Symbols and Geometric Elements

SegmentA B AB BAor

Ray A B AB��������������

LineFE EF

�������������� �or FE�������������� �

Ray Notation

Notice the position of the end point and the ray above.

AB��������������

A B C D

AC��������������

AD��������������

Line Variations

A B C D E

AB�������������� �

AC�������������� �

AD�������������� �

AE�������������� �

or BC�������������� �

CE�������������� �

DA�������������� �

Any two letters can be used to name the line.

Therefore, there can be multiple correct answers and confusion.

More SymbolsMathematical Shorthand

BA

SegmentsSet of Points

Length or Measurement Or Distance

A Number NOT a set of points

AB

BA l BA

l AB

mBA

m AB AB

BA

• Inches, centimeters, feet, meters, etc.

• If a coordinate system is used on a line, then ALGEBRA comes into play.

• Just use a ruler. Measurements are arbitrary because the units of measurements are arbitrary.

Measurements

Measure the Lines Below

Tolerance•Measurements are never exact.

•They are always open to interpretation.

•Answers are sometimes rounded up.

•Answers are sometimes rounded down.

•Some visual interpretations are different. There may be a scale.

•The degree of accuracy depends on the accuracy of the equipment.

•The degree of accuracy depends on the accuracy of measurer.

Coordinate Systems

-3 -2-4 -1 2 63 51 40

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Coordinate are numbers.

Points are letters

The name of the red pt. is ___The coordinate of the red pt. is ___

The name of the grey pt. is ___The coordinate of the grey pt. is __

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E

0J

5

Find the coordinates

J = ___

A = ___

C = ___

K = ___

-3 -2-4 -1 2 63 51 40

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5

-4

-2

6

Calculation of DistanceUsing Coordinates

3 5

You could simply count the blocks.

The answer is 2.

Calculation of Distance Using Coordinates

3 33

Counting blocks would be time consuming.

The answer is 30.

You could simply subtract.

Subtraction means the difference between numbers.

Calculation of Distance Using Coordinates

-8 33

The answer is 41.

You could simply subtract.

33 – (-8) = 33 + 8 = 41

Note that negative numbers requires using algebra.

-8 -5You could simply subtract.

-5 – (-8) = -5 + 8 = 3

However if we subtract the numbers in reverse, then...

-8 – (-5) = -8 + 5 = - 3

Therefore to avoid negative numbers, we take the absolute value of the differences.

a b

You subtract the coordinates then take the absolute value of the difference.

a bDistance =

AB = CE = EH = Why or How?

HK = GK = FJ = Why or How?

EF = CH = BK = Why or How?

-3 -2-4 -1 2 63 51 40

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1

3 4 4

1 5 9

3 ( 4)

Method 1: Count the blocks.

Method 2: Subtract coordinates and take the absolute value.

3 4 1

AB = CE = EH = Why or How?

HK = GK = FJ = Why or How?

EF = CH = BK = Why or How?

-3 -2-4 -1 2 63 51 40

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1 2

3 4 4

1 5 9

2 (0)

Method 1: Count the blocks.

Method 2: Subtract coordinates and take the absolute value.

2 2

AB = CE = EH = Why or How?

HK = GK = FJ = Why or How?

EF = CH = BK = Why or How?

-3 -2-4 -1 2 63 51 40

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1 2 3

3 4 4

1 5 9

3 0

Method 1: Count the blocks.

Method 2: Subtract coordinates and take the absolute value.

3 3

AB = CE = EH = Why or How?

HK = GK = FJ = Why or How?

EF = CH = BK = Why or How?

-3 -2-4 -1 2 63 51 40

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1 2 3

3

1 5 9

6 3

Method 1: Count the blocks.

Method 2: Subtract coordinates and take the absolute value.

3

3

AB = CE = EH = Why or How?

HK = GK = FJ = Why or How?

EF = CH = BK = Why or How?

-3 -2-4 -1 2 63 51 40

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1 2 3

4

4

1 5 9

6 2

Method 1: Count the blocks.

Method 2: Subtract coordinates and take the absolute value.

4

3

AB = CE = EH = Why or How?

HK = GK = FJ = Why or How?

EF = CH = BK = Why or How?

-3 -2-4 -1 2 63 51 40

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1 2 3

4

4 4

1 5 9

5 1

Method 1: Count the blocks.

Method 2: Subtract coordinates and take the absolute value.

4

3

AB = CE = EH = Why or How?

HK = GK = FJ = Why or How?

EF = CH = BK = Why or How?

-3 -2-4 -1 2 63 51 40

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1 2 3

3 4 4

1 5

3 ( 2)

Method 1: Count the blocks.

Method 2: Subtract coordinates and take the absolute value.

3 2 5

AB = CE = EH = Why or How?

HK = GK = FJ = Why or How?

EF = CH = BK = Why or How?

-3 -2-4 -1 2 63 51 40

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1 2 3

3 4 4

1 5 9

6 ( 3)

Method 1: Count the blocks.

Method 2: Subtract coordinates and take the absolute value.

6 3 9

Ruler Postulate• The points on a line can be paired up

with real numbers in such a way that any two points can have coordinates of 0 or 1.

• Once the coordinates have been chosen in this way, the distance between the any two points is the absolute value of the difference of their coordinates.

The measuring scale is arbitrary.

