1.
3
Before
Now
Why?
Key Vocabulary
midpoint
segment bisector
Use Midpoint and
Distance Formulas
You found lengths of segments.
You will find lengths of segments in the coordinate plane.
So you can find an unknown length, as in Example 1.
ACTIVITY FOLD A SEGMENT BISECTOR
STEP 1
STEP 2
STEP 3
Draw }
AB on a piece of
paper.
Fold the paper so that
B is on top of A.
Label point M. Compare
AM, MB, and AB.
MIDPOINTS AND BISECTORS The midpoint of a segment is the point that
divides the segment into two congruent segments. A segment bisector is
a point, ray, line, line segment, or plane th at intersects the segment at its
midpoint. A midpoint or a segment bisector bisects a segment.
M
M
]
CD is a segment bisector of }
AB .
So, }
AM > }
MB and AM 5 MB .
M is the midpoint of }
AB .
So, }
AM > }
MB and AM 5 MB.
EXAMPLE 1
Find segment lengths
SKATEBOARD In the skateboard design, }
VW bisects }
XY at
point T, and XT 5 39.9 cm. Find XY.
6
Solution
Point T is the midpoint of }
XY. So, XT 5 TY 5 39.9 cm.
XY 5 XT 1 TY
Segment Addition Postulate
5 39.9 1 39.9
Substitute.
5 79.8 cm
Add.
8
1.3 Use Midpoint and Distance Formulas
15
EXAMPLE 2
Use algebra with segment lengths
ALGEBRA Point M is the midpoint
VW. Find the length of }
VM.
of }
4x 2 1
3x 1 3
Solution
REVIEW ALGEBRA
STEP 1 Write and solve an equation. Use the fact that that VM 5 MW.
For help with solving
equations, see p. 875.
VM 5 MW
Write equation.
4x 2 1 5 3x 1 3
Substitute.
x2153
Subtract 3x from each side.
x54
Add 1 to each side.
STEP 2 Evaluate the expression for VM when x 5 4.
VM 5 4x 2 1 5 4(4) 2 1 5 15
c So, the length of }
VM is 15.
CHECK Because VM 5 MW, the length of }
MW should be 15. If you evaluate
the expression for MW, you should find that MW 5 15.
MW 5 3x 1 3 5 3(4) 1 3 5 15
READ DIRECTIONS
Always read direction
lines carefully. Notice
that this direction line
has two parts.
GUIDED PRACTICE
for Examples 1 and 2
In Exercises 1 and 2, identify the segment bisector of }
PQ. Then find PQ.
1 78
1.
P
2.
M
P
N
5x 2 7
11 2 2x
COORDINATE PLANE You can use the coordinates of the endpoints of a
segment to find the coordinates of the midpoint.
For Your Notebook
KEY CONCEPT
The Midpoint Formula
The coordinates of the midpoint of a
segment are the averages of the
x-coordinates and of the y-coordinates
of the endpoints.
If A(x1, y1) and B(x2, y 2) are points in a
coordinate plane, then the midpoint M
of }
AB has coordinates
, } 2.
1}
2
2
x1 1 x 2 y 1 1 y 2
16
Chapter 1 Essentials of Geometry
y2
B(x2, y2)
y1 1 y2
2
y1
x1 1 x2 y 1 1 y 2
2 ,
2
A(x1, y1)
x1
x1 1 x2
2
x2
EXAMPLE 3
Use the Midpoint Formula
a. FIND MIDPOINT The endpoints of }
RS are R(1, 23) and S(4, 2). Find
the coordinates of the midpoint M.
JK is M(2, 1). One endpoint is
b. FIND ENDPOINT The midpoint of }
J(1, 4). Find the coordinates of endpoint K.
Solution
S (4, 2)
a. FIND MIDPOINT Use the Midpoint Formula.
12
1 4 , 23 1 2 5 M 5 , 2 1
M 1}
}
}
}
2
M (?, ?)
c The coordinates of the midpoint M
5
1
, 2}
are 1 }
2.
2
R (1, 23)
b. FIND ENDPOINT Let (x, y) be the coordinates
of endpoint K. Use the Midpoint Formula.
CLEAR FRACTIONS
Multiply each side of
the equation by the
denominator to clear
the fraction.
STEP 1 Find x.
STEP 2 Find y.
11x
}52
2
41y
}51
2
11x54
41y52
x53
J (1, 4)
M (2, 1)
1
1
K (x, y )
y 5 22
c The coordinates of endpoint K are (3, 22).
