Mathematics is a complex science. Studying it, one has not only to solve examples and problems, but also to work with various shapes, and even planes. One of the most used in mathematics is the plane coordinate system. Correct work children have been taught with her for more than one year. Therefore, it is important to know what it is and how to work with it correctly.
Let's see what constitutes this system, what actions can be performed with its help, as well as find out its main characteristics and features.
Definition of the concept
The coordinate plane is the plane on which a specific coordinate system is defined. Such a plane is defined by two straight lines intersecting at right angles. The origin of coordinates is at the point of intersection of these lines. Every point on coordinate plane given by a pair of numbers called coordinates.
V school course mathematics, schoolchildren have to work quite closely with the coordinate system - build figures and points on it, determine which plane a particular coordinate belongs to, and also determine the coordinates of a point and write or name them. Therefore, let's talk in more detail about all the features of coordinates. But first, let's touch on the history of creation, and then we'll talk about how to work on the coordinate plane.
Historical reference
Ideas for creating a coordinate system were already in the time of Ptolemy. Even 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 yet, and scientists had to use other systems.
Initially, they set points by specifying latitude and longitude. Long time it was one of the most used ways to map information. But in 1637 Rene Descartes created his own coordinate system, later named after the "Cartesian" one.
Already at the end of the 17th 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 theory, here are a few illustrative examples coordinate plane so you can imagine it. The coordinate system is primarily used in chess. On the board, each square has its own coordinates - one letter coordinate, the second digital. With its help, you can determine the position of a particular piece on the board.
The second most a shining example can serve as a game loved by many " Sea battle". Remember how, while playing, you name the coordinate, for example, B3, thus indicating exactly where to aim. At the same time, placing the ships, you set points on the coordinate plane.
This coordinate system is widely used not only in mathematics, logic 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, as 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 as ( Oy). It is these two axes that form the coordinate system, dividing the plane into four quarters. The origin is at the point of 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 of the axes is subscribed.
Quarters
Now let's say a few words about such a concept as a quarter 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 of the quarters has its own characteristics. So, in the first quarter the abscissa and ordinate are positive, in the second quarter the abscissa is negative, the ordinate is positive, in the third both the abscissa and the ordinate are negative, in the fourth the abscissa is positive, and the ordinate is negative.
Remembering these features, you can easily determine to which quarter this or that point belongs. In addition, this information can be useful to you in the event that you have to do calculations using the Cartesian system.
Work with a coordinate plane
When we figured out the concept of a plane and talked about its quarters, we can move on to such a problem as working with this system, and also talk about how to apply points, coordinates of figures to it. 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 designations are applied to it. Then we work directly with points or shapes. In this case, even when constructing figures, points are first drawn on the plane, and then the figures are drawn.
Plane construction rules
If you decide to start marking shapes and points on paper, you need a coordinate plane. The coordinates of the points are applied to it. In order to build a coordinate plane, you only need a ruler and a pen or pencil. First, the horizontal abscissa is drawn, then the vertical - ordinate. It is important to remember that the axes intersect at right angles.
The next mandatory item is marking. On each of the axes in both directions, the units-line segments 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 basics you need to know to successfully place on a plane. various figures, and even mark equations.
When plotting points, remember how their coordinates are recorded correctly. So, usually by specifying a period, two numbers are written in brackets. The first number denotes the coordinate of the point along the abscissa axis, the second - along the ordinate axis.
The point should be built in this way. First mark on the axis Ox target point, then mark the point on the axis Oy... Next, draw imaginary lines from these designations and find the place of their intersection - this will be the given point.
You just have to mark and sign it. As you can see, everything is quite simple and does not require any special skills.
Place the shape
Now let's move on to such a question as the construction of figures on a coordinate plane. In order to build any shape on the coordinate plane, you need to know how to place points on it. If you know how to do this, then it is not so difficult to place a shape on a plane.
