Video lesson “Coordinate plane. "Coordinate plane" - video lessons in mathematics (grade 6) How to mark coordinates on the coordinate plane

A rectangular coordinate system on a plane is formed by two mutually perpendicular coordinate axes X’X and Y’Y. The coordinate axes intersect at point O, which is called the origin, a positive direction is selected on each axis. The positive direction of the axes (in a right-handed coordinate system) is chosen so that when the X'X axis is rotated counterclockwise by 90°, its positive direction coincides with the positive direction of the Y'Y axis. The four angles (I, II, III, IV) formed by the coordinate axes X'X and Y'Y are called coordinate angles (see Fig. 1).

The position of point A on the plane is determined by two coordinates x and y. The x coordinate is equal to the length of the segment OB, the y coordinate is equal to the length of the segment OC in the selected units of measurement. Segments OB and OC are defined by lines drawn from point A parallel to the Y'Y and X'X axes, respectively. The x coordinate is called the abscissa of point A, the y coordinate is called the ordinate of point A. It is written as follows: A(x, y).

If point A lies in coordinate angle I, then point A has a positive abscissa and ordinate. If point A lies in coordinate angle II, then point A has a negative abscissa and a positive ordinate. If point A lies in coordinate angle III, then point A has a negative abscissa and ordinate. If point A lies in coordinate angle IV, then point A has a positive abscissa and a negative ordinate.

Rectangular coordinate system in space is formed by three mutually perpendicular coordinate axes OX, OY and OZ. The coordinate axes intersect at point O, which is called the origin, on each axis a positive direction is selected, indicated by arrows, and a unit of measurement for the segments on the axes. The units of measurement are the same for all axes. OX - abscissa axis, OY - ordinate axis, OZ - applicate axis. The positive direction of the axes is chosen so that when the OX axis is rotated counterclockwise by 90°, its positive direction coincides with the positive direction of the OY axis, if this rotation is observed from the positive direction of the OZ axis. Such a coordinate system is called right-handed. If the thumb of the right hand is taken as the X direction, the index finger as the Y direction, and the middle finger as the Z direction, then a right-handed coordinate system is formed. Similar fingers of the left hand form the left coordinate system. It is impossible to combine the right and left coordinate systems so that the corresponding axes coincide (see Fig. 2).

The position of point A in space is determined by three coordinates x, y and z. The x coordinate is equal to the length of the segment OB, the y coordinate is the length of the segment OC, the z coordinate is the length of the segment OD in the selected units of measurement. The segments OB, OC and OD are defined by planes drawn from point A parallel to the planes YOZ, XOZ and XOY, respectively. The x coordinate is called the abscissa of point A, the y coordinate is called the ordinate of point A, the z coordinate is called the applicate of point A. It is written as follows: A(a, b, c).

Orty

A rectangular coordinate system (of any dimension) is also described by a set of unit vectors aligned with the coordinate axes. The number of unit vectors is equal to the dimension of the coordinate system and they are all perpendicular to each other.

In the three-dimensional case, such unit vectors are usually denoted i j k or e x e y e z. In this case, in the case of a right-handed coordinate system, the following formulas with the vector product of vectors are valid:

[i j]=k ;
[j k]=i ;
[k i]=j .

Story

The rectangular coordinate system was first introduced by Rene Descartes in his work “Discourse on Method” in 1637. Therefore, the rectangular coordinate system is also called - Cartesian coordinate system. The coordinate method of describing geometric objects marked the beginning of analytical geometry. Pierre Fermat also contributed to the development of the coordinate method, but his works were first published after his death. Descartes and Fermat used the coordinate method only on the plane.

The coordinate method for three-dimensional space was first used by Leonhard Euler already in the 18th century.

Links

Wikimedia Foundation. 2010.

See what “Coordinate plane” is in other dictionaries:

cutting plane- (Pn) Coordinate plane tangent to the cutting edge at the point under consideration and perpendicular to the main plane. [...

In topography, a network of imaginary lines encircling the globe in the latitudinal and meridional directions, with the help of which you can accurately determine the position of any point on the earth's surface. Latitudes are measured from the equator - the great circle... ... Geographical encyclopedia

This term has other meanings, see Phase diagram. Phase plane is a coordinate plane in which any two variables (phase coordinates) are plotted along the coordinate axes, which uniquely determine the state of the system... ... Wikipedia

main cutting plane- (Pτ) Coordinate plane perpendicular to the intersection of the main plane and the cutting plane. [GOST 25762 83] Topics: cutting processing General terms: coordinate plane systems and coordinate planes... Technical Translator's Guide

instrumental main cutting plane- (Pτi) Coordinate plane perpendicular to the intersection of the instrumental main plane and the cutting plane. [GOST 25762 83] Topics: cutting processing General terms: coordinate plane systems and coordinate planes... Technical Translator's Guide

tool cutting plane- (Pni) Coordinate plane tangent to the cutting edge at the point under consideration and perpendicular to the instrumental main plane. [GOST 25762 83] Subjects of cutting processing General terms of coordinate plane system and... ... Technical Translator's Guide

kinematic principal cutting plane- (Pτк) Coordinate plane perpendicular to the line of intersection of the kinematic main plane and the cutting plane ... Technical Translator's Guide

kinematic cutting plane- (Pnк) Coordinate plane tangent to the cutting edge at the point under consideration and perpendicular to the kinematic main plane ... Technical Translator's Guide

main plane- (Pv) The coordinate plane drawn through the point of interest on the cutting edge perpendicular to the direction of the speed of the main or resulting cutting movement at this point. Note In the instrumental coordinate system, the direction... ... Technical Translator's Guide

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. Children have been taught how to work with it correctly 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 find out its main characteristics and features.

