The Cartesian Plane
Math 1090-001 (Spring 2001)
Friday Jan 12, 2001
The Cartesian Plane is a standard graphical representation of
2
1
, the set of ordered pairs of real numbers.
• The Cartesian Plane has a pair of mutually perpendicular coordinate axes. Usually, these two axes are drawn
horizontally and vertically as in the diagram below.
• The point of intersection of these two axes is called the origin.
• The arrow on each axis indicates the direction of increase of the coordinate of that axis. Usually we give names
to the axes. For the discussion below, we will call the horizontal axis the x-axis and the vertical one the y-axis.
• Each point in the Plane determines an ordered pair of real numbers and vice versa as follows:
1. Suppose a point P on the plane is given.
2. Draw a horizontal and vertical line through P .
3. Denote the intersection of this horizontal line with the y-axis (the vertical axis) by B and denote the intersection of the vertical line with the x-axis (the horizontal axis) by A.
4. We now define the y-coordinate, denoted by yP , of the point P to be the signed distance from the origin to
the point A. Similarly, we define the x-coordinate, denoted by xP , of P to be the signed distance from the
origin to the point B. (See the diagram below.)
5. Now, we associate to the point P — which is in the Cartesian Plane — to the ordered pair (x P , yP ) of real
numbers.
6. Conversely, given any ordered pair (a, b), we assoicate to it the point in the Plane with x-coordinate a and
y-coordinate b.
y
6
(−4, 3)
(2, 4)
B
6
P ↔ (xP , yP )
0 < yP
A
-
xP > 0
(−3, −2)
(3, −3)
- x
The Cartesian Plane
Math 1090-001 (Spring 2001)
Friday Jan 12, 2001
2
Exercises
1. a)
Plot the points (−4, −11) and (5, 7).
b)
Draw a line through the two points.
c)
Which of the following points also lie on the line? (−2, −7), (−3, −6), (−1, −4), (0, −3), (2, 3), (3, 3), (1, 1),
(4, 6).
2. a)
Plot the points (−5, −3), (−3, −1), (−1, 1), (3, 5), (5, 7), (7, 9).
b)
Draw a line through the plotted points, and list several other points on the line.
c)
Express the relation between the x- and y-coordinates of each point on the line in a verbal statement.
d)
Express the same relation in the form y = f (x).
e)
Will this line pass through the point (−103, −102)? Give a reason for your anwser.
3. a)
Plot each of the following pairs of points, locate the midpoint of the line segment joining the pair of points,
and determine the coordinates of each midpoint.
i)
ii)
iii)
iv)
v)
vi)
(0, 0), (4, 10)
(0, 0), (−6, 2)
(−1, 3), (3, 11)
(−2, −3), (6, 5)
(−5, −4), (−9, −2)
(−2, −1), (−10, −9)
b)
How is the x-coordinate of the midpoint related to the x-coordinates of the given end-points? What about
the y-coordinate?
c)
Without plotting, determine the coordinates of the midpoint of the line segment joining the points (1, 3) and
(19, −17).
d)
What is the midpoint of the line segment joining the points (x1 , y1 ) and (x2 , y2 )?