Lesson: Segments and Rays 1 Geometry Segments and Rays.

Lesson: Segments and Rays 1

Geometry

Segments

and Rays

Lesson 1-2: Segments and Rays 2

Postulates

Definition: An assumption that needs no explanation.

Examples:

• Through any two points there is exactly one line.

• Through any three points, there is exactly one plane.

• A line contains at least two points.

• A plane contains at least three points.


The Ruler Postulate

The Ruler Postulate: Points on a line can be paired with the real numbers in such a way that:

• Any two chosen points can be paired with 0 and 1.

• The distance between any two points on a number line is the absolute value of the difference of the real numbers corresponding to the points.

Formula: Take the absolute value of the difference of the two coordinates a and b: │a – b │


Ruler Postulate : Example

-5 5

SRQPOLKJIHG M N

PK = | 3 - -2 | = 5 Remember : Distance is always positive

Find the distance between P and K.

Note: The coordinates are the numbers on the ruler or number line! The capital letters are the names of the points.

Therefore, the coordinates of points P and K are 3 and -2 respectively.

Substituting the coordinates in the formula │a – b │

Measuring Segment Lengths

What is ST? What is SV? What is UV? What is TV?

Measuring Segment Lengths

ST = | -4 – 8 | = | -12| = 12 SV = |-4 – 14 | = | -18| = 18 UV = | 10 – 14| = | -4| = 4 TV = |8 – 14| = | -6 | = 6

Postulate: Segment Addition Postulate If three points A, B, and C are collinear and B is

between A and C, then AB + BC = AC.


Between

Definition: X is between A and B if AX + XB = AB.

A BX

AX + XB = AB AX + XB > AB

A BX


Segment

Part of a line that consists of two points called the endpoints and all points between them.

How to sketch:

How to name:

Definition:

AB

AB or BA

The symbol AB is read as "segment AB".

AB (without a symbol) means the length of the segment or the distance between points A and B.


The Segment Addition Postulate

AB

C

If C is between A and B, then AC + CB = AB.Postulate:

Example: If AC = x , CB = 2x and AB = 12, then, find x, AC and CB.

AC + CB = AB

x + 2x = 12

3x = 12

x = 4

2xx

12

x = 4AC = 4CB = 8

Step 1: Draw a figure

Step 2: Label fig. with given info.

Step 3: Write an equation

Step 4: Solve and find all the answers


Congruent Segments

Definition:

If numbers are equal the objects are congruent.

AB: the segment AB ( an object )

AB: the distance from A to B ( a number )

AB

D

C

Congruent segments can be marked with dashes.

Correct notation:

Incorrect notation:

AB = CD AB CD

AB = CDAB CD

Segments with equal lengths. (congruent symbol: )


Midpoint

a b

2

1 1( , )x y 2 2( , )x y

A point that divides a segment into two congruent segments

Definition:

EDFIf DE EF , then E is the midpoint of DF.

On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is .

In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and

is .1 2 1 2,2 2

x x y y

Formulas:


Midpoint on Number Line - Example

Find the coordinate of the midpoint of the segment PK.

-5 5

SRQPOLKJIHG M N

a b 3 ( 2) 10.5

2 2 2

Now find the midpoint on the number line.


Segment BisectorAny segment, line or plane that divides a segment into two congruent parts is called segment bisector.

Definition:

B

E

D

FA

BE

D

FA

E

D

A F

B

AB bisects DF. AB bisects DF.

AB bisects DF.Plane M bisects DF.

Lesson: Segments and Rays 1 Geometry Segments and Rays.

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Transcript of Lesson: Segments and Rays 1 Geometry Segments and Rays.