GRE Exam: A Simple Guide To Solving Geometry Problems - The Coordinate Plane

The coordinate plane is used to designate points in a two-dimensional plane. It is created by two perpendicular lines that meet at the origin. The horizontal line is called the x-axis and the vertical line is called the y-axis.

These lines split the plane into four quadrants: I, II, III, and IV.


Points on the coordinate plane are designated by assigning two numbers: an x-coordinate and a y-coordinate. A point is written in the form (x,y).

The x-coordinate designates the horizontal position of a point.

If a point is to the right of the origin, then its x-coordinate is positive.

If a point is to the left of the origin, then its x-coordinate is negative.

If a point is on the y-axis, then its x-coordinate is zero.

The y-coordinate designates the vertical position of a point.

If a point is above the origin, then its y-coordinate is positive.

If a point is below the origin, then its y-coordinate is negative.

If a point is on the x-axis, then its y-coordinate is zero.


Properties of a Line
On the coordinate plane, a line is formed by joining two points. Any line on the coordinate plane will be defined by the following equation:
y = mx + b
x and y can refer to any point on the line, but m and b are constants—they define properties of a line.

Slope
In the equation y = mx + b, m refers to the slope of the line. The slope of a line indicates its steepness and whether it is rising or falling.



To determine the slope of a line, you are looking for the ratio of how much the line rises vertically to how much the line moves horizontally.
If you are given two points on a line, you can always calculate the slope.

Given points (x1,y1) and (x2,y2), the formula for the slope of a line is:

What is the slope of a line that contains the points (2,3) and (5,7)?
SOLUTION: Use the preceding formula: .

Line k passes through the points (2,3) and (4,6).


SOLUTION: Plug the above values into the slope formula: . , so Quantity B is greater.

Y-intercept
In the equation y = mx + b, b refers to the y-intercept of the line. The y-intercept of a line refers to where a line crosses the y-axis. Since this point is on the y-axis, the x-coordinate will always be zero.


Points and the Equation for a Line
Once you know the slope and y-intercept of a line, you can substitute those values into the equation for the line. For example, if you are told that a line has a y-intercept of 3 and a slope of 7, plug in 3 for b and 7 for m: y = 7x + 3.
Sometimes, the GRE will define a line using a different form than the previous one. Always manipulate the equation to be in y = mx + b form.

If the equation for line k is 2x + 3y = 6, what is the slope of line k?
SOLUTION: Isolate y so that the equation is in y = mx + b form. Subtract 2x:

Divide both sides by 3:

The slope of the line is .

Using Two Points to Determine a Line
Generally, if the GRE asks you to determine the equation for a line, it will do so by providing you with two points. Let’s look at how to do so:

What is the equation for the line that passes through (2,11) and (5,20)?
Step 1: Use the slope formula to determine m:

Step 2: Substitute the value for m in the equation for the line:

Step 3: Solve for b: Since (2,11) and (5,20) lie on the line, both points will satisfy the equation for the line. Thus you can plug in the coordinates for either point to solve for b. Use the point (2,11):

Step 4: Substitute b into the equation:


Using a Line to Determine a Point
In the previous example, you used two points on a line to determine its equation. In other situations, you will be given the equation for a line and will be asked to find a point that lies on the line.

Let’s look at an example:

If line k is defined by the equation y = 7x + 3, then which of the following points must lie on the line?
(3,7)
(7,3)
(3,0)
(2,17)
(17,2)

SOLUTION: Plug each point into the equation. Whichever point keeps the equation true will be your answer.
A: 7 = 7(3) + 3?        → False
B: 3 = 7(7) + 3?        → False
C: 0 = 7(3) + 3?        → False
D: 17 = 7(2) + 3?      → True
E: 2 = 7(17) + 3?      → False
The correct answer is D.

Horizontal and Vertical Lines
Any horizontal line will have a slope of zero. Because such lines have no slope, they will always be written in the form: y = b.

For example, the equation for the line below is y = 7.

Any vertical line will have an undefined slope.

Such lines will be written in the form: x = a, where a represents the x-intercept of the line.

For example, the equation for the line below is x = 4.


Parallel and Perpendicular Lines
If two lines are parallel, then their slopes are equal and the lines never intersect. (See figure.)

If two lines are perpendicular, then the product of their slopes is −1.

Another way to say this is that their slopes are negative reciprocals. (See figure.)

Line k is defined by the equation y = 2x – 3.

Lines l and m are each perpendicular to line k.

SOLUTION: Since the equation for line k is y = 2x – 3, you know that the slope of line k is 2.

Since lines l and m are perpendicular to line k, each of their slopes is the negative reciprocal of 2: .

Quantity A is greater.

Distance Between Two Points
You can use the Pythagorean theorem to calculate the distance between two points on the coordinate plane.

For example:

What is the distance between (1,3) and (4,7)?
Step 1: Plot the two points.

Step 2: Connect the two points to create a right triangle.

Step 3: Calculate the lengths of each side of the resulting triangle:

The length of the horizontal leg will be the difference between the x-coordinates of the given points: 4 – 1 = 3.

The length of the vertical leg will be the difference between the y-coordinates of the two points: 7 – 3 = 4.

Now that you have two legs of a right triangle, you can determine the distance between the two points by calculating the hypotenuse:

The distance between the two points is 5.