GMAT Exam: A Simple Guide To Geometry - Coordinate Geometry

Basic Concepts
The coordinate plane is defined by two real number lines, one horizontal and one vertical, intersecting at right angles at their zero points.
The two real number lines are the coordinate axes. Commonly, the horizontal axis with positive direction to the right is the x-axis, and the vertical axis with positive direction upward is the y-axis.
The two axes determine a plane. Their point of intersection is the origin.

The axes divide the coordinate plane into four quadrants.

The quadrants are numbered with Roman numerals—I, II, III, and IV—beginning in the upper right and going around counterclockwise.


Remember, the quadrants are numbered counterclockwise.

In the coordinate plane, each point P is identified by its coordinates, an ordered pair (x, y) of real numbers x and y.

The ordered pair (0, 0) is the origin.
The order in the ordered pair (x, y) that corresponds to a point P is important.
The absolute value of the first coordinate, x, is the perpendicular horizontal distance (right or left) of the point P from the y-axis.
If x is positive, P is to the right of the y-axis; if x is negative, it is to the left of the y-axis.
The absolute value of the second coordinate, y, is the perpendicular vertical distance (up or down) of the point P from the x-axis.
If y is positive, P is above the x-axis; if y is negative, it is below the x-axis.

For instance, in the coordinate plane shown below, the point P has coordinates (–6, 9), the point Q has coordinates (2, 4), the point R has coordinates (0, –3), the point S has coordinates (–4, –7), and the point T has coordinates (5, –8).


- Distance Between Two Points The distance between two points (x1, y1) and (x2, y2) in a coordinate plane is .

Here is an example.
The distance between (–6, 9) and (–4, –7) is

To avoid careless errors when using the distance formula, enclose substituted negative values in parentheses.
- Midpoint Between Two Points: The midpoint between two points (x1, y1) and (x2, y2) in a coordinate plane is the point with coordinates .

Here is an example.
The midpoint between (2, 4), and (0, –3) is

.

- Slope of a Line Through Two Points

The slope m of a line through two distinct points, (x1, y1) and (x2, y2), is , provided x1 ≠ x2.

Here is an example.
The slope of the line through (–4, –7) and (2, 4) is

.

When you use the slope formula, be sure to subtract the coordinates in the same order in both the numerator and the denominator.
That is, if x2 is the first term in the numerator, then y2 must be the first term in the denominator.
Also, it a good idea to enclose substituted negative values in parentheses to guard against careless errors.

- The slope describes the steepness or slant of the line between the two points. Lines that slant upward from left to right have positive slopes, and lines that slant downward from left to right have negative slopes.
 

= Horizontal lines have zero slope. The slope of a vertical line is undefined.

- If two lines are parallel, their slopes are equal; if two lines are perpendicular, their slopes are negative reciprocals of each other.

Graphs of Functions
The graph of a function is a set of ordered pairs in the coordinate plane. The equation that defines the function generates the ordered pairs.

The graphs of examples of four common functions are shown below.
 

In the graph of a quadratic function that has real roots, the graph will intersect the x- axis at the roots.

You can visually determine whether a graph is the graph of a function by using the vertical line test:
A graph is the graph of a function if and only if no vertical line crosses the graph in more than one point.
This test is the graphical equivalent of saying that no two different ordered pairs have the same first coordinate.

Distance from a Point to a Line
The distance from point (x1, y1) to the line whose equation is Ax + By + C = 0 is


For example, the distance from the point (–1, 2) to the line whose equation is 4x – 3y – 1 = 0 is