First of all, you need the coordinates of the points of the shape. It is on them that we will apply the coordinates chosen by you to our system of coordinates. Consider drawing a rectangle, triangle and circle.
Let's start with a rectangle. It is quite easy to apply. First, four points are drawn on the plane, denoting 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 it has three corners, which means that three points are applied to the plane, denoting its vertices.
Regarding the circle, here you should know the coordinates of the two points. The first point is the center of the circle, the second is the point that indicates its radius. These two points are plotted on the plane. Then a compass is taken, the distance between two points is measured. The point of the compass is placed at the center point and a circle is described.
As you can see, there is nothing complicated here either, the main thing is that you always have a ruler and compasses at hand.
Now you know how to plot the coordinates of the shapes. On the coordinate plane, this is not so difficult to do as it might seem at first glance.
conclusions
So, we have considered with you one of the most interesting and basic concepts for mathematics that every student has to deal with.
We have found out that the 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 a coordinate plane is the ability to correctly apply to it set points... To do this, you should know correct location axes, especially the quarters, as well as the rules by which the coordinates of the points are set.
We hope that the information we have provided was accessible and understandable, and was also useful for you and helped you better understand this topic.
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V Everyday life you can often hear the phrase: "Leave me your coordinates." In response, a person usually leaves his address or phone number, that is, the data by which he can be found.
Coordinates can be designated by a wide variety of sets of numbers or letters.
For example, the number of the car is the coordinates, because by the number of the car you can determine which city it is from and who is its owner.
Coordinates is a set of data used to determine the position of an object.
Examples of coordinates are: carriage number and seat on the train, latitude and longitude on geographic map, recording the position of the piece on the chessboard, the position of a point on the numerical axis, etc.
Whenever, according to certain rules, we unambiguously designate an object with a set of letters, numbers or other symbols, we set the coordinates of the object.
Cartesian coordinate system
The French mathematician Rene Descartes (1596 - 1650) proposed specifying the position of a point on a plane using two coordinates.
To find the coordinates, you need landmarks from which the countdown is conducted.
- On the plane, two numerical axes will serve as such reference points. In the drawing, the first axis is usually drawn horizontally, it is called the ABSCISS axis and is denoted by the letter X, the Ox axis is written. The positive direction on the abscissa axis is chosen from left to right and is shown with an arrow.
- The second axis is drawn vertically, it is called the ORDINATE axis and is denoted by the letter Y, the Oy axis is written. The positive direction on the ordinate axis is chosen from bottom to top and shown with an arrow.
The axes are mutually perpendicular (that is, the angle between them is 90 °) and intersect at a point denoted by O. Point O is the origin for each of the axes.
Coordinate system- these are two mutually perpendicular coordinate lines intersecting at a point that is the origin for each of them.
Coordinate axes are straight lines that form a coordinate system.
Abscissa axis(Ox) - horizontal axis.
Y-axis(Oy) - vertical axis.
Coordinate plane - the plane in which the coordinate system is built. The plane is designated as x0y.
We draw your attention to the choice of the length of the unit segments along the axes.
Numbers denoting numerical values on the axes can be positioned either to the right or to the left of the Oy axis. The numbers on the Ox axis are usually written at the bottom below the axis.
Typically, the unit line segment on the 0y axis is equal to the unit line segment on the 0x axis. But there are times when they are not equal to each other.
The coordinate axes divide the plane into 4 angles, which are called coordinate quarters... The quarter formed by the positive semiaxes (upper right corner) is considered the first (I).
We count the quarters (or coordinate angles) counterclockwise.
If you place a unit number circle on a coordinate plane, then coordinates can be found for its points. The numerical circle is positioned so that its center coincides with the point of origin of the plane, that is, the point O (0; 0).
Usually on a single number circle mark the points corresponding from the origin on the circle
- quarters - 0 or 2π, π / 2, π, (2π) / 3,
- mid-quarters - π / 4, (3π) / 4, (5π) / 4, (7π) / 4,
- thirds of quarters - π / 6, π / 3, (2π) / 3, (5π) / 6, (7π) / 6, (4π) / 3, (5π) / 3, (11π) / 6.
