6
CHAPTER 1
Graphs
THEOREM
Midpoint Formula
Themidpoint
ofthelinesegmentfrom
to
is
(2)
M
=
1
x
,
y
2
=
¢
x
1
+
x
2
2
,
y
1
+
y
2
2
≤
P
2
=
1
x
2
,
y
2
2
P
1
=
1
x
1
,
y
1
2
M
=
1
x
,
y
2
Finding the Midpoint of a Line Segment
Find the midpoint of the line segment from
to
. Plot the
points
and
and their midpoint.
P
2
P
1
P
2
=
1
3, 1
2
P
1
=
1

5, 5
2
EXAMPLE 4
Solution
Apply the midpoint formula (2) using
and
. Then
the coordinates
of the midpoint
M
are
That is,
. See Figure 10.
Now Work
P R O B L E M
3 5
M
=
1

1, 3
2
x
=
x
1
+
x
2
2
=

5
+
3
2
= 
1
and
y
=
y
1
+
y
2
2
=
5
+
1
2
=
3
1
x
,
y
2
y
2
=
1
x
1
= 
5,
y
1
=
5,
x
2
=
3,
x
y
5
5
–5
P
2
(3, 1)
P
1
(–5, 5)
M
(–1, 3)
Figure 10
7.
If
are the coordinates of a point
P
in the
xy
plane,
then
x
is called the
of
P
and
y
is the
of
P.
8.
The coordinate axes divide the
xy
plane into four sections
called
.
9.
If three distinct points
P
,
Q
, and
R
all lie on a line and if
then
Q
is called the
of the line segment from
P
to
R
.
d
1
P
,
Q
2
=
d
1
Q
,
R
2
,
1
x
,
y
2
10.
True or False
The distance between two points is some
times a negative number.
11.
True or False
The point
lies in quadrant IV of the
Cartesian plane.
12.
True or False
The midpoint of a line segment is found by
averaging the
x
coordinates and averaging the
y
coordinates
of the endpoints.
1

1, 4
2
Concepts and Vocabulary
‘Are You Prepared?’
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in
red
.
1.
On the real number line the origin is assigned the number
.
(p.A4)
2.
If
and 5 are the coordinates of two points on the real
number line, the distance between these points is
.
(pp.A5–A6)
3.
If 3 and 4 are the legs of a right triangle, the hypotenuse is
.
(p.A14)
4.
Use the converse of the Pythagorean Theorem to show that
a triangle whose sides are of lengths 11, 60, and 61 is a right
triangle.
(pp.A14–A15)

3
5.
The area
A
of a triangle whose base is
b
and whose altitude is
h
is
A
.
(p.A15)
6.
True or False
Two triangles are congruent if two angles
and the included side of one equals two angles and the
included side of the other.
(pp.A16–A17)
1.1
Assess Your Understanding
In Problems 13 and 14, plot each point in the xyplane.Tell in which quadrant or on what coordinate axis each point lies.
13.
(a)
(b)
(c)
C
=
1

2,

2
2
B
=
1
6, 0
2
A
=
1

3, 2
2
14.
(a)
(b)
(c)
C
=
1

3, 4
2
B
=
1

3,

4
2
A
=
1
1, 4
2
Skill Building
(d)
(e)
(f)
F
=
1
6,

3
2
E
=
1
0,

3
2
D
=
1
6, 5
2
(d)
(e)
(f)
F
=
1

3, 0
2
E
=
1
0, 1
2
D
=
1
4, 1
2
In Words
To find the midpoint of a line
segment, average the
x
coordinates
and average the
y
coordinates of
the endpoints.
SECTION 1.1
The Distance and Midpoint Formulas
7
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 Fall '15
 lolts
 Advanced Math, Midpoint Formula, Cartesian Coordinate System, Formulas, triangle, Plotting Points