Coordinate Geometry
Rectangular
Coordinates
A point P is positioned relative to
two perpendicular lines, called the coordinate axes. The
perpendicular distance from the yaxis to point P is the xcoordinate, the
perpendicular distance from the xaxis to point P. The coordinates x and y form
an ordered pair (x, y).

The point
(4, 3) is located 4 units from the yaxis to the right and 3 units above the
xaxis; the point (2, 1) is 2 units to the left of the yaxis and 1 unit above
the xaxis. The distance, d, between the two points can be found by the
Pythagorean Theorem. The horizontal leg is the total distance inthe xdirection: 4 
(2) = 6; the vertical leg is the distance in the ydirection: 3  1 = 2.
The distance is then
d= 6 + 2=
= 2
Example 1
A square has two corners of a diagonal at (6, 8) and (2,4). What is its area?
Solution
The difference in the xdirection is 6  2 = 4 and in the ydirection 8  4 =
4. The sides are both of length 4, so that the area is A = 4 x 4 = 16.
Slope
Formula:
How do you measure the slant of a line? By definition, it is the ratio of the vertical change to the horizontal
change (see figure below).
Forming the vertical change over the horizontal change (above) figure results in slope
formula (where m is the slope).
m = y  b
x  a
 use this formula to calculate slopes of lines
Example 2
What is the slope of this line?
Solution
To solve this problem, plug the digits in the line into the slope intercept
equation.
m = y  b
x  a
y= 4, b = 2
x
= 5, a = 1
m = (4  2)
(5  1)
m = 2 = 1
4
2
The slope is 1/2.
Slope
Intercept Formula
If
you have a formula, such as x  2y = 4, how do you calculate the slope of the
line?
If you want
to graph a line, the formula to use is
y = mx + b
In this equation, m is the
slope of the line and b is the yintercept.
The yintercept is when x = 0 in an equation.
The xintercept is when y = 0 in an equation.
The xintercept is the point where the line crosses the xaxis. It is found by
setting y = 0 and solving the resulting equation. The yintercept is the point
where the line crosses the yaxis. It is found by setting x = 0 and solving the
resulting equation.
Example 3
Graph the equation 4x  y = 5.
Solution
Try to convert the equation 4x  y = 5 into the format
y = mx + b
4x  y = 5 can be converted into
y = 4x  5
This means that m = 4 (the slope is 4) and b = 5 (the yintercept is 5)
In the above graph, the slope is 4 and the yintercept is 5.
In the above graph, the blue dot is the yintercept at (0,5).
The line slopes up at a rate of 4 up for every 1 across (slope of 4). It
intersects lines at (1,1) and (2,3).