Download - WARM UP!
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Find three cities on this map that appear to be collinear.
Chicago, Bloomington, SpringfieldANSWER
WARM UP!
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Front Side – True or False
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Back Side
Use the diagram to answer the questions.
9. Name one pair of opposite rays.
____________ & ____________
Opposite Rays
- Share the same end point
- The 2 rays are on the same line
- They go in opposite directions
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Lesson 1.2
Use Segments and Congruence
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Congruent Segments:
exactly the same size and length
XY= PQ
segment XY is congruent to segment PQ
II
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Between:
A point is between 2 points if it is on the line (collinear with) that connects those two points.
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Broken Pencil
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Segment Addition Postulate:
Segment Addition Postulate
Thus, XY + YZ = XZ
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EXAMPLE 1
Apply the the Segment Addition Postulate
SOLUTION
MapsThe cities shown on the map lie approximately in a straight line. Use the given distances to find the distance from Lubbock, Texas, to St. Louis, Missouri.
Because Tulsa, Oklahoma, lies between Lubbock and St. Louis, you can apply the Segment Addition Postulate.
LS = LT + TS = 380 + 360 = 740
The distance from Lubbock to St. Louis is about 740 miles.
ANSWER
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GUIDED PRACTICE for Examples 1 and 2
In the diagram, WY = 30. Can you use the Segment Addition Postulate to find the distance between points W and Z?
4.
NO; Because w is not between x and z.
ANSWER
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GUIDED PRACTICE
1. Use the Segment Addition Postulate to find XZ.
xz = xy + yz= 23 + 50= 73
Segment addition postulate
Substitute 23 for xy and 50 for yzAdd
SOLUTION
ANSWER xz = 73
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EXAMPLE 2 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
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GUIDED PRACTICE
Use the segment addition postulate to write an equation. Then solve the equation to find WX
Use the diagram at the right to find WX.5.
vx = vw + wx144 = 37 + wx107 = wx
Segment addition postulate
Subtract 37 from each side
SOLUTION
ANSWER WX = 107
Substitute 37 for vw and 144 for vx
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Example 3
Point S is between point R and point T. Use the given information to write an equation in terms of x. Solve the equation. Then find both RS and ST.
RS = 3x – 16 ST = 4x – 8RT = 60
3x-16
I-----------4x-8-----------I
I---------------60 -------------------I
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WARM-UP
Directions:
Find x.
What do you notice about the relationship between segment AB and segment BC?
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1.3 Lesson
Use Midpoint and Distance Formulas
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Midpoint
The midpoint of a segment is a point that divides a segment into 2 congruent
segments.
I IA B
So….. AM = MB
M
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Segment Bisector
A point, segment, line, or plane that divides a line segment into two equal parts
I I I I I I
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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.
Bisect: to cut in 1/2
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EXAMPLE 2 Use algebra with segment lengths
Point M is the midpoint of VW . Find the length of VM .
ALGEBRA
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GUIDED PRACTICE
Identify the segment bisector of .
PQ
Then find PQ.
line l
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Warm Up: Find the coordinates of the midpoint
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MIDPOINT FORMULA
The midpoint of two points P(x1, y1) and Q(x2, y2) is
M(X,Y) = M(x1 + x2, y2 +y2)
Think of it as taking the average of the x’s and the average of the y’s to make a new point.
2 2
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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.
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EXAMPLE 3 Use the Midpoint Formula
252
1 + 4 2
– 3 + 2 2 =, M , – 1M
The coordinates of the midpoint M are 1,–5
2 2
ANSWER
SOLUTION
a. FIND MIDPOINT Use the Midpoint Formula.
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EXAMPLE 3 Use the Midpoint Formula
FIND ENDPOINT Let (x, y) be the coordinates of endpoint K. Use the Midpoint Formula.
STEP 1 Find x.
1+ x 22
=
1 + x = 4
x = 3
STEP 2 Find y.
4+ y 12
=
4 + y = 2
y = – 2
The coordinates of endpoint K are (3, – 2).ANSWER
b. FIND ENDPOINT The midpoint of JK is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K.
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Guided Practice
A. The endpoints of are A(1, 2) and B(7, 8). Find the coordinates of the midpoint M.
B. The midpoint of is M(– 1, – 2). One endpoint is W(4, 4). Find the coordinates of endpoint V.