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1.05 Segments and the Coordinate Plane

 

 Blueprints for this Lesson:
 
  • Learn about the coordinate plane.
  • Graph points in the coordinate plane.
  • Find the distance between two points using the distance formula.
  • Find the midpoint between two points using the midpoint formula.

Foundational Knowledge:

Simplify the following:

  1. 52

  2. The square root of 16 is equal to what?

  3. The square root of 40 is equal to what?

  4. (4 - 7)2 + (-2 - 8)2

check your answersClick here to check your answers.


Framework for Understanding:

Imagine you are planning to build a home on a piece of property. Do you think you might want to use graph paper in some way? Would you want to know the size and location of the property for the home before you started? Of course! This information would help you design the home to meet building requirements for that property. The graph paper would help you map out the exact measurements and location before you started.

In math, we often use a coordinate plane to draw figures and to provide locations for them. It is important that you are able to read coordinates and know how to graph them. In this lesson you will review how to graph ordered pairs (coordinates), learn how to use a coordinate plane to measure distance, and find the midpoints of segments.

The Coordinate Plane - Web Interactivity

Read through the following interactive lesson. It is important that you are comfortable with the parts of the coordinate plane, how to graph ordered pairs, and how to name points.

  1. Go to the SAS® Curriculum Pathways®.

  2. You will be asked for a user name and password. Visit the materials list in the Course Information area for this information.

  3. Find the Quick Launch box. Type in the Web Inquiry number 963.

  4. Click Go. You can also locate activities by subject and title.

  5. When a new screen is displayed, click on "Coordinate Plane" on the Graphers menu.

  6. A new screen will be displayed. You can use the given ordered pairs or type in your own ordered pairs. (You can also move the existing ordered pairs.)

  7. There are four check boxes under the "graph paper" on the right of the screen. Click on Label all points and Show quadrants to see these items illustrated in the graph.

  8. Click on "Practice" on the left of the screen, to practice plotting points given the ordered pairs (Part 1) and identifying the ordered pairs given graphed points (Part 2).

Distance Formula:

When two points are graphed on a coordinate plane, it is important that you can find the distance between them. Sometimes this process is easy, because the two points are on the same vertical or horizontal line. Look at the examples below. segment A B is a vertical segment. It is easy to use the y-axis as a number line and determine the length of the segment. segment C D is a horizontal segment. You can use the x-axis as a number line and determine the length of the segment. Roll your cursor over the image to see the lengths of each segment.

Cartesian Coordinate Plane with ray BA and segment CD. Point A lies at left parenthesis -4, 4 right parenthesis. Point B lies at left parenthesis -4, -2 right parenthesis. Point C lies at left parenthesis 2,2 right parenthesis. Point D lies at left parenthesis 5,2 right parenthesis.

Not all segments graphed on a coordinate plane are vertical or horizontal. Some are diagonal. Because these segments do not follow a number line, we have to do more than just count boxes to determine their length. Mathematicians have devised a distance formula to be used in this situation. To see the derivation of this formula (where it came from) click here.

 

Distance Formula

Given the points (x1, y1) and (x2, y2), the distance between the two points is found using the following formula:

Distance equals the square root of left parenthesis x sub 2 minus x sub 1 rigth parenthesis quantity squared plus left parenthesis y sub 2 minus y sub 1 right parenthesis quantity squared.


Here is an example of using the distance formula given two ordered pairs.
Example 1:
Find the distance between the ordered pairs (5, 8) and (3, -2).
x 1 = 5          x2 = 3
y1 = 8           y2 = -2

Step 1: Determine the values for x1, y1, x2, and y2 before you begin.

