In coordinate geometry, the locations of points in a plane are indicated by ordered pairs of real numbers.
Plane: A flat surface that extends indefinitely in any direction.
x-axis and y-axis: The horizontal (x) and vertical (y) lines that intersect perpendicularly to indicate location on a coordinate plane. Each axis is a number line.
Ordered pair: Two numbers or quantities separated by a comma and enclosed in parentheses. An example would be (8,7). All the ordered pairs that you’ll see in GRE coordinate geometry problems will be in the form (x,y), where the first quantity, x, tells you how far the point is to the left or right of the y-axis, and the second quantity, y, tells you how far the point is above or below the x-axis.
Coordinates: The numbers that designate distance from an axis in coordinate geometry. The first number is the x-coordinate; the second is the y-coordinate. In the ordered pair (8,7), 8 is the x-coordinate and 7 is the y-coordinate.
Origin: The point where the x- and y-axes intersect; its coordinates are (0,0).
Here’s what a coordinate plane looks like:
Any point in a coordinate plane can be identified by an ordered pair consisting of its x-coordinate and its y-coordinate. Every point that lies on the x-axis has a y-coordinate of 0, and every point that lies on the y-axis has an x-coordinate of 0.
When you start at the origin and move:
| to the right | . . . . . . . . . . . . . . . . . . . . . | x is positive |
| to the left | . . . . . . . . . . . . . . . . . . . . . | x is negative |
| up | . . . . . . . . . . . . . . . . . . . . . | y is positive |
| down | . . . . . . . . . . . . . . . . . . . . . | y is negative |
Therefore, the coordinate plane can be divided into four quadrants, as shown below.
The distance between two points is equal to the length of the straight-line segment that has those two points as endpoints.
If a line segment is parallel to the x-axis, the y-coordinate of every point on the line segment will be the same. Similarly, if a line segment is parallel to the y-axis, the x-coordinate of every point on the line segment will be the same.
Therefore, to find the length of a line segment parallel to one of the axes, all you have to do is find the difference between the endpoint coordinates that do change. In the diagram that follows, the length of AB equals x2 − x1.
You can find the length of a line segment that is not parallel to one of the axes by treating the line segment as the hypotenuse of a right triangle. Simply draw in the legs of the triangle parallel to the two axes. The length of each leg will be the difference between the x- or y-coordinates of its endpoints. Once you’ve found the lengths of the legs, you can use the Pythagorean theorem to find the length of the hypotenuse (the original line segment).
In the diagram below, (DE)2 = (EF)2 + (DF)2.
Example:
If the coordinates of point A are (3,4) and the coordinates of point B are (6,8), what is the distance between points A and B?
You don’t have to draw a diagram to use the method just described, but drawing one may help you to visualize the problem. Plot points A and B and draw in line segment AB. The length of AB is the distance between the two points. Now draw a right triangle, with AB as its hypotenuse. The missing vertex will be the intersection of a line segment drawn through point A parallel to the x-axis and a line segment drawn through point B parallel to the y-axis. Label the point of intersection C. Since the x- and y-axes are perpendicular to each other, AC and BC will also be perpendicular to each other.
Point C will also have the same x-coordinate as point B and the same y-coordinate as point A. That means that point C has coordinates (6,4).
To use the Pythagorean theorem, you’ll need the lengths of AC and BC. The distance between points A and C is simply the difference between their x-coordinates, while the distance between points B and C is the difference between their y-coordinates. So AC = 6 − 3 = 3, and BC = 8 − 4 = 4. If you recognize these as the legs of a 3:4:5 right triangle, you’ll know immediately that the distance between points A and B must be 5. Otherwise, you’ll have to use the Pythagorean theorem to come to the same conclusion.
Straight lines can be described by linear equations.
Commonly:
where m is the slope
and b is the point where the line intercepts the y-axis, that is, the value of y where x = 0.
Lines that are parallel to the x-axis have a slope of zero and therefore have the equation y = b. Lines that are parallel to the y-axis have the equation x = a, where a is the x-intercept of that line.
If you’re comfortable with linear equations, you’ll sometimes want to use them to find the slope of a line or the coordinates of a point on a line. However, many such questions can be answered without determining or manipulating equations. Check the answer choices to see if you can eliminate any by common sense.
Example:
Line r is a straight line as shown above. Which of the following points lies on line r?
Line r intercepts the y-axis at (0,−2), so you can plug −2 in for b in the slope-intercept form of a linear equation. Line r has a rise (∆y) of 2 and a run (∆x) of 5, so its slope is
That makes the slope-intercept form
The easiest way to proceed from here is to substitute the coordinates of each answer choice into the equation in place of x and y; only the coordinates that satisfy the equation can lie on the line. Choice (E) is the best answer to start with, because 10 is the only x-coordinate that will not create a fraction on the right side of the equal sign. Plugging in (10,2) for x and y in the slope-intercept equation gives you
which simplifies to 2 = 4 − 2.
That’s true, so the correct answer choice is (E).