Mathematics is a rather complicated science. Studying it, you have to not only solve examples and tasks, but also work with various figures, and even planes. One of the most used in mathematics is the coordinate system on the plane. Correct work with her children have been taught for more than one year. Therefore, it is important to know what it is and how to work with it correctly.
Let's figure out what this system is, what actions can be performed with its help, and also learn its main characteristics and features.
Definition of a concept
A coordinate plane is a plane on which a specific coordinate system is defined. Such a plane is defined by two straight lines intersecting at right angles. At the intersection of these lines is the origin. Each point on the coordinate plane is defined by a pair of numbers called coordinates.
In the school course of mathematics, students have to work quite closely with the coordinate system - build shapes and points on it, determine which plane this or that coordinate belongs to, and also determine the coordinates of the point and write or name them. Therefore, we will talk in more detail about all the features of the coordinates. But before we touch on the history of creation, and then we’ll talk about how to work on the coordinate plane.
History reference
The idea of creating a coordinate system was in the time of Ptolemy. Already then astronomers and mathematicians were thinking about how to learn how to set the position of a point on a plane. Unfortunately, at that time there was no coordinate system known to us, and scientists had to use other systems.
Initially, they set the points by indicating latitude and longitude. For a long time, this was one of the most used ways of plotting this or that information on a map. But in 1637, Rene Descartes created his own coordinate system, later named after the great mathematician "Cartesian."
After the publication of the work "Geometry", the coordinate system of Rene Descartes won recognition in the scientific community.
Already at the end of the XVII century. the concept of "coordinate plane" has become widely used in the world of mathematics. Despite the fact that several centuries have passed since the creation of this system, it is still widely used in mathematics and even in life.
Coordinate Plane Examples
Before talking about the theory, we give some illustrative examples of the coordinate plane so that you can imagine it. First of all, the coordinate system is used in chess. On the board, each square has its own coordinates - one coordinate is alphabetic, the second is digital. With its help, you can determine the position of a particular piece on the board.
The second most striking example is the beloved game “Sea Battle”. Remember how, when playing, you name the coordinate, for example, B3, thus indicating where exactly you aim. At the same time, when arranging the ships, you set the points on the coordinate plane.
This coordinate system is widely used not only in mathematics, logical games, but also in military affairs, astronomy, physics and many other sciences.
Coordinate axes
As already mentioned, two axes are distinguished in the coordinate system. Let's talk a little about them, since they are of considerable importance.
The first axis - abscissa - is horizontal. It is denoted as ( Ox ). The second axis is the ordinate, which runs vertically through the reference point and is denoted by ( Oy ). It is these two axes that form the coordinate system, dividing the plane into four quarters. The origin is at the intersection of these two axes and takes the value 0 . Only if the plane is formed by two axes intersecting perpendicularly having a reference point, is it a coordinate plane.
Also note that each of the axes has its own direction. Usually, when constructing a coordinate system, it is customary to indicate the direction of the axis in the form of an arrow. In addition, when constructing a coordinate plane, each axis is signed.
Quarter
Now let's say a few words about such a concept as quarters of the coordinate plane. The plane is divided by two axes into four quarters. Each of them has its own number, while the numbering of the planes is counterclockwise.
Each quarter has its own characteristics. So, in the first quarter of the abscissa and the ordinate is positive, in the second quarter of the abscissa is negative, the ordinate is positive, in the third and the abscissa and ordinate are negative, in the fourth is the abscissa and negative is the ordinate.
By remembering these features, you can easily determine which quarter a particular point belongs to. In addition, this information may be useful to you even if you have to do calculations using the Cartesian system.
Work with the coordinate plane
When we figured out the concept of a plane and talked about its quarters, we can move on to a problem such as working with this system, as well as talk about how to put points on it, the coordinates of the figures. On the coordinate plane, this is not as difficult as it might seem at first glance.
First of all, the system itself is built; all important notations are applied to it. Then, work begins directly with points or figures. Moreover, even when constructing figures, points are first drawn on the plane, and then figures are already drawn.
Next, we’ll talk in more detail about building a system and directly drawing dots and shapes.
Rules for building a plane
If you decide to start marking shapes and points on paper, you will need a coordinate plane. The coordinates of the points are plotted on it. In order to build a coordinate plane, you only need a ruler and a pen or pencil. First, the horizontal abscissa axis is drawn, then the vertical axis is the ordinate. It is important to remember that the axes intersect at right angles.
Further on each axis indicate the direction and sign them using the generally accepted notation x and y . The intersection point of the axes is also marked and signed with the number 0 .
The next mandatory point is marking. On each axis in both directions, unit units are marked and signed. This is done so that you can then work with the plane with maximum convenience.
Mark the point
Now let's talk about how to plot the coordinates of points on the coordinate plane. This is the foundation that you should know in order to successfully place various figures on the plane, and even mark equations.
When building points, you should remember how their coordinates are recorded correctly. So, usually asking a point, two numbers are written in brackets. The first digit indicates the coordinate of the point along the abscissa, the second - along the ordinate.
To build a point follows this way. First mark a given point on the Ox axis, then mark a point on the Oy axis. Next, draw imaginary lines from these signs and find the place of their intersection - this will be the given point.
You just have to mark it and sign it. As you can see, everything is quite simple and does not require special skills.
Place the figure
Now we turn to such a question as the construction of figures on the coordinate plane. In order to build any shape on the coordinate plane, you should know how to place points on it. If you know how to do this, then placing the figure on a plane is not so difficult.
First of all, you will need the coordinates of the points of the figure. It is for them that we will apply the geometrical figures you have chosen to our coordinate system . Consider drawing a rectangle, triangle, and circle.
Let's start with the rectangle. Applying it is quite simple. First, four points are drawn on the plane, indicating the corners of the rectangle. Then all the points are connected in series with each other.
Drawing a triangle is no different. The only thing is that he has three angles, which means that three points are applied to the plane, indicating his vertices.
Regarding the circle, you should know the coordinates of two points. The first point is the center of the circle, the second is the point indicating its radius. These two points are plotted on a plane. Then a compass is taken, the distance between two points is measured. The point of the compass is placed at a point denoting the center, and a circle is described.
As you can see, there is also nothing complicated here, the main thing is to always have a ruler and a compass on hand.
Now you know how to apply the coordinates of the figures. On the coordinate plane, this is not so difficult as it might seem at first glance.
conclusions
So, we have examined with you one of the most interesting and basic concepts for mathematics, which every student has to deal with.
We have found out that a coordinate plane is a plane formed by the intersection of two axes. With its help, you can set the coordinates of points, apply shapes to it. The plane is divided into quarters, each of which has its own characteristics.
The main skill that should be developed when working with the coordinate plane is the ability to correctly apply specified points to it. To do this, you should know the correct location of the axes, the features of the quarters, as well as the rules by which the coordinates of the points are set.
We hope that the information presented by us was accessible and understandable, and was also useful to you and helped to better understand this topic.