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Mathematics LibreTexts

8.2: The Cartesian Coordinate System

  • Page ID
    22509
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    Let’s begin with the concept of an ordered pair of whole numbers.

    Ordered Pairs of Whole Numbers

    The construct (x, y), where x and y are whole numbers, is called an ordered pair of whole numbers.

    Examples of ordered pairs of whole numbers are (0, 0), (2, 3), (5, 1), and (4, 9).

    Order Matters

    Pay particular attention to the phrase “ordered pairs.” Order matters. Consequently, the ordered pair (x, y) is not the same as the ordered pair (y, x), because the numbers are presented in a different order.

    We’ve seen how to plot whole numbers on a number line. For example, in Figure 8.1, we’ve plotted the whole numbers 2, 5, and 7 as shaded “dots” on the number line.

    Screen Shot 2019-09-27 at 2.29.55 PM.png
    Figure 8.1: Plotting the whole numbers 2, 5, and 7 on a number line.

    To plot ordered pairs, we need two number lines, called the horizontal and vertical axes, that intersect at the zero location of each line and are at right angles to one another, as shown in Figure 8.2(a). The point where the zero locations touch is called the origin of the coordinate system and has coordinates (0, 0). In Figure 8.2(b), we’ve added a grid. The resulting construct is an example of a Cartesian Coordinate System.

    Screen Shot 2019-09-27 at 2.30.01 PM.png
    Figure 8.2: A Cartesian coordinate system.

    Now, consider the ordered pair of whole numbers (5, 6). To plot this point on the “coordinate system” in Figure 8.3(a), start at the origin (0, 0), then move 5 units in the horizontal direction, then 6 units in the vertical direction, then plot a point. The result is shown in Figure 8.3(a). Adding a grid of horizontal and vertical lines at each whole number makes plotting the point (5, 6) much clearer, as shown in Figure 8.3(b).

    Screen Shot 2019-09-27 at 2.30.55 PM.png
    Figure 8.3: Plotting the Point (5, 6) in a Cartesian Coordinate System.

    The numbers in the ordered pair (5, 6) are called the coordinates of the plotted point in Figure 8.3(b). The first number of the ordered pair is called the abscissa and measures the horizontal distance to the plotted point. The second number is called the ordinate and measures the vertical distance to the plotted point. The combination of axes and grid in Figure 8.3(b) is called a coordinate system.

    The grid in Figure 8.3(b) is a visualization that greatly eases the plotting of ordered pairs. However, you do not have to draw these gridlines yourself. Instead, you should work on graph paper.

    Graph Paper Requirement

    All plotting should be done on graph paper.

    Example 1

    Plot the following ordered pairs of whole numbers: (3, 2), (8, 6), and (2, 7).

    Solution

    Create a Cartesian coordinate system on graph paper, then:

    • To plot the ordered pair (3, 2), start at the origin, then move 3 units to the right and 2 units up.
    • To plot the ordered pair (8, 6), start at the origin, then move 8 units to the right and 6 units up.
    • To plot the ordered pair (2, 7), start at the origin, then move 2 units to the right and 7 units up.

    The results are shown on the following Cartesian coordinate system.

    Screen Shot 2019-09-27 at 1.58.14 PM.png

    Exercise

    Plot the following ordered pairs of whole numbers: (2, 2), (5, 5), and (7, 4).

    Answer

    Screen Shot 2019-09-27 at 1.58.08 PM.png

    Allowing for Negative Numbers

    Again, we’ve seen how to plot both positive and negative numbers on a number line. For example, in Figure 8.4, we’ve plotted the numbers −4, −3/2, 2.2 and 4.

    Screen Shot 2019-09-27 at 1.59.24 PM.png
    Figure 8.4: Plotting the numbers −4, −3/2, 2.2, and 4.

    Note that the positive direction is to the right, the negative to the left. That is, to plot the number 2.2, we move 2.2 units to the right on the line, but to plot the number −3/2, we move 3/2 units to the left.