GUIDED PRACTICE
for Example 3
3. The endpoints of }
AB are A(1, 2) and B(7, 8). Find the coordinates of the
midpoint M.
VW is M(21, 22). One endpoint is W(4, 4). Find the
4. The midpoint of }
coordinates of endpoint V.
DISTANCE FORMULA The Distance Formula is a formula for computing the
distance between two points in a coordinate plane.
For Your Notebook
KEY CONCEPT
The Distance Formula
READ DIAGRAMS
The red mark at one
corner of the triangle
shown indicates a
right triangle.
If A(x1, y1) and B(x2, y 2) are points in
a coordinate plane, then the distance
between A and B is
}}
AB 5 (x2 2 x1) 1 (y2 2 y1) .
2
B(x2, y2)
z y 2 2 y1 z
A(x1, y1)
z x2 2 x1 z
C(x2, y1)
x
1.3 Use Midpoint and Distance Formulas
17
The Distance Formula is based on the Pythagorean Theorem, which you will
see again when you work with right triangles in Chapter 7.
Distance Formula
2
(AB) 5 (x2 2 x1) 1 (y2 2 y1)
Pythagorean Theorem
2
c 2 5 a2 1 b2
B(x2, y2)
c
z y 2 2 y1 z
A(x1, y1)
z x2 2 x1 z
C(x2, y1)
a
x
EXAMPLE 4
ELIMINATE CHOICES
Drawing a diagram
can help you eliminate
choices. You can see
that choice A is not
large enough to be RS.
Standardized Test Practice
What is the approximate length of }
RS with endpoints R(2, 3)
and S(4, 21)?
A 1.4 units
B 4.0 units
C 4.5 units
D 6 units
Solution
Use the Distance Formula. You may find it helpful
to draw a diagram.
}}
RS 5 (x2 2 x1)2 1 (y2 2 y1)2
}}}
5 [(4 2 2)]2 1 [(21) 2 3]2
}}
5 (2) 1 (24)
2
Distance Formula
1
Substitute.
S(4, 21)
Evaluate powers.
The symbol means is
approximately equal to.
R(2, 3)
Subtract.
5 4 1 16
READ SYMBOLS
5 20
Add.
4.47
Use a calculator to approximate
the square root.
c The correct answer is C. A B C D
GUIDED PRACTICE
for Example 4
5. In Example 4, does it matter which ordered pair you choose to substitute
for (x1, y1) and which ordered pair you choose to substitute for (x2, y 2)?
Explain.
6. What is the approximate length of }
AB, with endpoints A(23, 2) and
B(1, 24)?
A 6.1 units
18
Chapter 1 Essentials of Geometry
B 7.2 units
C 8.5 units
D 10.0 units
1.3
HOMEWORK
KEY
EXERCISES
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 15, 35, and 49
5 STANDARDIZED TEST PRACTICE
Exs. 2, 23, 34, 41, 42, and 53
SKILL PRACTICE
1. VOCABULARY Copy and complete: To find the length of }
AB, with
endpoints A(27, 5) and B(4, 26), you can use the ? .
2.
EXAMPLE 1
on p. 15
for Exs. 310
WRITING Explain what it means to bisect a segment. Why is it
impossible to bisect a line?
FINDING LENGTHS Line l bisects the segment. Find the indicated length.
5
1
in.
4. Find UW if VW 5 } in.
5. Find EG if EF 5 13 cm.
3. Find RT if RS 5 5}
8
8
l
l
l
R
6. Find BC if AC 5 19 cm.
1
7. Find QR if PR 5 9}
in.
2
l
A
8. Find LM if LN 5 137 mm.
l
3
4
9. SEGMENT BISECTOR Line RS bisects }
PQ at point R. Find RQ if PQ 5 4} inches.
7
8
UV. Find UV if UT 5 2} inches.
10. SEGMENT BISECTOR Point T bisects }
EXAMPLE 2
on p. 16
for Exs. 1116
ALGEBRA In each diagram, M is the midpoint of the segment. Find the
indicated length.
11. Find AM.
12. Find EM.
x15
A
7x
2x
14. Find PR.
6x 1 7
8x 2 6
15. Find SU.
6x 2 11
13. Find JM.
16. Find XZ.
x 1 15
10x 2 51
4x 1 5
2x 1 35
4x 2 45
EXAMPLE 3
FINDING MIDPOINTS Find the coordinates of the midpoint of the segment
on p. 17
for Exs. 1730
with the given endpoints.
5x 2 22
17. C(3, 5) and D(7, 5)
18. E(0, 4) and F(4, 3)
19. G(24, 4) and H(6, 4)
20. J(27, 25) and K(23, 7)
21. P(28, 27) and Q(11, 5)
22. S(23, 3) and T(28, 6)
23.
