How the quarters will be located on the coordinate axis. Coordinate plane: what is it? How to mark points and construct figures on the coordinate plane
If you place the unit number circle on coordinate plane, then coordinates can be found for its points. The number circle is positioned so that its center coincides with the origin of the plane, i.e., point O (0; 0).
Usually on the unit number circle the points corresponding to the origin of the circle are marked
- quarters - 0 or 2π, π/2, π, (2π)/3,
- middle 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 the y coordinate is 0. We can denote it 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 semi-axis: C (π) = C (-1; 0).
End of third quarter: D ((2π)/3) = D (0; -1).
But how to 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 there is a unit circle, the hypotenuse is equal to 1. Next, draw a perpendicular from a point on the circle to any axis. Let it be towards the x axis. The result is a right triangle, the lengths of the legs of which are the x and y coordinates of the point on the circle.
A quarter circle is 90º. And half a quarter is 45º. Since the hypotenuse is drawn to the midpoint of the quadrant, 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. Consequently, the angle between the hypotenuse and the other leg also remains 45º. This results in 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 simplifies to x 2 + x 2 = 1. Solving it, we get x = √½ = 1/√2 = √2/2.
Thus, the coordinates of the point M 1 (π/4) = M 1 (√2/2; √2/2).
In the coordinates of the points of the midpoints of the other quarters, only the signs will change, and the modules of the values will remain the same, since the right triangle will only be turned over. 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 a circle, a right triangle is also constructed. 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 30º is equal to half the hypotenuse. This means that we have found the y coordinate, it is equal to ½.
Knowing the lengths of the hypotenuse and one of the legs, using the Pythagorean theorem we find the other 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 a perpendicular to the axis to the y axis. Then the angle at the origin 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 of the third quarters, the signs and order of the coordinate values will change. All points that are closer to the x axis will have a modulus x coordinate value equal to √3/2. Those points that are closer to the y axis will have a modulus y value equal to √3/2.
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)
IN 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, data by which he can be found.
Coordinates can be indicated by a variety of sets of numbers or letters.
For example, a car number is coordinates, because by the car number you can determine what city it is from and who its owner is.
Coordinates is a set of data from which the position of an object is determined.
Examples of coordinates are: car and seat number on the train, latitude and longitude on geographical map, recording the position of a piece on a chessboard, the position of a point on a number line, etc.
Whenever, according to certain rules, we unambiguously designate an object with a set of letters, numbers or other symbols, we specify 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 coordinates, you need landmarks from which to count.
- On a 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 designated by the letter X, and the Ox axis is written. The positive direction on the x-axis is chosen from left to right and shown with an arrow.
- The second axis is drawn vertically, it is called the ORDINATE axis and is designated by the letter Y, the Oy axis is written. The positive direction on the ordinate axis is chosen from bottom to top and is shown with an arrow.
The axes are mutually perpendicular (i.e. the angle between them is 90°) and intersect at a point designated O. Point O is the origin for each of the axes.
Coordinate system- these are two mutually perpendicular coordinate lines intersecting at a point, which is the origin of reference 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 is the plane in which the coordinate system is constructed. The plane is designated as x0y.
We draw your attention to the choice of the length of single segments along the axes.
Numbers indicating numeric values on the axes can be located both to the right and to the left of the Oy axis. The numbers on the Ox axis are usually written below the axis.
Typically, a unit segment on the 0y axis is equal to a unit 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 semi-axes (upper right corner) is considered the first (I).
We count the quarters (or coordinate angles) counterclockwise.
A rectangular coordinate system on a plane is defined by two mutually perpendicular straight lines. Straight lines are called coordinate axes (or coordinate axes). The point of intersection of these lines is called the origin and is designated by the letter O.
Usually one of the lines is horizontal, the other is vertical. The horizontal line is designated as the x-axis (or Ox) and is called the abscissa axis, the vertical line is the y-axis (Oy), called the ordinate axis. The entire coordinate system is designated xOy.
Point O divides each of the axes into two semi-axes, one of which is considered positive (denoted by an arrow), the other - negative.
Each point F of the plane is assigned a pair of numbers (x;y) - its coordinates.
The x coordinate is called the abscissa. It is equal to Ox, taken with the appropriate sign.
The y coordinate is called the ordinate and is equal to the distance from point F to the Oy axis (with the appropriate sign).
Axle distances are usually (but not always) measured in the same unit of length.
Points located to the right of the y-axis have positive abscissas. Points that lie to the left of the ordinate axis have negative abscissas. For any point lying on the Oy axis, its x coordinate is zero.
Points with a positive ordinate lie above the x-axis, and points with a negative ordinate lie below. If a point lies on the Ox axis, its y coordinate is zero.
Coordinate axes divide the plane into four parts, which are called coordinate quarters (or coordinate angles or quadrants).
1 coordinate quarter located in the upper right corner of the xOy coordinate plane. Both coordinates of points located in the first quarter are positive.
