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Symbols and Geometric Elements

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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.

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.

A

B

Length or Measurement

Or Distance

A Number NOT a set of points

Segments

Set of Points

Measurements

- 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.

- 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

Points are letters

Coordinate are numbers.

E

0

J

5

Find the coordinates

J = ___

A = ___

C = ___

K = ___

5

-4

-2

6

5

3

You could simply count the blocks.

The answer is 2.

33

3

Counting blocks would be time consuming.

You could simply subtract.

Subtraction means the difference between numbers.

The answer is 30.

-8

33

You could simply subtract.

33 – (-8) =

33 + 8 =

41

Note that negative numbers requires using algebra.

The answer is 41.

-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.

a

b

You subtract the coordinates

then take the absolute value

of the difference.

Distance =

1

Method 1: Count the blocks.

3

4

4

Method 2: Subtract coordinates and

take the absolute value.

1

5

9

1

1

2

Method 1: Count the blocks.

3

4

4

Method 2: Subtract coordinates and

take the absolute value.

1

5

9

2

1

2

3

Method 1: Count the blocks.

3

4

4

Method 2: Subtract coordinates and

take the absolute value.

1

5

9

3

1

2

3

3

Method 1: Count the blocks.

1

5

9

Method 2: Subtract coordinates and

take the absolute value.

3

1

2

3

3

4

Method 1: Count the blocks.

1

5

9

Method 2: Subtract coordinates and

take the absolute value.

4

1

2

3

3

4

4

Method 1: Count the blocks.

1

5

9

Method 2: Subtract coordinates and

take the absolute value.

4

Method 1: Count the blocks.

1

2

3

Method 2: Subtract coordinates and

take the absolute value.

3

4

4

1

5

5

Method 1: Count the blocks.

1

2

3

Method 2: Subtract coordinates and

take the absolute value.

3

4

4

1

5

9

9

- 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.

M is a midpoint of

IfM is on the segment and

It 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.

The midpoint

Must be on the

Segment!

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

D is midpoint of

E is midpoint of

C is midpoint of

B is midpoint of

Find the value of their coordinates.

8

4

8

12

2

Line

Ray

Segment

length of

Segment

Length of

Segment

Measure of

Segment

Same Size and Shape

Not the same shape

Not the same size

Not the same size

NO

Yes

-2 12

-12 -8

-5 7

0 18

0 -15

-12 12

4 12

12 8

-5 7

0 12

0 -12

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

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

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.

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.

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.

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?

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

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

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

You must be able to do these

complex algebraic problems.

They will be in the chapter test and

the marking period exam (QPA)

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

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

The letters are the names of the points.

The numbers are the coordinates that indicate

the relative position of each point.

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.

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

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.

C’est fini.

Good day and good luck.