Section 2.1

Using Segments and Congruence

Midpoint Formula

Objectives – What we’ll learn…

Apply the properties of real numbers to the measure of segments.

Segments

A

B

C

D

Where is B located?

Between A and C

Where is D located?

Not between A and C

For a point to be between two other points, all three points must be collinear. Segments can be defined using the idea of betweenness of points.

Measure of SegmentsABC

What is a segment?

A part of a line that consists of two endpoints and all the points between them.

What is the measure of a segment?

The distance between the two endpoints.

In the above figure name three segments:

CB BA AC

Postulate 2-1Ruler PostulateThe distance between points A and B, written as AB, is the

absolute value of the difference of the coordinates of A and B.

Since x is at -2 and Y is at 4, we can say the distance from X to Y or Y to X is:

-2 – 4 = 6 or 4 – (-2) = 6

X Y

Use |Absolute Value|!!!

EXAMPLE 1 Apply the Ruler Postulate

Measure the length of ST to the nearest tenth of a centimeter.

SOLUTION

Align one mark of a metric ruler with S. Then estimate the coordinate of T. For example, if you align S with 2, T appears to align with 5.4.

Use Ruler Postulate.ST = 5.4 – 2 = 3.4

The length of ST is about 3.4 centimeters.ANSWER

Summary

What do we use to find the distance between two points?

|Absolute Value|

Section 2.2

Using Segments and Congruence

Distance and Midpoint Formula

Postulate 2-2 Segment Addition Postulate

If Q is between P and R, then

PQ + QR = PR.

If PQ +QR = PR, then Q is between P and R.

P Q R

2x 4x + 6

PQ = 2x QR = 4x + 6 PR = 60

Use the Segment Addition Postulate find the measure of PQ and QR.

PQ + QR = PR (Segment Addition)

2x + 4x + 6 = 60

6x + 6 = 60

6x = 54

x =9

PQ = 2x = 2(9) = 18

QR =4x + 6 = 4(9) + 6 = 42

Step 1:

Step 2:

Step 3:

Step 4:

Steps

1. Draw and label the Line Segment.

2. Set up the Segment Addition/Congruence Postulate.

3. Set up/Solve equation.

4. Calculate each of the line segments.

EXAMPLE 3 Find a length

Use the diagram to find GH.

Use the Segment Addition Postulate to write an equation. Then solve the equation to find GH.

SOLUTION

Segment Addition Postulate.

Substitute 36 for FH and 21 for FG.

Subtract 21 from each side.

21 + GH=36

FG + GH=FH

=15 GH

EXAMPLE 4 Compare segments for congruence

SOLUTION

To find the length of a horizontal segment, find the absolute value of the difference of the x-coordinates of the endpoints.

Use Ruler Postulate.JK = 2 – (– 3) = 5

Plot J(– 3, 4), K(2, 4), L(1, 3), and M(1, – 2) in a coordinate plane. Then determine whether JK and LM are congruent.

EXAMPLE 4 Compare segments for congruence

To find the length of a vertical segment, find the absolute value of the difference of the y-coordinates of the endpoints.

Use Ruler Postulate.LM = – 2 – 3 = 5

JK and LM have the same length. So, JK LM.

Remember when we speak of length the bar does not go over the letters but it does when we speak of congruence.

=~

ANSWER

Section 2.5

Midpoint Formula: Finding the midpoint and endpoint.

What is midpoint?

The midpoint M of PQ is the point between P and Q such that PM = MQ.

P M Q

Endpoint: P

Endpoint: Q

Midpoint: M

How do you find the midpoint?

On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is (a + b)/2. Find the

AVERAGE!

1.) Find the midpoint of AC:

Examples:

0-5 6

(-5 + 6)/2

Endpoint: -5

Endpoint: 6

Midpoint: 1/2

(Finding Average of two numbers)

2.) If M is the midpoint of AZ, 2.) If M is the midpoint of AZ,

AM = 3x + 12 and MZ = 6x –9; find AM = 3x + 12 and MZ = 6x –9; find the measure of AM and MZ.the measure of AM and MZ.

