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2.1 Goal Bisect a segment. Find the coordinates of the midpoint of a segment.
Segment Bisectors Geo-Activity
Folding a Segment Bisector
1 On a piece of paper, draw AB &*. ●
2 Fold the paper so that point B ● is on top of point A.
Key Words • midpoint • segment bisector
A
• bisect
B
B
3 Label the point where the fold ● intersects AB &* as point M.
A
4 Use a ruler to measure the ●
lengths of AM &**, MB &*, and AB &*. What do you notice?
A
M
B
M
B
&*. The midpoint of a In the Geo-Activity, M is called the midpoint of AB segment is the point on the segment that divides it into two congruent segments.
Student Help VOCABULARY TIP
Bi- means “two,” and -sect means “to cut.” So, bisect means “to cut in two.”
A segment bisector is a segment, ray, line, or plane that intersects a segment at its midpoint. To bisect a segment means to divide the segment into two congruent segments. C M A
M
M is the midpoint of AB &*.
B
A
D
B
^&*( is a bisector of AB CD &*.
2.1
Segment Bisectors
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Student Help
EXAMPLE
&*. Find AM and MB. M is the midpoint of AB
STUDY TIP
The midpoint of a segment divides the segment in half. A
26 A
M
B
Solution
M
B
If you know the length 1 2
of AB &*, multiply AB by to find AM and MB.
Find Segment Lengths
1
&*, so AM and MB are each half the length of AB &*. M is the midpoint of AB 1 2
1 2
AM MB p AB p 26 13 ANSWER
AM 13 and MB 13.
EXAMPLE
Find Segment Lengths
2
&*. Find PS and RS. P is the midpoint of RS
R
7
P
S
Solution &*, so PS RP. Therefore, PS 7. P is the midpoint of RS &*. You know that RS is twice the length of RP
RS 2 p RP 2 p 7 14 ANSWER
PS 7 and RS 14.
Find Segment Lengths 1. Find DE and EF.
2. Find NP and MP.
18 D
E
EXAMPLE
3
F
M
N
11
Use Algebra with Segment Lengths
&*. Line l is a segment bisector of AB Find the value of x.
l A
5x
M
Solution AM MB
Line l bisects AB &* at point M.
5x 35
Substitute 5x for AM and 35 for MB.
5x 35 5 5
Divide each side by 5.
x7 CHECK
Simplify.
✓Check your solution by substituting 7 for x. 5x 5(7) 35
54
Chapter 2
Segments and Angles
P
35
B
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Midpoints If you know the coordinates of the endpoints of a line
segment in a coordinate plane, you can find the coordinates of the midpoint of the segment using the Midpoint Formula.
THE MIDPOINT FORMULA Words
Student Help READING TIP
The numbers 1 and 2 in x 1 and y2 are called subscripts. You read x 1 as “x sub 1” and y2 as ”y sub 2.”
The coordinates of the midpoint of a segment are the averages of the x-coordinates and the y-coordinates of the endpoints.
Symbols
y
y2 y 1 y2 2
M
y1
A(x1, y1)
The midpoint of the segment joining A(x1, y1) and B(x2, y2)
x x 2
y y 2
B(x2, y2)
1 1 2 , 2 . is M
4
EXAMPLE
x1 x2 2
x1
x 1 x2 y 1 y2 , 2 2
x2
x
Use the Midpoint Formula
&*. Find the coordinates of the midpoint of AB b. A(2, 3), B(5, 1)
a. A(1, 2), B(7, 4)
Solution
Student Help SKILLS REVIEW
For help plotting points in a coordinate plane, see p. 664.
First make a sketch. Then use the Midpoint Formula. a.
b.
y
y
A(2, 3) B(7, 4) 1
M
1
M
1
A(1, 2)
x
B(5, 1) 1
x
Let (x1, y1) (1, 2) and (x2, y2) (7, 4). x x
y y
2 2 17 24 , 2 2
1 1 2 , 2 M
Let (x1, y1) (2, 3) and (x2, y2) (5, 1). x x
y y
2 2 2 5 3 (1) , 2 2 3 , 1 2
1 1 2 , 2 M
(4, 3)
Use the Midpoint Formula &*. Then find the coordinates of its midpoint. Sketch PQ 3. P(2, 5), Q(4, 3)
4. P(0, 2), Q(4, 0)
5. P(1, 2), Q(4, 1) 2.1
Segment Bisectors
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2.1 Exercises Guided Practice Vocabulary Check
l
1. In the diagram shown at the right,
name the midpoint and a segment &*. bisector of AB
Skill Check
A
C
B
M is the midpoint of the segment. Find the segment lengths. 2. Find RM and MS.
3. Find FM and MG.
6 R
50
M
S
F
4. Find MQ and PQ.
P
G
5. Find YM and YZ.
P
M
4
M
Y
M
20
Z
48
K
&*. Find the value of the variable. M is the midpoint of JK 6.
7.
x 14
J
M
33
K
J
M
16r
&*. Find the coordinates of the midpoint of PR 8.
9.
y
P (2, 1) P(2, 3)
10.
y
y
R (5, 6)
1 1
4
x
R (4, 1)
1
P (3, 0) 1
x
2
R (1, 3)
Practice and Applications Extra Practice See p. 677.
Recognizing Midpoints In Exercises 11–14, determine whether M is &*. Explain your reasoning. the midpoint of AB 11.
12.
A 13.
B
A
M
B M
A 14.
M
B
A M B
15.