On the coordinate plane with the above location of the unit circle on it, you can find the coordinates corresponding to these points of the circle.
The coordinates of the ends of the quarters are very easy to find. At point 0 of the circle, the x coordinate is 1, and y is 0. It can be denoted as A (0) = A (1; 0).
The end of the first quarter will be located on the positive y-axis. Therefore, B (π / 2) = B (0; 1).
The end of the second quarter is on the negative semiaxis: C (π) = C (-1; 0).
End of the third quarter: D ((2π) / 3) = D (0; -1).
But how do you find the coordinates of the midpoints of the quarters? For this they build right triangle... Its hypotenuse is a segment from the center of the circle (or origin) to the midpoint of the quarter circle. This is the radius of the circle. Since the circle is unit, the hypotenuse is 1. Next, a perpendicular is drawn from the point of the circle to any axis. Let it be towards the x-axis. It turns out a right-angled triangle, the lengths of the legs of which are the x and y coordinates of the point of the circle.
The quarter circle is 90º. And half a quarter is 45 degrees. Since the hypotenuse is drawn to the point of the middle of the quarter, the angle between the hypotenuse and the leg extending from the origin is 45º. But the sum of the angles of any triangle is 180º. Therefore, the angle between the hypotenuse and the other leg is also 45º. It turns out an isosceles right triangle.
From the Pythagorean theorem we obtain the equation x 2 + y 2 = 1 2. Since x = y and 1 2 = 1, the equation is simplified to x 2 + x 2 = 1. Solving it, we get x = √½ = 1 / √2 = √2 / 2.
Thus, the coordinates of the point are M 1 (π / 4) = M 1 (√2 / 2; √2 / 2).
In the coordinates of the points of the midpoints of other quarters, only the signs will change, and the moduli of the values will remain the same, since the right-angled triangle will only be inverted. We get:
M 2 ((3π) / 4) = M 2 (-√2 / 2; √2 / 2)
M 3 ((5π) / 4) = M 3 (-√2 / 2; -√2 / 2)
M 4 ((7π) / 4) = M 4 (√2 / 2; -√2 / 2)
When determining the coordinates of the third parts of the quarters of the circle, a right-angled triangle is also built. If we take the point π / 6 and draw a perpendicular to the x-axis, then the angle between the hypotenuse and the leg lying on the x-axis will be 30º. It is known that a leg lying opposite an angle of 30 degrees is equal to half of the hypotenuse. So, we found the y-coordinate, it is equal to ½.
Knowing the lengths of the hypotenuse and one of the legs, according to the Pythagorean theorem, we find another leg:
x 2 + (½) 2 = 1 2
x 2 = 1 - ¼ = ¾
x = √3 / 2
Thus, T 1 (π / 6) = T 1 (√3 / 2; ½).
For the point of the second third of the first quarter (π / 3), it is better to draw the perpendicular to the axis to the y axis. Then the angle at the origin of coordinates will also be 30º. Here, the x coordinate will be equal to ½, and y, respectively, √3 / 2: T 2 (π / 3) = T 2 (½; √3 / 2).
For other points in the third quarters, the signs and order of the coordinate values will change. All points that are closer to the x-axis will have an x-coordinate modulo √3 / 2. Those points that are closer to the y-axis will have a y-value of √3 / 2 in absolute value.
T 3 ((2π) / 3) = T 3 (-½; √3 / 2)
T 4 ((5π) / 6) = T 4 (-√3 / 2; ½)
T 5 ((7π) / 6) = T 5 (-√3 / 2; -½)
T 6 ((4π) / 3) = T 6 (-½; -√3 / 2)
T 7 ((5π) / 3) = T 7 (½; -√3 / 2)
T 8 ((11π) / 6) = T 8 (√3 / 2; -½)