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Help with solving exercises on Cartesian Plane Elements

    We will analyze the main elements that we must take into account to construct a Cartesian plane. Furthermore, we will understand its importance both in calculus, physics, and engineering.

    What is a Cartesian Plane?

    The term Cartesian plane is named after the philosopher, mathematician, and physicist René Descartes (1596 – 1650), who was one of the first to use it in the study of Analytic Geometry.

    We understand the Cartesian plane as a two-dimensional space where points can be represented by ordered pairs of numbers (a, b), called the coordinates of the point. With several of these points on a Cartesian plane, due to a function or algebraic equation, we can represent or trace its curve through a graph.

    To construct a Cartesian plane

    1. Draw two perpendicular lines of real numbers.
    2. Ensure that these two lines intersect at the zero (0) of each line.
    3. Typically, one line is horizontal, called the abscissa or x-axis. The other line is vertical, called the ordinate or y-axis.
    4. The point where the x-axis and the y-axis intersect is called the origin (O).
    5. The x-axis, extending to the right from the origin, is positive.
    6. The y-axis, extending upwards from the origin, is positive.
    7. The two axes divide the plane into four quadrants, labeled I, II, III, and IV.
    8. The scale of each axis can be arbitrarily chosen but must be proportional along the entire axis.proporcionales en todo el eje.
    Cartesian plane, with its quadrants

    Exercise 1: Describe the position of some of the points on the following Cartesian plane.

    Points on the Cartesian plane

    Solution:

    • The point (-2, 2) is located in Quadrant II. The x-coordinate is positioned two units to the left of the origin, and the y-coordinate is positioned two units upward from the origin.
    • The point (2, -4) is located in Quadrant IV. The x-coordinate is positioned two units to the right of the origin, and the y-coordinate is positioned four units downward from the origin.
    • The point (-3, -2) is located in Quadrant III. The x-coordinate is positioned three units to the left of the origin, and the y-coordinate is positioned two units downward from the origin.

    Distance between two points in the Cartesian Plane

    We will present a formula to calculate the distance between two distant points in the Cartesian plane. These points are A(x1, y1) and B(x2, y2), which are arbitrarily located on the plane. The aforementioned points can be represented in the following figure of the plane:

    Distancia entre dos puntos en el plano cartesiano o plano coordenado

    The distance between two points in the plane is given by the formula:

    Formula for the distance between two points in the Cartesian plane

    Where:

    • (x1,y1) and (x2,y2) are the coordinates of the two points.
    • d represents the distance between the two points.

    Exercise 2: Calculate the distance between points A(3, 5) and B(-1, -3) located on the coordinate plane.

    Solution

    Where we have x1= 3 , y1= 5 , x2= -1 , y2=-3 . Using the distance formula, we can substitute the coordinates of points A and B:

    Solution to the problem of the distance between two points on the coordinate plane

    So, the distance between points A(3, 5) and B(-1, -3) is approximately 8.94 units.

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    Exercise 3: Which of the points P(1, -2) or Q(8, 9) is closer to the point A(5, 3)?

    Solution

    Calculamos la distancia del punto P al punto A, x1= 1 , y1= -2 , x2= 5 , y2=3

    Problems involving the distance between two points in the Cartesian plane.

    We calculate the distance from point Q to point A, x1= 8 , y1= 9 , x2= 5 , y2=3

    Problems involving the distance between two points in the Cartesian plane.

    Demostramos que d(P, A) < d(Q, A). Por lo tanto, P está más cerca al punto A.

    Exercise 4: Graph the points on the coordinate plane manually (explanation) and represent them in GeoGebra.

    (2, 3), (-2, 3), (4, 5), y (-4, 5)

    Solution

    The point (2, 3) is located in Quadrant I. The x-coordinate is two units to the right of the origin, and the y-coordinate is three units upward from the origin.

    The point (-2, 3) is located in Quadrant II. The x-coordinate is two units to the left of the origin, and the y-coordinate is three units upward from the origin.

    The point (4, 5) is located in Quadrant I. The x-coordinate is four units to the right of the origin, and the y-coordinate is five units upward from the origin.

    The point (-4, 5) is located in Quadrant II. The x-coordinate is four units to the left of the origin, and the y-coordinate is five units upward from the origin.

    Plotting points on the coordinate plane to determine distance.

    Exercise 5: Graph the points on the Cartesian plane and determine the distance between them.

    (-2, 5), (10, 0)

    Solution

    First, we represent the points on the Cartesian plane using GeoGebra.

    Plotting points on the coordinate plane to determine distance.

    Now, we determine the distance between the two points.

    To find the distance between two points in the Cartesian plane

    The distance between points A and B is 13 units.

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    To find the distance between two points in the Cartesian plane using GeoGebra, follow these steps:

    1. Open GeoGebra on your device.
    2. Click on the “Point” tool to create the two points representing the coordinates.
    3. Once the points are created, click on the “Segment between two points” tool to draw the line segment connecting the two points.
    4. Click on the “Move” tool and select the line segment.
    5. Right-click on the line segment and choose “Length” to display the length of the segment, which represents the distance between the two points.

    By following these steps, you can easily find the distance between two points in the Cartesian plane using GeoGebra.

    The first thing we need to find the distance between two points is access to the GeoGebra executable (Version 6) on the computer or its online version (Executables, Installation, and GeoGebra Online).

    To provide the explanation, we will use the points from the previous example.

    Steps to calculate the distance between two points in the Cartesian plane:

    • We open GeoGebra on our computer, and we will see something like this:
    Using GeoGebra to plot points on the Cartesian plane.
    • We select the Input and then enable the Integrated Keyboard of GeoGebra. This will allow us to write and insert any expression; in our case, we put parentheses.
    Using GeoGebra to plot points on the Cartesian plane.
    • We write each element of the point separated by a comma (-2, 5) and (10, 0), and then press the Enter button.
    Plotting points on the coordinate plane to determine distance in GeoGebra.
    • Finally, we select the Segment tool and start from point (-2, 5) to point (10, 0).
    Calculating the distance between two points in GeoGebra.

    As we can observe, the distance between the two points is 13 units. This is the same result as Exercise 5, where we applied the distance formula.

    For more information, we recommend this YouTube video. It explains step by step how to create the graph of the distance between the two points in GeoGebra.

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    References

    • Stewart, J.; Redlin,l.; Watson,S., 2012. Pre-Calculus. Sixth Edition. Thomson Learning. Mexico.
    • For further illustration, visit: Wikipedia: Cartesian Coordinates.
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