    To plot ordered pairs having both positive and negative numbers, we need two such number lines that intersect at the zero location of each line and are at right angles to one another, as shown in Figure 8.5(a). As before, adding a grid of horizontal and vertical lines at each integer will be extremely helpful when plotting points (see Figure 8.5(b)). The system of axes and grid in Figure 8.5(b) is called the Cartesian Coordinate System, named after its inventor, René Descartes.

    Screen Shot 2019-09-27 at 1.59.33 PM.png
    Figure 8.5: The Cartesian coordinate system.

    Plotting Points in the Cartesian Coordinate System

    On the horizontal axis, the positive direction is to the right, negative is to the left. On the vertical axis, the positive direction is up, negative is down. The point (0, 0) is called the origin of the coordinate system, and is the starting point for all point plotting.

    Example 2

    Sketch the points (4, 3), (−3, 2), (−2, −4), and (3, −3) on a Cartesian coordinate system.

    Solution

    Set up a Cartesian coordinate system on graph paper.

    • To plot the point (4, 3), start at the origin, move 4 units to the right, then 3 units up.
    • To plot the point (−3, 2), start at the origin, move 3 units to the left, then 2 units up.
    • To plot the point (−2, −4), start at the origin, move 2 units to the left, then 4 units down.
    • To plot the point (3, −3), start at the origin, move 3 units to the right, then 3 units down.

    These points are plotted and shown in Figure 8.6.

    Screen Shot 2019-09-27 at 1.59.43 PM.png
    Figure 8.6: Plotting points in the Cartesian coordinate system.

    Exercise

    Sketch the points (3, 4), (−4, 3), (−3, −4), and (4, −3) on a Cartesian coordinate system.

    Answer

    Screen Shot 2019-09-27 at 1.59.52 PM.png

    Example 3

    What are the coordinates of the points P, Q, R, and S in the Cartesian coordinate system that follows?

    Screen Shot 2019-09-27 at 2.00.02 PM.png

    Solution

    Make all measurements from the origin.

    • To obtain the coordinates of point P, start at the origin, move 3 units to the right, then 3 units up. Hence, the coordinates of the point P are (3, 3).
    • To obtain the coordinates of point Q, start at the origin, move 4 units to the left, then 3 units up. Hence, the coordinates of the point Q are (−4, 3).
    • To obtain the coordinates of point R, start at the origin, move 3 units to the left, then 4 units down. Hence, the coordinates of the point R are (−3, −4).
    • To obtain the coordinates of point S, start at the origin, move 4 units to the right, then 3 units down. Hence, the coordinates of the point S are (4, −3).

    These results are shown on the following Cartesian coordinate system.

    Screen Shot 2019-09-27 at 2.00.12 PM.png

    Exercises

    1. Identify the coordinates of the point P.

    Screen Shot 2019-09-27 at 2.22.07 PM.png

    2. Identify the coordinates of the point P.

    Screen Shot 2019-09-27 at 2.22.21 PM.png

    3. Identify the coordinates of the point P.

    Screen Shot 2019-09-27 at 2.22.26 PM.png

    4. Identify the coordinates of the point P.

    Screen Shot 2019-09-27 at 2.22.32 PM.png

    5. Identify the coordinates of the point P.

    Screen Shot 2019-09-27 at 2.22.39 PM.png

    6. Identify the coordinates of the point P.

    Screen Shot 2019-09-27 at 2.22.44 PM.png

    7. Identify the coordinates of the point P.

    Screen Shot 2019-09-27 at 2.22.50 PM.png

    8. Identify the coordinates of the point P.

    Screen Shot 2019-09-27 at 2.22.56 PM.png


    9. The points A(−1, 1), B(1, 1), C(1, 2), and D(−1, 2) are the vertices of a rectangle. Plot these points, draw the rectangle ABCD, then compute the area of rectangle ABCD.

    10. The points A(−3, −4), B(4, −4), C(4, −1), and D(−3, −1) are the vertices of a rectangle. Plot these points, draw the rectangle ABCD, then compute the area of rectangle ABCD.