Note: The values of x1 and y1 are the x and y values from the first ordered pair. The values of x2 and y2 are the x and y values from the second ordered pair.
Distance equals the square root of left parenthesis 3 minus 5 right parenthesis quantity squared plus left parenthesis negative 2 minus 8 right parenthesis quantity squared. Next line. Distance equals the square root of left parenthesis negative 2 right parenthesis quantity squared plus left parenthesis negative 10 right parenthesis quantity squared. Next line. Distance equals square root of 4 plus 100. Next line. Distance equals square root of 104 or square root of 2 square root of 26.
Step 2: Substitute the values for x1, y1, x2, and y2 into the distance formula and simplify according to the order of operations.
In order to understand this next example, please review the following important information about circles.
  1. The distance from the center to any point on the circle is called the radius.
  2. The distance from one side of the circle to the other passing through the center is called the diameter.
  3. If you multiply the radius by 2, you will know the length of the diameter. So, if the radius is 4, the diameter is 8.
Circle with diameter drawn from side to side of circle through the center point. A radius is also drawn from the center point to the side of the circle.
Example 2:
What is the length of the radius that connects the point (-1, 3) on the circle to the center point (2, 8)?



x1 = -1          x2 = 2
y1 = 3           y2 = 8

A circle is drawn with a radius in it. The center point lies at left parenthesis 2, 8 right parenthesis. The end of the radius lies at the point left parenthesis -1, 3 right parenthesis.

Step 1: Determine the values for x1, y1, x2, and y2 before you begin.

 
Distance equals the square root of left parenthesis 2 minus negative 1 right parenthesis quantity squared plus left parenthesis 8 minus 3 right parenthesis quantity squared. Next line. Distance equals the square root of left parenthesis 3 right parenthesis squared plus left parenthesis 5 right parenthesis squared. Next line. Distance equals square root of 9 plus 25. Next line. Distance equals square root of 34.
Step 2: Substitute the values for x1, y1, x2, and y2 into the distance formula and simplify according to the order of operations.

Midpoint Formula

Given the points (x1, y1) and (x2, y2), the midpoint between the two points is found using the following formula:

Midpoint equals left parenthesis quantity x sub 1 plus x sub 2 divided by 2, quantity y sub 1 plus y sub 2 divided by 2 right parenthesis.

Notice: The midpoint is an actual ordered pair.

The midpoint formula allows us to find the midpoint between two points on a line segment.

Cartesian Coordinate Plane with segment drawn in the first quadrant. One endpoint of the segment is labeled left parenthesis x sub 1, y sub 1 right parenthesis. The other endpoint of the segment is labeled left parenthesis x sub 2, y sub 2 right parenthesis. A arrow points to the location on the segment that is labeled midpoint.
Example 3:
Find the midpoint between the points (-2, 2) and (6, 4).

x1 = -2          x2 = 6
y1 = 2          y2 = 4

Step 1: Indentify the values for x1, y1, x2, and y2 before you begin.

Midpoint equals left parenthesis quantity negative 2 plus 6 divided by 2, quantity 2 plus 4 divided by 2 right parenthesis. Next line. Midpoint equals left parenthesis 4 divided by 2, 6 divided by 2 right parenthesis. Next line. Midpoint equals left parenthesis 2, 3 right parenthesis.
 Step 2: Substitute the values for x1, y1, x2, and y2 into the midpoint formula and simplify following the order of operations.
Example 4:

The midpoint of line A B is M (4, 1).  One endpoint is A (-2, 3). Find the coordinate of the other endpoint B.

x-coordinate of M= x coordinate of the midpoint equals quantity x sub 1 plus x sub 2 divided by 2.
y-coordinate of M= y coordinate of the midpoint equals quantity y sub 1 plus y sub 2 divided by 2.
x-coordinate of M=4
y-coordinate of M=1
x1 = -2             y1 = 3		 Step 1: Identify the values for x1, y1, x2, and y2 before you begin.
4 equals quantity negative 2 plus x sub 2 divided by 2. 1 equals quantity 3 plus y sub 2 divided by 2. Next line. Multiply both sides by 2. Next line. 8 equals minus 2 plus x sub 2. 2 equals 3 plus y sub 2. Next lin. 10 equals x sub 2. Minus 1 equals y sub 2. Step 2: Substitute the values for the x and y-coordinates of M, x1, y1, and simplify.
Endpoint B has coordinates (10, -1) Step 3: Write the coordinates as an ordered pair.

Move on to the practice page to try some on your own.