BellworkBellwork

• Find the average of -11 & 5• Solve

• Simplify• Find to the nearest hundredth

75

2

x

300

5 20

Clickers

Bellwork SolutionBellwork Solution

• Find the average of -11 & 5

11 5

26

32

Bellwork SolutionBellwork Solution

• Solve7

52

7 10

3

x

x

x

Bellwork SolutionBellwork Solution

• Simplify 300

Bellwork SolutionBellwork Solution

• Find to the nearest hundredth5 20

Segments and Segments and Congruence & Use Congruence & Use

Midpoint and Distance Midpoint and Distance FormulasFormulas

Sections 1.2 &1.3Sections 1.2 &1.3

The ConceptThe Concept• Today we’re going to start with the idea of

congruence and continue onto two monumental yet simple postulates

• We will then see the definition for a midpoint and the formula for finding it

• We’ll also learn the formula for finding the length of a line via the distance formula

DefinitionsDefinitions• Postulate

– Rule that is accepted without proof– Can also be called an axiom

Example or IllustrationWritten DefinitionSection & Page Number

Word or Concept

Example or IllustrationWritten DefinitionSection & Page Number

Word or Concept

Postulate 1.2

Rule that is accepted without proof

Ruler postulateRuler postulate• This postulate explains that two points on a line can be

explained as two coordinates– The distance between the two coordinates

correlates to the length of the segment

A B

x1 x2

distance12 xx

Ruler Postulate usesRuler Postulate uses• One use is if we assign values to the coordinates

A B

x1 x2

01212 (2,0) (12,0)

• Or it can be used to allow the use of a ruler to find the length of a segment– Seems nonsensical, but again is something that must be

explained in order to base further exploration

Segment Addition PostulateSegment Addition Postulate• This postulate explains that two connected segments

created by a point between two others can be added together to get the full distance

• It can also be used to explain interior points– If B is between A & C then AB+BC=AC– If AB+BC=AC, then B is between A and C

A CB

Segment Addition Postulate useSegment Addition Postulate use• Based on this postulate find the length of BC, if AC=32

A CB

13

CongruenceCongruence• Congruence

– The same measure as– AB is congruent to CD– Written as

– Important that congruence is used in lieu of equals

AB CD

NomenclatureNomenclature• At this point we should also discuss the two different

nomenclatures you may see regarding segments

• This denotes the segment AB

• This denotes the length of segment ABAB

AB

Use of CongruenceUse of Congruence• Given the following points are XY and WZ congruent?

– X: (-2, -5)– Y: (-2, 3)– W: (-4,3)– Z: (4, 3)

DefinitionsDefinitions• Midpoint

– The point on a line segment that lies exactly halfway between the two endpoints

– Divides the segment into two congruent pieces

Example or IllustrationWritten DefinitionSection & Page Number

Word or Concept

Example or IllustrationWritten DefinitionSection & Page Number

Word or Concept

Midpoint 1.3

The point on a line segment that lies exactly halfway between the two endpointsDivides the segment into two congruent pieces

AB

C

AB BC

Finding a midpointFinding a midpoint• How do we find a midpoint?

– We simply divide the length by two– AB is 20– What is AC?

A BC

ExampleExampleIf point X is the midpoint of segment JK and the length of

JX is 14.5, what length of segment JK?

J KX .13

. 14.5

. 29

. 58

A

B

C

D

BisectorsBisectors• If a line, ray or segment goes through the midpoint of another

segment, it is called the bisector of the segment

A B

C

D

E

Showing congruenceShowing congruence• We are able to show congruence of segments in a figure through the use of slash marks

• Using the same diagram, in which segment AB is bisected

A B

C

D

E

MidpointsMidpoints• What happens if we put a line on the coordinate plane?

– How do we find the midpoint?

• We can use a derivation of the ruler principle to find the midpoint of a line on the coordinate axis…– The formula is

(x1,y1)

(x2,y2)

B

1 2 1 2,2 2

x x y y

Where does this come from?Where does this come from?• How did we get this formula?

B1 2 1 2,

2 2

x x y y

(x1,y1)

½(x1+x2)

(x2,y2)

x2

½ (y1+y2)

y2

Click-InClick-In• What is the midpoint of a line segment that goes from

(1, 2) to (11,20)?

.(5,9)

. (6,11)

. (10,18)

. (12,22)

A

B

C

D

Long Distance CallLong Distance CallIn addition to finding the midpoint of a line when on the

coordinate plane, we can also find the distance or length of the segment using the ruler postulate and the pythagorean theorem

• The ruler postulate gives us length, but only in one dimension

• The Pythagorean Theorem gives us the length of the hypotenuse of a triangle if we have the length of the two sides

• So…we use the ruler postulate to figure the two lengths and then apply the Pythagorean theorem

• Let’s take a look…

Where does this come from?Where does this come from?• How did we get this formula?

(x1,y1)

(x2,y2)

x2-x1

y2-y1

2 2 2

2 2 22 1 2 1

2 22 1 2 1

( ) ( )

( ) ( )

a b c

x x y y c

c x x y y

a=

b=c

Why is there not a plus or minus in front of

this?

ExampleExample• Find the distance between (-10,2) & (4,1)

2 2

2 2

(4 10) (1 2)

(14) ( 1)

196 1

197

d

On your ownOn your own• Find the distance between (-1,-1) & (10,2)

. 130

. 82

. 112 4 7

. 14

A

B

C or

D

On your ownOn your own• Find the distance between (3,3) & (-2,-2)

.0

. 2

. 10

. 50 5 2

A

B

C

D or

HomeworkHomework

• 1.2– 1,6-12 even, 16-30

• 1.3– 2-8, 20-22, 28-34 even, 43-45, 49

Practical ExamplePractical Example

Most Important PointsMost Important Points• Definition of Congruence• Segment addition postulate• Definition of Midpoint• Definition of Bisector• Showing congruency in segments• Midpoint Formula• Distance Formula