Using Segments and Congruence

1
Section 2.1

• Using Segments and Congruence
• Midpoint Formula

2
Objectives What well learn

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

3
Segments
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.
4
Measure of Segments
A
B
C
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
5
Postulate 2-1Ruler Postulate

• The distance between points A and B, written as
  AB, is the absolute value of the difference of
  the coordinates of A and B.

Use Absolute Value!!!
X
Y
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
6
EXAMPLE 1
Apply the Ruler Postulate
SOLUTION
Use Ruler Postulate.
7
Summary

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

Absolute Value
8
Section 2.2

• Using Segments and Congruence
• Distance and Midpoint Formula

9
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.

2x
4x 6
R
P
Q
PQ 2x QR 4x 6 PR 60
Use the Segment Addition Postulate find the
measure of PQ and QR.
10
Step 1

• 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 2
Step 3
Step 4
11
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.

12
EXAMPLE 3
Find a length
SOLUTION
Use the Segment Addition Postulate to write an
equation. Then solve the equation to find GH.
Segment Addition Postulate.
Substitute 36 for FH and 21 for FG.
Subtract 21 from each side.
13
EXAMPLE 4
Compare segments for congruence
SOLUTION
Use Ruler Postulate.
14
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.
15
Section 2.5

• Midpoint Formula Finding the midpoint and
  endpoint.

16
What is midpoint?

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

Endpoint P
Endpoint Q
Midpoint M
17

• 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!
18
Examples
1.) Find the midpoint of AC
Endpoint -5
Endpoint 6
(Finding Average of two numbers)
(-5 6)/2
Midpoint 1/2
19

• 2.) If M is the midpoint of AZ,
• AM 3x 12 and MZ 6x 9 find the measure of
  AM and MZ.
• AM MZ (Def. of Midpoint)
• 3x 12 6x 9
• 21 3x
• X 7
• AM 3x 12 3(7) 12 33
• MZ 6x 9 6(7) 9 33

Step 2
Step 3
Step 4
20
Steps of finding midpoint.
Find midpoint of (-3, 7) and (8, -4).

• Endpoint 1 ( -3 , 7 )
• Endpoint 2 ( 8 , -4 )
• Midpoint ( , )

(Average of x)
(Average of y)
21
Steps

• Draw and label the Line Segment.
• 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.

22
Steps of finding Endpoint!
Find the other endpoint with endpoint (-5, 6)
midpoint (3/2, 5).

• Endpoint 1 ( -5 , 6 )
• Endpoint 2 ( x , y )
• Midpoint ( , )

(8, 4)
Solve Equations
23
Steps of finding Midpoint

• Write down the order pair.
• Find the AVERAGE of the x1 and x2.
• (x1 x2)/2
• Find the AVERAGE of the y1 and y2.
• (y1 y2) /2
• Write them as an order pair.

24
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 -gt
lt-Find AVERAGE of y
25

• 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)

26
EXAMPLE 3
Use the Midpoint Formula
27
EXAMPLE 2
Use algebra with segment lengths
SOLUTION
VM MW
Write equation.
4x 1 3x 3
Substitute.
x 1 3
Subtract 3x from each side.
x 4
Add 1 to each side.
28
EXAMPLE 2
Use algebra with segment lengths
VM 4x 1 4(4) 1 15
29
Bisectors

• What is a segment bisector?
• - Any segment, line, or plane that intersects a
  segment at its midpoint.

If B is the midpoint of AC, then MN bisects AC.
30
EXAMPLE 1
Find segment lengths
SOLUTION
XY XT TY
Segment Addition Postulate
39.9 39.9
Substitute.
79.8 cm
Add.
31
for Examples 1 and 2
GUIDED PRACTICE
32
Distance Formula

• The Distance Formula was developed from the
  Pythagorean Theorem

Where d distance x x-coordinate and
yy-coordinate