56
Chapter 2
&*, that is bisected by Sketch a line segment, PQ line l at point R.
Visualize It!
Segments and Angles
x
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Homework Help Example 1: Example 2: Example 3: Example 4:
Exs. 16–19 Exs. 20–23 Exs. 26–29 Exs. 30–35
Finding Segment Lengths M is the midpoint of the segment. Find the segment lengths. 16. Find KM and ML.
17. Find DM and ME.
38 K
82
M
L
18. Find YM and MZ.
D
M
19. Find AM and MB.
17 Y
E
2.7
M
Z
A
M
B
Finding Segment Lengths Line l bisects the segment. Find the segment lengths. 20. Find CB and AB.
21. Find MP and MN.
l
l A
36
C
B
22. Find FE and DE.
M
P
23. Find UT and ST.
l D
l
F
29.5
N
15
S
E
3.6
U
T
Biking The Minuteman Bikeway is a 10.5 mile bike path that runs from Arlington to Bedford, Massachusetts. 24. Caitlin and Laurie begin at opposite ends of the Minuteman
Bikeway and meet at the halfway point on the path. How far does each rider bike? 25. Caitlin starts on the path 4.3 miles from the Arlington end. Laurie
starts on the path 3 miles from the Bedford end. How far will each rider bike before reaching the halfway point on the path? Using Algebra Find the value of the variable. 26.
IStudent Help ICLASSZONE.COM
HOMEWORK HELP Extra help with problem solving in Exs. 30–35 is at classzone.com
28.
A
6p
M
72
B
A
r3
M
15
B
27.
29.
A
19
M
q7
B
A
4
M
2s 6
B
&*. Midpoint Formula Find the coordinates of the midpoint of PQ 30. P(0, 0), Q(4, 6)
31. P(3, 8), Q(7, 6)
32. P(5, 6), Q(9, 7)
33. P(12, 0), Q(6, 1)
34. P(4, 4), Q(4, 0)
35. P(3, 2), Q(7, 4) 2.1
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Student Help LOOK BACK
For more about baseball, see p. 50.
Strike Zone In Exercises 36 and 37, use the information below.
In baseball, the strike zone is the region a baseball needs to pass through for the umpire to declare it a strike if the batter does not swing. The top of the strike zone is a horizontal plane passing through the midpoint of the top of the batter’s shoulders and the top of the uniform pants when the player is in a batting stance. Source: Major League Baseball
36. Find the coordinate of T.
37. Find the coordinate of T.
60
63
T
T 45
42
24
22
0
0
Student Help VOCABULARY TIP
Lines of latitude run parallel to the Equator. Lines of longitude run north-south.
EXAMPLE
Latitude-Longitude Coordinates
Find the coordinates of the place halfway between San Francisco (37.8N, 122.4W) and Los Angeles (34.1N, 118.2W).
122°W
120°W
118°W
116°W
NEVADA 38°N
San Francisco (37.8N, 122.4W)
Solution x x 2
y y 2
37.8 34.1 122.4 118.2 , 2 2
1 1 2 , 2 M
(35.95, 120.3W)
ANSWER
C ALIFOR NIA 36°N
34°N
Los Angeles (34.1N, 118.2W)
The place halfway between San Francisco and Los Angeles has coordinates (35.95N, 120.3W).
Latitude-Longitude Coordinates Find the coordinates of the place halfway between the two cities in California. 38. Fresno: (36.7N, 119.8W)
Napa: (38.3N, 122.3W)
39. Bishop: (37.4N, 118.4W)
Los Angeles: (34.1N, 118.2W)
40. San Francisco: (37.8N, 122.4W) 41. Santa Barbara: (34.4N, 119.7W)
Palo Alto: (37.4N, 122.1W) 58
Chapter 2
Segments and Angles
Oakland: (37.8N, 122.3W)
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&*, 42. Using Midpoints In the diagram below, B is the midpoint of AC
AB 9, and AD 25. Find CD. 25 A
9
B
C
D
&* is M(7, 5). The coordinates of 43. Challenge The midpoint of AB
point A are (4, 1). Find the coordinates of point B. Explain.
Standardized Test Practice
&*. What is the value of x? 44. Multiple Choice T is the midpoint of QR A C
B D
17 29.5
22 P
88
4x 10
T
R
78
45. Multiple Choice What is the midpoint of the segment joining
(2, 7) and (6, 2)? F
Mixed Review
2, 92
G
H
(4, 9)
J
(2, 4)
92, 2
Evaluating Statements Use the diagram at the right to determine whether the statement is true or false. (Lessons 1.3, 1.5) 46. Point A lies on line m.
l
47. Point E lies on line l.
C
48. Points B, E, and C are collinear.
E
A
49. Lines l and m intersect at point E.
m
B
F
D
50. Point E is between points B and C. 51. Point F is between points A and B.
Classifying Angles Name the vertex and sides of the angle. Then state whether it appears to be acute, right, obtuse, or straight. (Lesson 1.6) 52.
53.
54.
P
J
A
K
P
L
R B
Algebra Skills
C
Evaluating Expressions Evaluate the expression. (Skills Review, p. 670) 55. 2 p 15 40
1 2
56. 120 35 p 3
57. p 50 145
58. p 16 20
59. 6 3 p 5 2
60. 11 p 4 7 20
61. 12 p 2 3 p 4
62. 5 10 p 6 1
63. 2 (3 4) p 5
5 4
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Segment Bisectors
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