Algebal review 10 doc

Practice Question
Which of the five points on the graph above has coordinates (x,y) such that x ϩ y ϭ 1?
a. A
b. B
c. C
d. D
e. E
Answer
d. You must determine the coordinates of each point and then add them:
A (2,Ϫ4): 2 ϩ (Ϫ4) ϭϪ2
B (Ϫ1,1): Ϫ1 ϩ 1 ϭ 0
C (Ϫ2,Ϫ4): Ϫ2 ϩ (Ϫ4) ϭϪ6
D (3,Ϫ2): 3 ϩ (Ϫ2) ϭ 1
E (4,3): 4 ϩ 3 ϭ 7
Point D is the point with coordinates (x,y) such that x ϩ y ϭ 1.
Lengths of Horizontal and Vertical Segments
The length of a horizontal or a vertical segment on the coordinate plane can be found by taking the absolute value
of the difference between the two coordinates, which are different for the two points.
A
E
B
D
1
C
1
–GEOMETRY REVIEW–
139
Example
Find the length of A
ෆ
B

ෆ
and B
ෆ
C
ෆ
.
A
ෆ
B
ෆ
is parallel to the y-axis, so subtract the absolute value of the y-coordinates of its endpoints to find its length:
A
ෆ
B
ෆ
ϭ |3 Ϫ (Ϫ2)|
A
ෆ
B
ෆ
ϭ |3 ϩ 2|
A
ෆ
B
ෆ
ϭ |5|
A
ෆ
B
ෆ

ϭ 5
B
ෆ
C
ෆ
is parallel to the x-axis, so subtract the absolute value of the x-coordinates of its endpoints to find its length:
B
ෆ
C
ෆ
ϭ |Ϫ3 Ϫ 3|
B
ෆ
C
ෆ
ϭ |Ϫ6|
B
ෆ
C
ෆ
ϭ 6
Practice Question
A
B
C
(Ϫ2,7)
(Ϫ2,Ϫ6)
(5,Ϫ6)
A
B

C
(Ϫ3,3)
(Ϫ3,Ϫ2) (3,Ϫ2)
–GEOMETRY REVIEW–
140
What is the sum of the length of A
ෆ
B
ෆ
and the length of B
ෆ
C
ෆ
?
a. 6
b. 7
c. 13
d. 16
e. 20
Answer
e. A
ෆ
B
ෆ
is parallel to the y-axis, so subtract the absolute value of the y-coordinates of its endpoints to find
its length:
A
ෆ
B
ෆ

ϭ |7 Ϫ (Ϫ6)|
A
ෆ
B
ෆ
ϭ |7 ϩ 6|
A
ෆ
B
ෆ
ϭ |13|
A
ෆ
B
ෆ
ϭ 13
B
ෆ
C
ෆ
is parallel to the x-axis, so subtract the absolute value of the x-coordinates of its endpoints to find
its length:
B
ෆ
C
ෆ
ϭ |5 Ϫ (Ϫ2)|
B
ෆ
C

ෆ
ϭ |5 ϩ 2|
B
ෆ
C
ෆ
ϭ |7|
B
ෆ
C
ෆ
ϭ 7
Now add the two lengths: 7 ϩ 13 ϭ 20.
Distance between Coordinate Points
To find the distance between two points, use this variation of the Pythagorean theorem:
d ϭ ͙(x
2
Ϫ x
ෆ
1
)
2
ϩ (
ෆ
y
2
Ϫ y
1
ෆ
)

2
ෆ
Example
Find the distance between points (2,Ϫ4) and (Ϫ3,Ϫ4).
C
(2,4)
(Ϫ3,Ϫ4)
(5,Ϫ6)
–GEOMETRY REVIEW–
141
The two points in this problem are (2,Ϫ4) and (Ϫ3,Ϫ4).
x
1
ϭ 2
x
2
ϭϪ3
y
1
ϭϪ4
y
2
ϭϪ4
Plug in the points into the formula:
d ϭ ͙(x
2
Ϫ x
ෆ
1
)

2
ϩ (
ෆ
y
2
Ϫ y
1
ෆ
)
2
ෆ
d ϭ ͙(Ϫ3 Ϫ
ෆ
2)
2
ϩ
ෆ
(Ϫ4 Ϫ
ෆ
(Ϫ4))
ෆ
2
ෆ
d ϭ ͙(Ϫ3 Ϫ
ෆ
2)
2
ϩ
ෆ
(Ϫ4 ϩ