AM = MZ (Def. of Midpoint)AM = MZ (Def. of Midpoint)

3x + 12 = 6x – 93x + 12 = 6x – 9

21 = 3x21 = 3x

X = 7X = 7

AM =3x + 12 =3(7) + 12 = 33 AM =3x + 12 =3(7) + 12 = 33

MZ = 6x – 9 = 6(7) – 9 = 33MZ = 6x – 9 = 6(7) – 9 = 33

Step 2:Step 3:

Step 4:

Steps of finding midpoint.

1. Endpoint 1: ( -3 , 7 )

Endpoint 2: ( 8 , -4 )

Midpoint: ( , )

€

−3+ 8

2

€

5

2€

7 + (−4)

2

€

3

2

(Average of x) (Average of y)

Find midpoint of (-3, 7) and (8, -4).

Steps

1. Draw and label the Line Segment.

2. Set up the GEOMETRY Expression. a) Segment Addition Postulate

b) Definition of Midpoint

c) Definition of Congruence

3. Set up/Solve equation.

4. Calculate each of the line segments.

Steps of finding Endpoint!

1. Endpoint 1: ( -5 , 6 )

Endpoint 2: ( x , y )

Midpoint: ( , ) €

−5 + x

2

€

3

2€

6 + y

2

€

5

Solve Equations:

€

−5 + x

2=

3

2

€

x = 8

€

6 + y

2= 5

€

y = 4

(8, 4)

Find the other endpoint with endpoint (-5, 6) & midpoint (3/2, 5).

Steps of finding Midpoint:

1. Write down the order pair.

2. Find the AVERAGE of the x1 and x2.

(x1 + x2)/2 =

1. Find the AVERAGE of the y1 and y2.

(y1 + y2) /2 =

1. Write them as an order pair.

Example:1.) Find the midpoint, M, of A(2, 8) and B(4, -4).

x = (2 + 4) ÷ 2 = 3

y = (8 + (-4)) ÷ 2 = 2

M = (3, 2)

2.) Find M if N(1, 3) is the midpoint of MP where the coordinates of P are (3, 6).

M = (-1, 0)

Find AVERAGE of x -><-Find AVERAGE of y

Q. How do you find the midpoint of 2 ordered pairs?

A. In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates (x1, y1) and (x2, y2) are ((x1 + x2)/2), (y1 + y2)/2)

EXAMPLE 3 Use the Midpoint Formula

a. FIND MIDPOINT The endpoints of RS are R(1,–3) and S(4, 2). Find the coordinates of the midpoint M.

SOLUTION

EXAMPLE 2 Use algebra with segment lengths

STEP 1 Write and solve an equation. Use the fact that VM = MW.

VM = MW4x – 1 = 3x + 3

x – 1 = 3x = 4

Write equation.

Substitute.

Subtract 3x from each side.Add 1 to each side.

Point M is the midpoint of VW . Find the length of VM .ALGEBRA

EXAMPLE 2 Use algebra with segment lengths

STEP 2 Evaluate the expression for VM when x = 4.

VM = 4x – 1 = 4(4) – 1 = 15

So, the length of VM is 15.

Check: Because VM = MW, the length of MW should be 15. If you evaluate the expression for MW, you should find that MW = 15.

MW = 3x + 3 = 3(4) +3 = 15

Bisectors

What is a segment bisector?

- Any segment, line, or plane that intersects a segment at its midpoint.

A B C

M

N

If B is the midpoint of AC, then MN bisects AC.

In the skateboard design, VW bisects XY at point T, and XT = 39.9 cm. Find XY.

Skateboard

SOLUTION

EXAMPLE 1 Find segment lengths

Point T is the midpoint of XY . So, XT = TY = 39.9 cm.

XY = XT + TY= 39.9 + 39.9= 79.8 cm

Segment Addition PostulateSubstitute.

Add.

GUIDED PRACTICE for Examples 1 and 2

2.

In Exercises 1 and 2, identify the segment bisectorof PQ . Then find PQ.

line l ; 11 57

ANSWER

Distance Formula

The Distance Formula was developed from the Pythagorean Theorem

Where d = distance

x =x-coordinate and y=y-coordinate