Quadrant IV

(+, –)


An ordered pair of real numbers, called coordinates of a point, locates apoint in the coordinate plane.

Each ordered pair corresponds to EXACTLY ________ in the coordinate plane.

one point

The point in the coordinate plane is called the graph of the ordered pair.

Locating a point on the coordinate plane is called _______ the ordered pair.graphing

Postulate 2 – 4

Completeness Property for Points

in the Plane

Each point in a coordinate plane corresponds to exactly one __________________________.

Each ordered pair of real numbers corresponds to exactly one __________________________.

ordered pair of real numbers

point in the coordinate plane


y

x5-4 -2 1 3 5

5

-4

-2

1

3

5

-5 -1 4

-5

-1

4

-3

-5

2

2-5

-3

Graphing an ordered pair, (point): (x, y)

Graph point A at (4, 3)

The first number, 4, is called the___________.x-coordinate

It tells the number of units the point lies tothe __________ of the origin.

The second number, 3, is called the___________.y-coordinate

It tells the number of units the point lies _____________ the origin.

left or right

above or below

(4, 3)

What is the coordinate of the origin?

(0, 0)


y

x5-4 -2 1 3 5

5

-4

-2

1

3

5

-5 -1 4

-5

-1

4

-3

-5

2

2-5

-3

Graphing an ordered pair, (point): (x, y)

Graph point B at (2, –3)

The first number, 2, is called the___________.x-coordinate

It tells the number of units the point lies tothe __________ of the origin.

The second number, –3, is called the___________.y-coordinate

It tells the number of units the point lies _____________ the origin.

left or right

above or below

(2, –3)


y

x5-4 -2 1 3 5

5

-4

-2

1

3

5

-5 -1 4

-5

-1

4

-3

-5

2

2-5

-3

Name the points A, B, C, & D

Point A(x, y) =

Point B(x, y) =

Point C(x, y) =

Point D(x, y) =

A(3, 2) (–3, 2) B

(–3, –2)C (3, –2)D

(3, 2)A

DC

B

(–3, –2)

(–3, 2)

(3, –2)


y

x5-4 -2 1 3 5

5

-4

-2

1

3

5

-5 -1 4

-5

-1

4

-3

-5

2

2-5

-3

Consider these questions:

Point A(x, y) = A(2, 4)

Point B(x, y) = B(2, 0)

Point C(x, y) = C(2, –3)

Point D(x, y) = D(2, –5)

On a piece of grid paper draw lines representing the x-axis and the y-axis.

Graph :

1) What do you notice about the graphs of these points?

They lie on a vertical line.

2) What do you notice about the x-coordinates of these points?

They are the same number.

Write a general statement about ordered pairs that have the same x-coordinate.

They lie on a vertical line that intersects the x-axis at the x-coordinate.

x = 2

y

x5-4 -2 1 3 5

5

-4

-2

1

3

5

-5 -1 4

-5

-1

4

-3

-5

2

2-5

-3


Consider these questions:

Point W(x, y) = W(–4, –4)

Point X(x, y) = X(–2, –4)

Point Y(x, y) = Y(0, –4)

Point Z(x, y) = Z(3, –4)

Graph :

1) What do you notice about the graphs of these points?

They lie on a horizontal line.

2) What do you notice about the y-coordinates of these points?

They are the same number.

Write a general statement about ordered pairs that have the same y-coordinate.

They lie on a horizontal line that intersects the y-axis at the y-coordinate.

y = – 4

On the same coordinate plane


Theorem 2 – 4

If a and b are real numbers,

a vertical line contains all points (x, y) such that _____

and

a horizontal line contains all points (x, y) such that _____

x = a

y = b

y

x5-4 -2 1 3 5

5

-4

-2

1

3

5

-5 -1 4

-5

-1

4

-3

-5

2

2-5

-3

Graph the lines: x = –3

y = 2

Graph the point of intersection of theselines.

(–3, 2)

1) Name the coordinates of each point.


0

y

x

5-1 2 4

4

0

-2 1-3

2

3-2

-2

A

B

A = (2, 0)

B = (3, –2)

2) Graph point C at (0, –4).

4) Graph y = 2.

3) Graph x = –4.

5) Graph and label the intersection of x = –4 and y = 2

y

x5-4 -2 1 3 5

5

-4

-2

1

3

5

-5 -1 4

-5

-1

4

-3

-5

2

2-5

-3 C(0, -4)

x = –4

y = 2(–4, 2)

MidpointsMidpoints

You will learn to find the coordinates of the midpoint of a segment.

bisectsThe midpoint of a line segment, , is the point C that ______ the segment.AB

BA C

-7 -6 -5 -4 -2-3 0-1 1 2 3 4 5 6 7

A BC

C = [3 + (-5)] ÷ 2

= (-2) ÷ 2

= -1

MidpointsMidpoints

Theorem 2 – 5

On a number line, the coordinate of the midpoint of a segmentwhose endpoints have coordinates a and b is

2

a b

A B

2

a b

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

Consider the x-coordinate: x = 1 x = 9

y = 7

y = 3

It will be (midway) between the lines

x = 1 and x = 9

Consider the y-coordinate:

It will be (midway) between the lines

y = 3 and y = 7

A

B

x

yC(x, y)

MidpointsMidpoints

Theorem 2 – 6

On a coordinate plane, the coordinates of the midpoint of a

segment whose endpoints have coordinates (x1, y1) and (x2, y2) are

1 2 1 2,2 2

x x y y

O

y

x

1 1( , )x y

2 2( , )x y

1 2 1 2,2 2

x x y y

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Find the midpoint, C(x, y), of a segment on the coordinate plane.

x = 1 x = 9

y = 7

y = 3

A(1, 7)

B(9, 3)

x

yC(5, 5)

,C

1 2

2

y y1 2

2

x x

,C

7 3

2

1 9

2

,C

10

2

10

2

5,5C

MidpointsMidpoints

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

Graph A(1, 1) and B(7, 9)

C

Draw AB

B(7, 9)

A(1, 1)

Estimate the midpointof AB.

Check your answerusing the midpoint formula.

C(4, 5)

1+7 1+9,

2 2C

8 10,

2 2C

1 2 1 2,2 2

x x y yC

MidpointsMidpoints

x-coordinate of B

1 2 32

x x

273

2

x

27 6x

2 1x

0

y

0 x

10-1 2 4 6 8 10

10

-1

2

4

6

8

10

-2 3 7-2

1

5

9

1 9

3

-2-2

5

7

y-coordinate of B

1 2 52

y y

225

2

y

22 10y

2 8y

Replace x1 with 7and y1 with 2

Multiply each side by 2

Add or subtract toisolate the variable

Suppose C(3, 5) is the midpoint of AB. Find the coordinate of B.

A(7, 2)

B(x, y) is somewhere over there.

B(-1, 8)

midpoint

C(3, 5)

MidpointsMidpoints

0 21 43 65 7 98 10

A

0 21 43 65 7 98 10

1 2 3

1 2 43 5 76 8 9 10 11

§ 2.1 Real Numbers and Number Lines Real Numbers and Number LinesReal Numbers and Number Lines §...

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