precalc midpoint formulas - 6 CHAPTER 1 Graphs THEOREM...

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 xy-plane.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
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