    11. The points A(−2, −1), B(3, −1), C(3, 3), and D(−2, 3) are the vertices of a rectangle. Plot these points, draw the rectangle ABCD, then compute the area of rectangle ABCD.

    12. The points A(−3, −1), B(2, −1), C(2, 2), and D(−3, 2) are the vertices of a rectangle. Plot these points, draw the rectangle ABCD, then compute the area of rectangle ABCD.


    13. The points A(−4, −2), B(1, −2), C(1, 1), and D(−4, 1) are the vertices of a rectangle. Plot these points, draw the rectangle ABCD, then compute the perimeter of rectangle ABCD.

    14. The points A(−4, −4), B(1, −4), C(1, −3), and D(−4, −3) are the vertices of a rectangle. Plot these points, draw the rectangle ABCD, then compute the perimeter of rectangle ABCD.

    15. The points A(−1, 2), B(3, 2), C(3, 3), and D(−1, 3) are the vertices of a rectangle. Plot these points, draw the rectangle ABCD, then compute the perimeter of rectangle ABCD.

    16. The points A(−4, 2), B(3, 2), C(3, 4), and D(−4, 4) are the vertices of a rectangle. Plot these points, draw the rectangle ABCD, then compute the perimeter of rectangle ABCD.


    17. The points A(−3, −1), B(1, −1), and C(−3, 0) are the vertices of a triangle. Plot these points, draw the triangle ABC, then compute the area of the triangle ABC.

    18. The points A(−3, −2), B(1, −2), and C(−3, 2) are the vertices of a triangle. Plot these points, draw the triangle ABC, then compute the area of the triangle ABC.

    19. The points A(−1, −2), B(0, −2), and C(−1, 0) are the vertices of a triangle. Plot these points, draw the triangle ABC, then compute the area of the triangle ABC.

    20. The points A(−2, −3), B(−1, −3), and C(−2, 1) are the vertices of a triangle. Plot these points, draw the triangle ABC, then compute the area of the triangle ABC.


    21. Plot the points A(−3, −3) and B(0, 0) and find the straight-line distance between the two points. Hint: Create a right triangle, then use the Pythagorean Theorem.

    22. Plot the points A(−2, −3) and B(1, 2) and find the straight-line distance between the two points. Hint: Create a right triangle, then use the Pythagorean Theorem.

    23. Plot the points A(−2, −3) and B(0, 0) and find the straight-line distance between the two points. Hint: Create a right triangle, then use the Pythagorean Theorem.

    24. Plot the points A(−3, −2) and B(2, 2) and find the straight-line distance between the two points. Hint: Create a right triangle, then use the Pythagorean Theorem.


    25. Find the area of the shaded triangle.

    Screen Shot 2019-09-27 at 2.10.48 PM.png

    26. Find the area of the shaded triangle.

    Screen Shot 2019-09-27 at 2.10.54 PM.png

    27. Find the area of the shaded triangle.

    Screen Shot 2019-09-27 at 2.10.59 PM.png

    28. Find the area of the shaded triangle.

    Screen Shot 2019-09-27 at 2.11.03 PM.png


    Hint: In Exercises 29-32, surround the triangle with a rectangle, like those shown in Exercises 25-28.

    29. Find the area of the triangle with vertices at A(−4, −1), B(4, −2), and C(1, 3).

    30. Find the area of the triangle with vertices at A(−4, 2), B(3, 0), and C(0, 4).

    31. Find the area of the triangle with vertices at A(−3, 1), B(3, −3), and C(1, 4).

    32. Find the area of the triangle with vertices at A(1, 2), B(3, 0), and C(2, 4).


    Answers

    1. (1, 4)

    3. (4, 1)

    5. (−3, −1)

    7. (−1, 1)

    9. 2 square units

    11. 20 square units

    13. 16 units

    15. 10 units

    17. 2 square units

    19. 1 square units

    21. \(\sqrt{18}\)

    23. \(\sqrt{13}\)

    25. 6

    27. 7

    29. \(\frac{37}{2}\)

    31. 17

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