ෆ
4)
2
ෆ
d ϭ ͙(Ϫ5)
2
ෆ
ϩ (0)
2
ෆ
d ϭ ͙25
ෆ
d ϭ 5
The distance is 5.
Practice Question
What is the distance between the two points shown in the figure above?
a. ͙20
ෆ
b. 6
c. 10
d. 2͙34
ෆ
e. 4͙34
ෆ
(1,Ϫ4)
(Ϫ5,6)
–GEOMETRY REVIEW–
142
Answer
d. To find the distance between two points, use the following formula:

d ϭ ͙(x
2
Ϫ x
ෆ
1
)
2
ϩ (
ෆ
y
2
Ϫ y
1
ෆ
)
2
ෆ
The two points in this problem are (Ϫ5,6) and (1,Ϫ4).
x
1
ϭϪ5
x
2
ϭ 1
y
1
ϭ 6
y
2
ϭϪ4

Plug the points into the formula:
d ϭ ͙(x
2
Ϫ x
ෆ
1
)
2
ϩ (
ෆ
y
2
Ϫ y
1
ෆ
)
2
ෆ
d ϭ ͙(1 Ϫ (Ϫ
ෆ
5))
2
ϩ
ෆ
(Ϫ4 Ϫ
ෆ
6)
2
ෆ
d ϭ ͙(1 ϩ 5

ෆ
)
2
ϩ (Ϫ
ෆ
10)
2
ෆ
d ϭ ͙(6)
2
ϩ
ෆ
(Ϫ10)
ෆ
2
ෆ
d ϭ ͙36 ϩ 1
ෆ
00
ෆ
d ϭ ͙136
ෆ
d ϭ ͙4 ϫ 34
ෆ
d ϭ ͙34
ෆ
The distance is 2͙34
ෆ
.
Midpoint

A midpoint is the point at the exact middle of a line segment. To find the midpoint of a segment on the coordi-
nate plane, use the following formulas:
Midpoint x ϭ
ᎏ
x
1
ϩ
2
x
2
ᎏ
Midpoint y ϭ
ᎏ
y
1
ϩ
2
y
2
ᎏ
Example
Find the midpoint of A
ෆ
B
ෆ
.
B
A
Midpoint
(5,Ϫ5)

(Ϫ3,5)
–GEOMETRY REVIEW–
143
Midpoint x ϭ
ᎏ
x
1
ϩ
2
x
2
ᎏ
ϭ
ᎏ
Ϫ3
2
ϩ 5
ᎏ
ϭ
ᎏ
2
2
ᎏ
ϭ 1
Midpoint y ϭ
ᎏ
y
1
ϩ
2

y
2
ᎏ
ϭ
ᎏ
5 ϩ
2
(Ϫ5)
ᎏ
ϭ
ᎏ
0
2
ᎏ
ϭ 0
Therefore, the midpoint of A
ෆ
B
ෆ
is (1,0).
Slope
The slope of a line measures its steepness. Slope is found by calculating the ratio of the change in y-coordinates
of any two points on the line, over the change of the corresponding x-coordinates:
slope ϭ
ᎏ
ho
v
r
e
i

r
z
t
o
ic
n
a
t
l
a
c
l
h
c
a
h
n
a
g
n
e
ge
ᎏ
ϭ
ᎏ
x
y
2
2
Ϫ

Ϫ
y
x
1
1
ᎏ
Example
Find the slope of a line containing the points (1,3) and (Ϫ3,Ϫ2).
Slope ϭ
ᎏ
x
y
2
2
Ϫ
Ϫ
y
x
1
1
ᎏ
ϭ
ᎏ
3
1
Ϫ
Ϫ
(
(
Ϫ

Ϫ
2
3
)
)
ᎏ
ϭ
ᎏ
3
1
ϩ
ϩ
2
3
ᎏ
ϭ
ᎏ
5
4
ᎏ
Therefore, the slope of the line is
ᎏ
5
4
ᎏ
.
Practice Question
(5,6)
(1,3)
(1,3)

(Ϫ3,Ϫ2)
–GEOMETRY REVIEW–
144