WRITING Develop a formula for finding the midpoint of a segment
with endpoints A(0, 0) and B(m, n). Explain your thinking.
1.3 Use Midpoint and Distance Formulas
19
24. ERROR ANALYSIS Describe the error made in
8 2 2,
} 2 5 (3, 2)
1}
2
2
3 2 (21)
finding the coordinates of the midpoint of a
segment with endpoints S(8, 3) and T(2, 21).
FINDING ENDPOINTS Use the given endpoint R and midpoint M of RS to find
the coordinates of the other endpoint S.
25. R(3, 0), M(0, 5)
26. R(5, 1), M(1, 4)
27. R(6, 22), M(5, 3)
28. R(27, 11), M(2, 1)
29. R(4, 26), M(27, 8)
30. R(24, 26), M(3, 24)
EXAMPLE 4
DISTANCE FORMULA Find the length of the segment. Round to the nearest
on p. 18
for Exs. 3134
tenth of a unit.
31.
32.
33.
(23, 5)
S(21, 2)
(5, 4)
1
1
R(2, 3)
P(1, 2)
34.
T (3, 22)
1
MULTIPLE CHOICE The endpoints of }
MN are M(23, 29) and N(4, 8).
MN ?
What is the approximate length of }
A 1.4 units
B 7.2 units
C 13 units
D 18.4 units
NUMBER LINE Find the length of the segment. Then find the coordinate of
the midpoint of the segment.
35.
36.
24 22
38.
39.
230
41.
37.
0
28 26 24 22
220
210
220 210
28
24
10
20
30
40.
29
26
23
26
22
MULTIPLE CHOICE The endpoints of }
LF are L(22, 2) and F(3, 1).
JR are J(1, 21) and R(2, 23). What is the approximate
The endpoints of }
difference in the lengths of the two segments?
A 2.24
42.
B 2.86
C 5.10
D 7.96
SHORT RESPONSE One endpoint of PQ is P(22, 4). The midpoint of PQ
is M(1, 0). Explain how to find PQ.
COMPARING LENGTHS The endpoints of two segments are given. Find each
segment length. Tell whether the segments are congruent.
43. }
AB : A(0, 2), B(23, 8)
} C(22, 2), D(0, 24)
CD:
46.
44. }
EF: E(1, 4), F(5, 1)
45. }
JK: J(24, 0), K(4, 8)
} G(23, 1), H(1, 6)
GH:
} L(24, 2), M(3, 27)
LM:
ALGEBRA Points S, T, and P lie on a number line. Their coordinates
are 0, 1, and x, respectively. Given SP 5 PT, what is the value of x ?
x
8
3x
4
JK, JM 5 }, and JK 5 } 2 6. Find MK.
47. CHALLENGE M is the midpoint of }
20
5 WORKED-OUT SOLUTIONS
on p.. WS1
5 STANDARDIZED
TEST PRACTICE
PROBLEM SOLVING
T
1
feet. Find QR and MR.
18 }
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
49. DISTANCES A house and a school are 5.7 kilometers apart on the same
straight road. The library is on the same road, halfway between the house
and the school. Draw a sketch to represent this situation. Mark the locations
of the house, school, and library. How far is the library from the house?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
ARCHAEOLOGY The points on the diagram show the positions of objects at
an underwater archaeological site. Use the diagram for Exercises 50 and 51.
y
Distance (m)
on p. 15
for Ex. 48
}
48. WINDMILL In the photograph of a windmill, ST
}
}
bisects QR at point M. The length of QM is
50. Find the distance between each pair of objects. Round
to the nearest tenth of a meter if necessary.
4
B
2
A
0
a. A and B
b. B and C
c. C and D
d. A and D
e. B and D
f. A and C
2
4
6 x
Distance (m)
51. Which two objects are closest to each other? Which two are farthest apart?
(FPNFUSZ
at classzone.com
52. WATER POLO The diagram
shows the positions of three
players during part of a water
polo match. Player A throws
the ball to Player B, who
then throws it to Player C.
How far did Player A throw
the ball? How far did Player B
throw the ball? How far would
Player A have thrown the
ball if he had thrown it
directly to Player C? Round all
answers to the nearest tenth of
a meter.
Distance (m)
EXAMPLE 1
Distance (m)
1.3 Use Midpoint and Distance Formulas
21
53.
EXTENDED RESPONSE As shown, a path goes around a triangular park.
Y
a. Find the distance around the park to the
b. A new path and a bridge are constructed from
nearest yard.
