Symbols and geometric elements
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Symbols and Geometric Elements. Segment. or. A. B. Ray. A. B. Line. or. F. E. Ray Notation. A. B. C. D. Notice the position of the end point and the ray above. Line Variations. A. B. C. D. E. or. Any two letters can be used to name the line.

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Symbols and Geometric Elements

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Symbols and geometric elements

Symbols and Geometric Elements

Segment

or

A

B

Ray

A

B

Line

or

F

E


Symbols and geometric elements

Ray Notation

A

B

C

D

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


Line variations

Line Variations

A

B

C

D

E

or

Any two letters can be used to name the line.

Therefore, there can be multiple correct answers

and confusion.


More symbols mathematical shorthand

More SymbolsMathematical Shorthand

A

B

Length or Measurement

Or Distance

A Number NOT a set of points

Segments

Set of Points


Inches centimeters feet meters etc

Measurements

Inches, centimeters, feet, meters, etc.

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

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


Measure the lines below

Measure the Lines Below


Tolerance

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

Coordinate Systems

Points are letters

Coordinate are numbers.


Symbols and geometric elements

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 __

E

0

J

5


Symbols and geometric elements

Find the coordinates

J = ___

A = ___

C = ___

K = ___

5

-4

-2

6


Calculation of distance using coordinates

Calculation of DistanceUsing Coordinates

5

3

You could simply count the blocks.

The answer is 2.


Calculation of distance using coordinates1

Calculation of Distance Using Coordinates

33

3

Counting blocks would be time consuming.

You could simply subtract.

Subtraction means the difference between numbers.

The answer is 30.


Calculation of distance using coordinates2

Calculation of Distance Using Coordinates

-8

33

You could simply subtract.

33 – (-8) =

33 + 8 =

41

Note that negative numbers requires using algebra.

The answer is 41.


Symbols and geometric elements

-8

-5

You 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.


Symbols and geometric elements

a

b

You subtract the coordinates

then take the absolute value

of the difference.

Distance =


Symbols and geometric elements

1

Method 1: Count the blocks.

3

4

4

Method 2: Subtract coordinates and

take the absolute value.

1

5

9

1


Symbols and geometric elements

1

2

Method 1: Count the blocks.

3

4

4

Method 2: Subtract coordinates and

take the absolute value.

1

5

9

2


Symbols and geometric elements

1

2

3

Method 1: Count the blocks.

3

4

4

Method 2: Subtract coordinates and

take the absolute value.

1

5

9

3


Symbols and geometric elements

1

2

3

3

Method 1: Count the blocks.

1

5

9

Method 2: Subtract coordinates and

take the absolute value.

3


Symbols and geometric elements

1

2

3

3

4

Method 1: Count the blocks.

1

5

9

Method 2: Subtract coordinates and

take the absolute value.

4


Symbols and geometric elements

1

2

3

3

4

4

Method 1: Count the blocks.

1

5

9

Method 2: Subtract coordinates and

take the absolute value.

4


Symbols and geometric elements

Method 1: Count the blocks.

1

2

3

Method 2: Subtract coordinates and

take the absolute value.

3

4

4

1

5

5


Symbols and geometric elements

Method 1: Count the blocks.

1

2

3

Method 2: Subtract coordinates and

take the absolute value.

3

4

4

1

5

9

9


Ruler postulate

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.


Symbols and geometric elements

M is a midpoint of

IfM is on the segment and

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


Symbols and geometric elements

D, E, and F

Are equidistant from both A and B

But they are NOT midpoints.

The midpoint

Must be on the

Segment!


Bisectors

Bisectors

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


Symbols and geometric elements

D is midpoint of

E is midpoint of

C is midpoint of

B is midpoint of

Find the value of their coordinates.


Symbols and geometric elements

8

4

8

12

2


Symbol scramble

Symbol Scramble

Line

Ray

Segment

length of

Segment

Length of

Segment

Measure of

Segment


Congruent figures

Congruent Figures

Same Size and Shape

Not the same shape

Not the same size

Not the same size

NO

Yes


Distance between coordinates

Distance Between Coordinates

-2 12

-12 -8

-5 7

0 18

0 -15

-12 12

4 12

12 8

-5 7

0 12

0 -12


Sometime always never

Sometime, Always & Never

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

Always

Sometimes

Never


Sometime always never1

Sometime, Always & Never

Always

  • 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?

Always

Never

Never

Always


Segment addition postulate

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 postulate applications

Segment Addition PostulateApplications

AB = 8

BC = 22

AC = ?

A

B

C

22

8

x

8 + 22 = x

First, label the diagram.

30 = x

Second, find equation.

Third, solve equation.


Segment addition postulate applications1

Segment Addition PostulateApplications

B

AB = 8

AC = 22

BC = ?

A

C

8

x

22

8 + x = 22

First, label the diagram.

Second, find equation.

x = 14

Third, solve equation.


Segment addition postulate applications2

Segment Addition PostulateApplications

A

B

C

AB = 3x - 4

BC = 2x + 7

AC = 18

Find AB & BC

3x - 4

2x + 7

18

3x- 4 + 2x + 7 = 18

First, label the diagram.

5x + 3 = 18

Second, find equation.

5x = 15

Third, solve equation.

x = 3

Not done yet?


Segment addition postulate applications3

Segment Addition PostulateApplications

A

B

C

AB = 3x - 4

BC = 2x + 7

AC = 18

Find AB & BC

3x - 4

2x + 7

18

5

13

Substitute Back in.

3x- 4 + 2x + 7 = 18

3x - 4

2x + 7

5x + 3 = 18

3(3) - 4

2(3) + 7

5x = 15

6 + 7 = 13

9- 4 = 5

x = 3


Segment addition postulate applications4

Segment Addition PostulateApplications

AB = 3x - 13

BC = 16

AC = 4x + 14

Find AB & AC

A

B

C

3x - 13

16

4x - 4

Label diagram.

3x- 13 + 16 = 4x - 4

Find equation.

3x+3 = 4x - 4

Not Done Yet

NDY

- x = - 7

Solve equation.

x = 7


Segment addition postulate applications5

Segment Addition PostulateApplications

AB = 3x - 13

BC = 16

AC = 4x + 14

Find AB & AC

A

B

C

3x - 13

16

4x - 4

8

24

Substitute into expressions.

3x- 13 + 16 = 4x - 4

3x - 13

4x - 14

3x+3 = 4x - 4

3(7) - 13

4(7) - 14

21 - 13

- x = - 7

28 - 14

8

24

x = 7


Symbols and geometric elements

You must be able to do these

complex algebraic problems.

They will be in the chapter test and

the marking period exam (QPA)


Summary

Summary

There are several symbols for geometric terms.

A

B

C

D

F

E

No symbol means…

The distance from B to C.

A numerical value.

B

C


Summary 2

Summary 2

There are alternate symbols for

distance, length, or measurement.

A

B

Measurements are always arbitrary due to

the choice of units (meters, feet, etc.),

degree of accuracy and

scale.

5

5


Summary 3

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

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 5

Summary 5

The 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

Summary 6

A

B

C

AB = 3x - 4

BC = 2x + 7

AC = 18

Find AB & BC

3x - 4

2x + 7

18

You must be able to do these algebraic problems.


Symbols and geometric elements

C’est fini.

Good day and good luck.


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