The Cartesian Plane

When you are done with this section, you will be able to do the following:

Find the distance between two points
Find the midpoint of the line segment which joins two points
Determine which quadrant or on which axis a point resides
Solve application problems requiring the distance and/or midpoint formulas

What is the Cartesian Plane?

The Cartesian Plane (or coordinate plane) is made up of two directed real lines that perpendicularly intersection at their respective zero points.

The horizontal axis is called the x-axis. The real numbers on the x-axis increase as you move to the right. With respect to the vertical axis, the positive numbers are to the right of the vertical axis and the negative numbers are to the left of the vertical axis.

The vertical axis is called the y-axis. The real numbers on the y-axis increase as you move upward. With respect to the horizontal axis, the positive number to above the horizontal axis and the negative numbers are below the horizontal axis.

What is the origin?

The point where the horizontal axis and the vertical axis intersect is called the origin.

All points on the plane can be located by indicating where the point is horizontally (x-coordinate) and where the point is vertically (y-coordinate).

This location is written in a short hand of an ordered pair where

the first part of the ordered pair is the horizontal position of the point

and

the second part of the ordered pair is where the vertical position of the point.

What are quadrants?

The two axes break the Cartesian (or coordinate) Plane into four parts. These parts are called quadrants. The quadrants are numbered I, II, III, and IV started with the top right quadrant and working counter clockwise.

The points in quadrant I have x-coordinates and y-coordinates which are both positive.
The points in quadrant II have x-coordinates which are negative and y-coordinates which are positive.
The points in quadrant III have x-coordinates and y-coordinates which are both negative.
The points in quadrant IV have x-coordinates which are positive and y-coordinates which are negative.

It is assumed that you already know how to graph points in the plane. This is intended to be a quick review of the Cartesian Plane. There are several sources on-line with additional information about the Cartesian Plane.

Additional On-line Resources:

How do you find the distance between two points in the Cartesian Plane?

It is likely that you have been taught the distance formula in the past. It is also likely that you don't remember the distance formula very well. I expect however that you probably remember the Pythagorean Theorem. The Pythagorean Theorem tells you how to find the length of one side of a right triangle if you know the lengths of the other two sides.

Now let's look at how this is related to finding the distance between two points.

Example 1:

Find the distance between the points (-3, 1) and (7, 4).
Solution:

Let's start by graphing the two points on the Cartesian plane.

We want to find out how long the line segment (green line) connecting the two points is.

Since we are going to use the Pythagorean Theorem to answer this questions, we must have a right triangle. We will make a right triangle by running a vertical line through the point (7, 4) and a horizontal line through the point (-3, 1). Note that we could have switched the point through which we drew the vertical and horizontal lined and still would end up with a right triangle.

You can see that we have formed a right triangle. The legs are the yellow and purple line segments. We now need to determine how long the yellow and purple line segments are.

The yellow line segment goes straight up from 1 to 4 (following along the y-axis). The length then is the difference between 1 and 4 or |4 - 1| (read the absolute value of 4 minus 1). Since length must be non-negative, we must use the absolute value signs. This also allows us to subtract the numbers in either order.

The purple line segment goes horizontally from -3 to 7 (following along the x-axis). The length then is the difference between -3 and 7 or |-3 - 7| (read the absolute value of -3 minus 7). Since length must be non-negative, we must use the absolute value signs. This also allows us to subtract the numbers in either order.

Now we know that the legs of the right triangle have length 3 and 10. We put this into the Pythagorean Theorem to find out how long the hypotenuse is.

We are not done yet. Since all distances must be positive, we only take the positive value for c. Thus the distance between the points (-3, 1) and (7, 4) is

How does this relate to the distance formula? Let's put it all together in a more generic way. We will try to find the distance between the point

and

.
We will go through the process just like before.

Plot the points:

Draw the vertical and horizontal lines:

Find the point of intersection:

The third point in the triangle is

because it is in the same horizontal position (given by the x-coordinate) as

and the same vertical position (given by the y-coordinate) as

.

Now we determine the lengths of the legs of the right triangle.

The length of the vertical leg is the absolute value of the difference between the y-coordinates
(

)

The length of the horizontal leg is the absolute value of the difference between the x-coordinates
(

)

Now we put these distances into the Pythagorean Theorem to find the length of the green line segment.

We can get rid of the absolute value since when we square either the positive or negative number, we end up at the same place. We also only take the positive square root since we are finding a distance and distances cannot be negative.

Thus we have derived the Distance Formula between two points

and

Example 2: This one if for you to try.

Find the distance between the points (-1, 2) and (5, 4).
Solution: Click here for the solution.

Example 3: This one if for you to try.

Show that the following points form the vertices of a parallelogram: (1, 1), (8, 2), (9, 5), (2, 4).
Solution: Click here for the solution.

Example 4: This one if for you to try.

Find x so that the distance between (x, 8) and (-9, -4) is 15.
Solution: Click here for the solution.

It is assumed that you have already been introduced to the distance formula. This is intended to be a quick review of this formula. There are several sources on-line with additional information and examples related to the distance formula.

Additional On-line Resources:

How do you find the midpoint of the line segment joining the two points in the Cartesian Plane?

A midpoint, as you might guess, is a point half way between two points. You can also think of it as being a point whose x-coordinate is half way between the x-coordinates of the given points and whose y-coordinate is half way between the y-coordinates of the given points.

Way back in math, we learned that a number which is exactly half way between two other numbers is the same as the average of two numbers. This can help us to remember how to find the midpoint of a line segment joining tow points. We just average the coordinates. Since there are only 2 x-coordinates, we would add the x-coordinates together and divide that number by 2. Similarly, we would add the y-coordinates together and divide that number by 2.

This gives us the following formula for the midpoint of the line segment joining two points

and

Example 1:

Find the midpoint of the line segment joining the points (-3, 1) and (7, 4).

Solution:

First we will determine which point will be point 1 and which points will be point 2. It does not matter which we choose.

We will let (-3, 1) be point 1. This means that

and

.
We will let (7, 4) be point 2. This means that

and

.
Now we plug these numbers into the midpoint formula and we get

Remember that when you enter this in WeBWorK, you must enter the answer as an ordered pair including the parentheses.

Example 2: This one if for you to try.

Find the midpoint of the line segment joining the points (-1, 2) and (5, 4).
Solution: Click here for the solution.

Example 3: This one if for you to try.

Use the midpoint formula to estimate the sales for 1998 given the information in the table below.

Year	1996	2000
Sales	$4,200,000	$5,650,000

Solution: Click here for the solution.

Application:

Example: This one is for you to try

In a football game, the qurterback throws a pass from the 10-yard line, 12 yards from the sideline. The pass is cought on the 40-yard line 25 yards from the same sideline. How long was the pass?

Solution: Click here for the solution.

Cartesian Plane Homework:

The Cartesian Plane Homework contains 6 problems.
Things to remember:

The perimeter of a polygon is the sum of the lengths of the sides.
When you take the square root of both sides of an equation, you must use both the positive and negative square root.
Draw pictures to help you visualize word problems.

Homework examples:

Example 1: similar to problem 4
Example 2: similar to problem 6

Keep in mind that it is recommended that you also complete all the problems in the set called "also recommended" with the same number as the Cartesian Plane Homework. There are many different ways to ask the same questions. This will allow you to see additional problems that are related to this topic.