Coordinate Geometry

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

The point (4, 3) is located 4 units from the y-axis to the right and 3 units above the x-axis; the point (-2, 1) is 2 units to the left of the y-axis and 1 unit above the x-axis. The distance, d, between the two points can be found by the Pythagorean Theorem. The horizontal leg is the total distance inthe x-direction: 4 - (-2) = 6; the vertical leg is the distance in the y-direction: 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 x-direction is 6 - 2 = 4 and in the y-direction 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 y-intercept.

The y-intercept is when x = 0 in an equation.
The x-intercept is when y = 0 in an equation.

The x-intercept is the point where the line crosses the x-axis. It is found by setting y = 0 and solving the resulting equation. The y-intercept is the point where the line crosses the y-axis. 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 y-intercept is -5)

In the above graph, the slope is 4 and the y-intercept is -5.

In the above graph, the blue dot is the y-intercept 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).