Every point in space can be assigned three numbers with respect to a starting point. Those three numbers allow us to distinguish any point from any other point in space. Fortunately for you, we are not dealing here with three dimensions, but only with two.
Coordinates of a point: Each point on a number line is assigned a number. In the same way, each point in a plane is assigned a pair of numbers.
x‐axis and y‐axis: To locate points in a plane, two perpendicular lines are used—a horizontal line called the x‐axis and a vertical line called the y‐axis.
Origin: The point of intersection of the x‐axis and y‐axis.
Coordinate plane: The x‐axis, y‐axis, and all the points in the plane they determine.
Ordered pairs: Every point in a coordinate plane is named by a pair of numbers whose order is important; these numbers are written in parentheses and separated by a comma.
x‐coordinate : The number to the left of the comma in an ordered pair is the x‐coordinate of the point and indicates the amount of movement along the x‐axis from the origin. The movement is to the right if the number is positive and to the left if the number is negative.
y‐coordinate : The number to the right of the comma in an ordered pair is the y‐coordinate of the point and indicates the amount of movement perpendicular to the x‐axis. The movement is above the x‐axis if the number is positive and below the x‐axis if the number is negative.
Note: The coordinates [ordered pair] for the origin are (0, 0).
The x‐axis and y‐axis separate the coordinate plane into four regions called quadrants. (See Figure 1.) The upper right quadrant is quadrant 1; the upper left quadrant is quadrant II; the lower left quadrant is quadrant III; and the lower right quadrant is quadrant IV. Notice the following:
Figure 1 The coordinate axes separate the plane into four quadrants.
- In quadrant I, x is always positive and y is always positive.
- In quadrant II, x is always negative and y is always positive.
- In quadrant III, x is always negative and y is always negative.
- In quadrant IV, x is always positive and y is always negative.
The point associated with an ordered pair of real numbers is called the graph of the ordered pair.
Example 1: Identify the points A, B, C, D, E, and F on the coordinate graph in Figure .
Figure 2 Finding the coordinates of specific points in the plane.
Example 2: Rectangle ABCD has coordinates as follows: A(−5,2), B(8,2), and C(8, −4). Find the coordinates of D.
A graph is helpful in solving this problem. Refer to Figure . The coordinates of D must be (−5,−4) .
Figure 3 Finding the coordinates of the fourth vertex of a rectangle.
Example 3: Use Figure 3 to find the following distances: (a) from A to B (called AB) and (b) from B to C (called BC).