1. Plane coordinate geometry has many
uses in the practical world. The architects
and the engineers, the pilot and the
navigators, the businessmen and the
accountants, the artist and the sculptors,
and even the carpenters make use of
coordinate geometry in their respective field
of work.
2. 7.1 Cartesian Coordinate System
To fully understand the mathematical
characteristics of lines and curves represented by
their equations, it is important to picture them in a
coordinate plane. This coordinate plane used in
coordinate geometry is attributed to Renẻ Descartes,
the father of Modern Mathematics, who bridged the
gap between Algebra and Geometry.
The Cartesian coordinate system consist of
two number lines whish are perpendicular to each
other. One is vertical and the other is horizontal. The
horizontal number line is called the x-axis, while the
vertical number line is called the y-axis. These two
axes intersect at a point (0, 0) called the origin.
3. Look at the figure below.
II 6 I
4
2
-6 -4 -2 0 2 4 6
-2
-4
III -6 IV
4. The x-axis and y-axis divide the plane into four
quadrants, properly labeled in the figure as I, II, III, IV.
They are also referred to as FIRST QUADRANT,
SECOND QUADRANT, THIRD QUADRANT, FOURTH
QUADRANT .
Coordinates to the right of the y-axis are positive,
while those to the left of y-axis are negative.
Coordinates upward form the x-axis are positive and
those downward from the x-axis are negative.
Any point in a plane is identified by an ordered pair
of numbers denoted as (x, y) where x and y are called
the coordinate of a point. The x-coordinate is the
abscissa and the y-coordinate is the ordinate. The
abscissa is always the first coordinate in the ordered
pair, and the ordinate is always the second coordinate in
the ordered pair.
5. There is one-to-one correspondence between the set
of the ordered pairs and the points in the plane.
Y
8 B(2 ,7)
C(-4 ,6)
4
2 A(3 ,2)
D(-7 ,1)
X
-8 -4 -2 0 2 4 8
E(-3 ,-2
F(-6 ,-3) G(4, -4)
4
H(9, -5)
6
8
6. The abscissa of the x-coordinate represents the distance of
a point from the y-axis,
And its sign indicates whether it is to the left or to the right of
the y-axis .The ordinate or the y-coordinate represents the
distance of the point from the x-axis, and its sign indicates
whether it is above or below the x-axis.
Remember!!!
Each ordered pair of number corresponds to exactly
one point in the coordinate plane. Each point in the
coordinate plane corresponds to exactly one ordered
pair of numbers.
7. 7.1.1 Slope Of a Line
Highways, roads, and bridges have different degrees of
steepness. This steepness is also called Slope. Slope is
the ratio of the rise to the run,
written as rise
run
Consider the following Illustrations:
A. B.
3 (rise) 6 (rise)
12 (run) 12 (run)
Car A goes through a road w/slope 3 or 1 , while car B
12 4
goes through a road w/slope 6 or 1 .
12 2 Next Page!!!!!!!!!
8. Lines have slopes, too!!!!...
L m 6 (run)
2 (rise) 4 (rise)
6 (run)
The slope of line l is: The slope of line m is:
Rise = 2 = 1 Rise = -4 = -2
Run 6 3 Run 6 3
Consider the graph of y=2x. Observe that the 2
Slope of the non-vertical line is found by 1 4 u ni ts
comparing the vertical change (rise) to the
Horizontal Change (run). Consider the points
(-1,-2) and (1,2) . The vertical change is 3 2 1 0 1 2 3 4
4 unit while the horizontal change is 2 unit. 1
2 2 u n it s
9. Illustrative Example:
A.
Rise= 2-(-2)=4=2 The Slope is Positive when the line
Run 2-0 2 rises from left to right
B.
Rise=2-0= 2= -2 The slope is negative when the line
Run 0-3 -3 3 falls from left to right
C.
2-2 = 0 = 0
3-(-1) 4 The slope of a horizontal line is Zero
.It does not rise or fall
D.
3-(-3) = 6 (Undefined)
-2-(-2) o The slope of a vertical line is
Undefined
10. Remember!!!
The slope m of a non vertical line containing
two points with coordinates (x,y) and (x,y) is
given by the formula
M= Y2-Y1
X2-X1
The slope of a horizontal line is Zero “0”.The
slope of a vertical line is Undefined.
11. 7.1.2 Linear Equation
Plot the points in the following table of values. Then
connect the points with a straight line.
x -3 -2 -1 0 1 2
y -4 -2 0 2 4 6