The transition from one quarter to another is carried out counterclockwise.
2 coordinate quarter is located in the upper left corner. Points lying in the second quarter have a negative abscissa and a positive ordinate.
3 coordinate quarter lies in the lower left quadrant of the xOy plane. Both coordinates of the points belonging to the III coordinate angle are negative.
4 coordinate quarter is the lower right corner of the coordinate plane. Any point from the IV quarter has a positive first coordinate and a negative second.
An example of the location of points in a rectangular coordinate system:
Mathematics is a rather complex science. While studying it, you have to not only solve examples and problems, but also work with various shapes and even planes. One of the most used in mathematics is the coordinate system on a plane. Proper 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 figure out what it is this system, what actions can be performed with its help, and also learn its main characteristics and features.
Definition of the concept
The coordinate plane is the plane on which the specific system coordinates Such a plane is defined by two straight lines intersecting at right angles. At the point of intersection of these lines is the origin of coordinates. Each point on the coordinate plane is specified by a pair of numbers called coordinates.
In a school mathematics course, schoolchildren have to work quite closely with a coordinate system - construct figures and points on it, determine which plane a particular coordinate belongs to, as well as 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 background
Ideas about creating a coordinate system existed back in the time of Ptolemy. Even then, astronomers and mathematicians were thinking about how to learn 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 specified points using latitude and longitude. For a long time this was one of the most used methods of putting this or that information on a map. But in 1637, Rene Descartes created own system coordinates, later named after the “Cartesian”.
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.
Examples of a coordinate plane
Before talking about theory, let's give a few illustrative examples coordinate plane so you can visualize it. The coordinate system is primarily 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 game beloved by many “ Sea battle" Remember how, when playing, you name a coordinate, for example, B3, thus indicating exactly where you are aiming. At the same time, when placing ships, you specify points on the coordinate plane.
This coordinate system is widely used not only in mathematics and logic games, but also in military affairs, astronomy, physics and many other sciences.
Coordinate axes
As already mentioned, there are two axes in the coordinate system. Let's talk a little about them, as they are of considerable importance.
The first axis is abscissa - 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 located at the intersection point of these two axes and takes the value 0 . Only if the plane is formed by two axes intersecting perpendicularly and 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 signed.
Quarters
Now let's say a few words about such a concept as quarters of the coordinate plane. The plane is divided into four quarters by two axes. Each of them has its own number, and the planes are numbered 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 ordinate are negative, in the fourth the abscissa is positive and the ordinate is negative.
By remembering these features, you can easily determine which quarter a particular point belongs to. In addition, this information may be useful to you if you have to do calculations using the Cartesian system.
Working with the coordinate plane
When we have understood 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 put points and coordinates of figures on 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. Moreover, even when constructing figures, points are first drawn on the plane, and then the figures are drawn.
Rules for constructing 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 construct a coordinate plane, you only need a ruler and a pen or pencil. First, the horizontal x-axis is drawn, then the vertical axis is drawn. It is important to remember that the axes intersect at right angles.
Next mandatory item is marking. On each of the axes in both directions, unit segments are marked and labeled. This is done so that you can then work with the plane with maximum convenience.
Mark a 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 a variety of shapes on a plane, and even mark equations.
When constructing points, you should remember how their coordinates are correctly written. So, usually when specifying a point, two numbers are written in brackets. The first digit indicates the coordinate of the point along the abscissa axis, the second - along the ordinate axis.
The point should be constructed in this way. First mark on the axis Ox specified point, then mark the point on the axis Oy. Next, draw imaginary lines from these designations and find the place where they intersect - this will be the given point.
All you have to do is mark it and sign it. As you can see, everything is quite simple and does not require any special skills.
Place the figure
Now let's move on to the issue of constructing figures on a coordinate plane. In order to construct any figure on the coordinate plane, you should know how to place points on it. If you know how to do this, then placing a figure on a plane is not so difficult.
First of all, you will need the coordinates of the points of the figure. It is according to them that we will apply the ones you have chosen to our coordinate system. Let us consider the application of a rectangle, a triangle and a circle.
Let's start with a rectangle. It's quite easy to apply. First, four points are marked on the plane, indicating the corners of the rectangle. Then all the points are sequentially connected to each other.
Drawing a triangle is no different. The only thing is that it has three angles, which means that three points are marked on the plane, indicating its 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 the plane. Then take a compass and measure the distance between two points. The point of the compass is placed at the point marking the center, 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 compass at hand.
Now you know how to plot the coordinates of figures. Doing this on the coordinate plane is not as difficult as it might seem at first glance.
Conclusions
So, we have looked at one of the most interesting and basic concepts for mathematics that every schoolchild 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 and draw shapes on 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 plot given points on it. To do this you need to know correct location axes, features of the quarters, as well as the rules by which the coordinates of points are specified.
We hope that the information we presented was accessible and understandable, and was also useful to you and helped you better understand this topic.