A B

If M is on the segment and

MB = 4.01 cmMA = 4.01 cm

MA B

ABM is a midpoint of

AM MBIt is necessary for both conditions. Let’s see why?

D, E, and F

Are equidistant from both A and B

But they are NOT midpoints.

m FG = 10.24 cm

m FA = 10.24 cmDB = 4.79 cm

AD = 4.79 cm

m EG = 6.94 cm

m EA = 6.94 cm

MB = 4.01 cmMA = 4.01 cm

MA B

D

E

F

The midpointMust be on the Segment!

Bisectors

I L

DA B

F G K M

C

E

H

J

N

O

Bisectors can be any segment, ray, line, or plane if they go thru the midpoint of a segment.

160

B EC DA F

D is midpoint of E is midpoint of C is midpoint of B is midpoint of

Find the value of their coordinates.

AFDFADAC

160

B EC DA F

8

84 12

2

Symbol Scramble

AB CD��������������

EF�������������� �

GH m JKl MN

Segment

Length ofSegment

LineRay

length ofSegment

Measure ofSegment

Congruent FiguresSame Size and Shape

Yes NO

Not the same shape

Not the same size

Not the same size

or not

not

not

not

Distance Between Coordinates

4 12

12 8

-5 7

0 12

0 -12

-2 12 -12 -8-5 7 0 18 0 -15-12 12

12 4

8 12

7 5

12 0

12 0

12 2

12 8

7 5

18 0 15 0

12 12

Sometime, Always & Never

• The length of a segment is ______ negative.

• If point S is between points A and B, then S _____ lies on the segment.

• A coordinate can ______ be paired with a point on a number line.

• A bisector of a segment is __________ a line.

• A ray ______ has a midpoint.

Always

Never

Always

Never

Sometimes

Sometime, Always & Never• A ray _____ has an endpoint.

• Congruent segments ______ have equal lengths.

• and _____ denote the same rays.

• A line _____ has one midpoint.

• A ____ has many midpoints. Why?

AB��������������

BA��������������

Always

Always

Never

Never

Always

Segment Addition Postulate

A B C

If B is between A and C, then….

AB + BC = AC

Note:Between means that A, B, and C are collinear.

B must be on the segment AC.

Segment Addition PostulateApplications

A B C

22

AB = 8BC = 22AC = ?8

First, label the diagram.

x

Second, find equation.

Third, solve equation.

8 + 22 = x

30 = x

Segment Addition PostulateApplications

A B C

22

AB = 8AC = 22BC = ?8

First, label the diagram.

x

Second, find equation.

Third, solve equation.

8 + x = 22

x = 14

Segment Addition PostulateApplications

A B C

18

AB = 3x - 4BC = 2x + 7AC = 18Find AB & BC

3x - 4

First, label the diagram.

2x + 7

Second, find equation.

Third, solve equation.

3x- 4 + 2x + 7 = 18

5x + 3 = 18

5x = 15

x = 3Not done yet?

Segment Addition PostulateApplications

A B C

18

AB = 3x - 4BC = 2x + 7AC = 18Find AB & BC

3x - 4 2x + 7

3x- 4 + 2x + 7 = 18

5x + 3 = 18

5x = 15

x = 3

Substitute Back in.

3x - 4

3(3) - 4

9- 4 = 5

2x + 7

2(3) + 7

6 + 7 = 13

13 5

Segment Addition PostulateApplications

A B C

16

AB = 3x - 13BC = 16 AC = 4x + 14Find AB & AC

3x - 13

4x - 4

3x- 13 + 16 = 4x - 4

- x = - 7

Label diagram.

Find equation.

Solve equation.

3x+3 = 4x - 4

x = 7

Not Done YetNDY

Segment Addition PostulateApplications

A B C

16

AB = 3x - 13BC = 16 AC = 4x + 14Find AB & AC

3x - 13

4x - 4

3x- 13 + 16 = 4x - 4

- x = - 7

Substitute into expressions.

3x+3 = 4x - 4

x = 7

3x - 13

3(7) - 13

21 - 13

8

8

4(7) - 14

4x - 14

28 - 14

24

24

You must be able to do these complex algebraic problems.

They will be in the chapter test and the marking period exam (QPA)

Summary

A B

There are several symbols for geometric terms.

AB

C D CD��������������

FE EF�������������� �

B C BCNo symbol means…The distance from B to C.A numerical value.

Summary 2

A B

There are alternate symbols for distance, length, or measurement.

m AB

5 5

l ABAB

Measurements are always arbitrary due to the choice of units (meters, feet, etc.),degree of accuracy andscale.

AB = CE = EH = Why or How?

HK = GK = FJ = Why or How?

EF = CH = BK = Why or How?

-3 -2-4 -1 2 63 51 40

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Summary 3

The letters are the names of the points.

The numbers are the coordinates that indicate the relative position of each point.

Summary 4

The ruler postulate allows us to…

1. Build number lines at any scale.

2. Compute distance by taking the absolute value of the difference of the coordinates.

Summary 5The segment addition postulate allows us to conclude…

The distance on a line is the sum of its parts.

A B C

AB + BC = AC

Summary 6

A B C

18

AB = 3x - 4BC = 2x + 7AC = 18Find AB & BC

3x - 4 2x + 7

You must be able to do these algebraic problems.

C’est fini.

Good day and good luck.