Sat math Essentials

(3,5)
(+,+)
I
(–3,5)
(–,+)
II
(–3,–5)
(–,–)
III
(3,–5)
(+,–)
IV
(
3
,5)
(0,0)
Quadrant
II
Quadrant
I
Quadrant
III
Quadrant
IV
(
3
,5)
(
3
,
5)
(
3
,
5)
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Practice Question
Which of the five points on the graph above has coordinates (
x,y
) such that 
x
y
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,
4): 2 
(
4) 
2
B
(
1,1):
1 
1 
0
C
(
2,
4):
2 
(
4) 
6
D
(3,
2): 3 
(
2) 
1
E
(4,3): 4 
3 
7
Point 
D
is the point with coordinates (
x
,
y
) such that 
x
y
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
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Example
Find the length of
A
B
and 
B
C
.
A
B
is parallel to the 
y
-axis, so subtract the absolute value of the 
y
-coordinates of its endpoints to find its length:
A
B
|3 
(
2)|
A
B
|3 
2|
A
B
|5|
A
B
5
B
C
is parallel to the 
x
-axis, so subtract the absolute value of the 
x
-coordinates of its endpoints to find its length:
B
C
|
3 
3|
B
C
|
6|
B
C
6
Practice Question
A
B
C
(
2,7)
(
2,
6)
(5,
6)
A
B
C
(
3
,
3
)
(
3
,
2)
(
3
,
2)
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What is the sum of the length of
A
B
and the length of
B
C
?
a.
6
b.
7
c.
13
d.
16
e.
20
Answer
e.
A
B
is parallel to the 
y
-axis, so subtract the absolute value of the 
y
-coordinates of its endpoints to find
its length:
A
B
|7 
(
6)|
A
B
|7 
6|
A
B
|13|
A
B
13
B
C
is parallel to the 
x
-axis, so subtract the absolute value of the 
x
-coordinates of its endpoints to find
its length:
B
C
|5 
(
2)|
B
C
|5 
2|
B
C
|7|
B
C
7
Now add the two lengths: 7 
13 
20.
Distance between Coordinate Points
To find the distance between two points, use this variation of the Pythagorean theorem:
d
(
x
2
x
1
)
2
(
y
2
y
1
)
2
Example
Find the distance between points (2,
4) and (
3,
4).
C
(2,4)
(
3
,
4)
(5,
6)
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The two points in this problem are (2,
4) and (
3,
4).
x
1
2
x
2
3
y
1
4
y
2
4
Plug in the points into the formula:
d
(
x
2
x
1
)
2
(
y
2
y
1
)
2
d
(
3 
2)
2
(
4 
(
4))
2
d
(
3 
2)
2
(
4 
4)
2
d
(
5)
2
(0)
2
d
25
d
5
The distance is 5.
Practice Question
What is the distance between the two points shown in the figure above?
a.
20
b.
6
c.
10
d.
2
34
e.
4
34
(1,
4)
(
5,6)
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Answer
d.
To find the distance between two points, use the following formula:
d
(
x
2
x
1
)
2
(
y
2
y
1
)
2
The two points in this problem are (
5,6) and (1,
4).
x
1
5
x
2
1
y
1
6
y
2
4
Plug the points into the formula:
d
(
x
2
x
1
)
2
(
y
2
y
1
)
2
d
(1 
(
5))
2
(
4 
6)
2
d
(1 
5
)
2
(
10)
2
d
(6)
2
(
10)
2
d
36 
1
00
d
136
d
4 
34
d
34
The distance is 2
34
.
Midpoint_A_midpoint'>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
x
1
2
x
2
Midpoint 
y
y
1
2
y
2
Example
Find the midpoint of
A
B
.
B
A
Midpoint
(5,
5)
(
3
,5)
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Midpoint 
x
x
1
2
x
2
3
2
5
2
2
1
Midpoint 
y
y
1
2
y
2
5
2
(
5)
0
2
0
Therefore, the midpoint of
A
B
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 
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
x
y
2
2
y
x
1
1
Example
Find the slope of a line containing the points (1,3) and (
3,
2).
Slope 
x
y
2
2
y
x
1
1
3
1
(
(
2
3
)
)
3
1
2
3
5
4
Therefore, the slope of the line is 
5
4
.
Practice Question
(5,6)
(1,3)
(1,
3
)
(
3
,
2)
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What is the slope of the line shown in the figure on the previous page?
a.
1
2
b.
3
4
c.
4
3
d.
2
e.
3
Answer
b.
To find the slope of a line, use the following formula:
slope 
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
x
y
2
2
y
x
1
1
The two points shown on the line are (1,3) and (5,6).
x
1
1
x
2
5
y
1
3
y
2
6
Plug in the points into the formula:
slope 
6
5
3
2
slope 
3
4
Using Slope
If you know the slope of a line and one point on the line, you can determine other coordinate points on the line.
Because slope tells you the ratio of
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
, you can simply move from the coordinate point you know the
required number of units determined by the slope.
Example
A line has a slope of
6
5
and passes through point (3,4). What is another point the line passes through? 
The slope is 
6
5
, so you know there is a vertical change of 6 and a horizontal change of 5. So, starting at point
(3,4), add 6 to the 
y
-coordinate and add 5 to the 
x
-coordinate:
y:
4 
6 
10
x:
3 
5 
8
Therefore, another coordinate point is (8,10).
If you know the slope of a line and one point on the line, you can also determine a point at a certain coordi-
nate, such as the 
y
-intercept (
x
,0) or the 
x
-intercept (0,
y
).
Example
A line has a slope of
2
3
and passes through point (1,4). What is the 
y
-intercept of the line?
Slope 
x
y
2
2
y
x
1
1
, so you can plug in the coordinates of the known point (1,4) and the unknown point, the 
y
-intercept (
x
,0), and set up a ratio with the known slope,
2
3
, and solve for 
x
:
y
x
2
2
y
x
1
1
2
3
0
x
1
4
2
3
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0
x
1
4
2
3
Find cross products.
(
4)(3) 
2(
x
1)
12 
2
x
2
12 
2 
2
x
2 
2
1
2
0
2
2
x
1
2
0
x
5 
x
Therefore, the 
x
-coordinate of the 
y
-intercept is 
5, so the 
y
-intercept is (
5,0).
Facts about Slope
■
A line that 
rises to the right
has a positive slope.
■
A line that 
falls to the right
has a negative slope.
■
A horizontal line has a slope of 0.
slope 
 0
negative slope
positive slope
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■
A vertical line does not have a slope at all—it is undefined.
■
Parallel lines have equal slopes.
■
Perpendicular lines have slopes that are negative reciprocals of each other (e.g., 2 and 
1
2
).
Practice Question
A line has a slope of
3 and passes through point (6,3). What is the 
y
-intercept of the line?
a.
(7,0)
b.
(0,7)
c.
(7,7) 
d.
(2,0)
e.
(15,0) 
slopes are negative reciprocals
equal slopes
no slope
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Answer
a.
Slope 
y
x
2
2
y
x
1
1
, so you can plug in the coordinates of the known point (6,3) and the unknown point,
the 
y
-intercept (
x
,0), and set up a ratio with the known slope,
3, and solve for 
x
:
y
x
2
2
y
x
1
1
3
0
x
6
3
3
x
3
6
3
Simplify.
(
x
6) 
x
3
6
3(
x
6)
3 
3
x
18
3 
18 
3
x
18 
18
21 
3
x
2
3
1
3
3
x
2
3
1
x
7 
x
Therefore, the 
x
-coordinate of the 
y
-intercept is 7, so the 
y
-intercept is (7,0).