PR. Find QM
point Q to the midpoint M of }
to the nearest yard.
c. A man jogs from P to Q to M to R to Q and
back to P at an average speed of 150 yards
per minute. About how many minutes does
it take? Explain.
"
2
54. CHALLENGE }
AB bisects }
CD at point M, }
CD bisects }
AB at point M,
and AB 5 4 p CM. Describe the relationship between AM and CD.
MIXED REVIEW
The graph shows data about the number of children in the families of
students in a math class. (p. 888)
1 child 28%
55. What percent of the students in the class
belong to families with two or more children?
2 children 56%
3 or more
children 16%
56. If there are 25 students in the class, how
many students belong to families with two children?
PREVIEW
Solve the equation. (p. 875)
Prepare for
Lesson 1.4
in Exs. 5759.
57. 3x 1 12 1 x 5 20
58. 9x 1 2x 1 6 2 x 5 10
59. 5x 2 22 2 7x 1 2 5 40
In Exercises 6064, use the diagram at the right. (p. 2)
60. Name all rays with endpoint B.
A
B
61. Name all the rays that contain point C.
62. Name a pair of opposite rays.
63. Name the intersection of AB and BC .
]
64. Name the intersection of BC and plane P.
QUIZ for Lessons 1.11.3
1. Sketch two lines that intersect the same plane at two different points.
The lines intersect each other at a point not in the plane. (p. 2)
In the diagram of collinear points, AE 5 26, AD 5 15,
and AB 5 BC 5 CD. Find the indicated length. (p. 9)
2. DE
3. AB
4. AC
5. BD
6. CE
7. BE
RS are R(22, 21) and S(2, 3). Find the coordinates of the
8. The endpoints of }
midpoint of }
RS. Then find the distance between R and S. (p. 15)
22
EXTRA PRACTICE for Lesson 1.3, p. 896
ONLINE QUIZ at classzone.com
MIXED REVIEW of Problem Solving
STATE TEST PRACTICE
classzone.com
Lessons 1.11.3
1. MULTI-STEP PROBLEM The diagram shows
5. SHORT RESPONSE Point E is the midpoint of
} and the midpoint of }
AB
CD. The endpoints of
}
AB are A(24, 5) and B(6, 25). The coordinates
of point C are (2, 8). Find the coordinates of
point D. Explain how you got your answer.
]
]
existing roads (BD and DE ) and a new road
}) under construction.
(CE
Y
"
6. OPEN-ENDED The distance around a figure
is its perimeter. Choose four points in a
coordinate plane that can be connected to
form a rectangle with a perimeter of
16 units. Then choose four other points
and draw a different rectangle that has
a perimeter of 16 units. Show how you
determined that each rectangle has a
perimeter of 16 units.
a. If you drive from point B to point E on
7. SHORT RESPONSE Use the diagram of a box.
existing roads, how far do you travel?
b. If you use the new road as you drive from
B to E, about how far do you travel? Round
to the nearest tenth of a mile if necessary.
c. About how much shorter is the trip from
What are all the names that can be used to
describe the plane that contains points B, F,
and C ? Name the intersection of planes ABC
and BFE. Explain.
B to E if you use the new road?
2. GRIDDED ANSWER Point M is the midpoint
PQ. If PM 5 23x 1 5 and MQ 5 25x 2 4,
of }
PQ.
find the length of }
3. GRIDDED ANSWER You are hiking on a trail
that lies along a straight railroad track. The
total length of the trail is 5.4 kilometers.
You have been hiking for 45 minutes at an
average speed of 2.4 kilometers per hour.
How much farther (in kilometers) do you
need to hike to reach the end of the trail?
8. EXTENDED RESPONSE Jill is a salesperson
who needs to visit towns A, B, and C. On the
map below, AB 5 18.7 km and BC 5 2AB.
Assume Jill travels along the road shown.
Town
A
4. SHORT RESPONSE The diagram below shows
FH represents a vertical
the frame for a wall. }
board, and }
EG represents a brace. If
FH?
FG 5 143 cm, does the brace bisect }
}
If not, how long should FG be so that the
FH? Explain.
brace does bisect }
% &
Town
B
Town
C
a. Find the distance Jill travels if she starts
at Town A, visits Towns B and C, and then
returns to Town A.
b. About how much time does Jill spend
driving if her average driving speed is
70 kilometers per hour?
'
c. Jill needs to spend 2.5 hours in each town.
Can she visit all three towns and return to
Town A in an 8 hour workday ? Explain.
(
Mixed Review of Problem Solving
23