Definition of the concept

A coordinate plane is a plane on which a specific coordinate system is specified. 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 plotting this or that information on a map. 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.

Examples of a coordinate plane

Before we talk about the theory, we will give some visual examples of the coordinate plane so that you can imagine 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 beloved game “Battleship”. 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 to do 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.

The next mandatory item is applying markings. 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 tip 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 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 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.

Topic of this video lesson: Coordinate plane.

Goals and objectives of the lesson:

Get acquainted with rectangular coordinate system on a plane
- teach how to freely navigate the coordinate plane
- construct points according to their given coordinates
- determine the coordinates of a point marked on the coordinate plane
- understand coordinates well by ear
- perform geometric constructions clearly and accurately
- development of creative abilities
- fostering interest in the subject

The term " coordinates" comes from the Latin word - "orderly"

To indicate the position of a point on the plane, take two perpendicular lines X and Y.

X axis - abscissa axis
Y-axis ordinate axis
Point O - origin

The plane on which the coordinate system is specified is called coordinate plane.

Each point M on the coordinate plane corresponds to a pair of numbers: its abscissa and ordinate. On the contrary, each pair of numbers corresponds to one point on the plane, for which these numbers are coordinates.

Examples considered:

by constructing a point from its coordinates
finding the coordinates of a point located on the coordinate plane

Some additional information:

The idea of specifying the position of a point on a plane originated in ancient times, primarily among astronomers. In the II century. The ancient Greek astronomer Claudius Ptolemy used latitude and longitude as coordinates. He gave a description of the use of coordinates in the book “Geometry” in 1637.

A description of the use of coordinates was given in the book “Geometry” in 1637 by the French mathematician Rene Descartes, therefore the rectangular coordinate system is often called Cartesian.

Words " abscissa», « ordinate», « coordinates"was the first to use at the end of the 17th century.

For a better understanding of the coordinate plane, imagine that we are given: a geographical globe, a chessboard, a theater ticket.

To determine the position of a point on the earth's surface, you need to know the longitude and latitude.
To determine the position of a piece on a chessboard, you need to know two coordinates, for example: e3.
Seats in the auditorium are determined by two coordinates: row and seat.

Additional task.

After studying the video lesson, to consolidate the material, I suggest you take a pen and a piece of paper in a box, draw a coordinate plane and build figures according to the given coordinates:

Fungus
1) (6; 0), (6; 2), (5; 1,5), (4; 3), (2; 1), (0; 2,5), (- 1,5; 1,5), (- 2; 5), (- 3; 0,5), (- 4; 2), (- 4; 0).
2) (2; 1), (2,2; 2), (2,3; 4), (2,5; 6), (2,3; 8), (2; 10), (6; 10), (4,8; 12), (3; 13,3), (1; 14),
(0; 14), (- 2; 13,3), (- 3,8; 12), (- 5; 10), (2; 10).
3) (- 1; 10), (- 1,3; 8), (- 1,5; 6), (- 1,2; 4), (- 0,8;2).
Mouse 1) (3; - 4), (3; - 1), (2; 3), (2; 5), (3; 6), (3; 8), (2; 9), (1; 9), (- 1; 7), (- 1; 6),
(- 4; 4), (- 2; 3), (- 1; 3), (- 1; 1), (- 2; 1), (-2; - 1), (- 1; 0), (- 1; - 4), (- 2; - 4),
(- 2; - 6), (- 3; - 6), (- 3; - 7), (- 1; - 7), (- 1; - 5), (1; - 5), (1; - 6), (3; - 6), (3; - 7),
(4; - 7), (4; - 5), (2; - 5), (3; - 4).
2) Tail: (3; - 3), (5; - 3), (5; 3).
3) Eye: (- 1; 5).
Swan
1) (2; 7), (0; 5), (- 2; 7), (0; 8), (2; 7), (- 4; - 3), (4; 0), (11; - 2), (9; - 2), (11; - 3),
(9; - 3), (5; - 7), (- 4; - 3).
2) Beak: (- 4; 8), (- 2; 7), (- 4; 6).
3) Wing: (1; - 3), (4; - 2), (7; - 3), (4; - 5), (1; - 3).
4) Eye: (0; 7).
Camel
1) (- 9; 6), (- 5; 9), (- 5; 10), (- 4; 10), (- 4; 4), (- 3; 4), (0; 7), (2; 4), (4; 7), (7; 4),
(9; 3), (9; 1), (8; - 1), (8; 1), (7; 1), (7; - 7), (6; - 7), (6; - 2), (4; - 1), (- 5; - 1), (- 5; - 7),
(- 6; - 7), (- 6; 5), (- 7;5), (- 8; 4), (- 9; 4), (- 9; 6).
2) Eye: (- 6; 7).
Elephant
1) (2; - 3), (2; - 2), (4; - 2), (4; - 1), (3; 1), (2; 1), (1; 2), (0; 0), (- 3; 2), (- 4; 5),
(0; 8), (2; 7), (6; 7), (8; 8), (10; 6), (10; 2), (7; 0), (6; 2), (6; - 2), (5; - 3), (2; - 3).
2) (4; - 3), (4; - 5), (3; - 9), (0; - 8), (1; - 5), (1; - 4), (0; - 4), (0; - 9), (- 3; - 9),
(- 3; - 3), (- 7; - 3), (- 7; - 7), (- 8; - 7), (- 8; - 8), (- 11; - 8), (- 10; - 4), (- 11; - 1),
(- 14; - 3), (- 12; - 1), (- 11;2), (- 8;4), (- 4;5).
3) Eyes: (2; 4), (6; 4).
Horse
1) (14; - 3), (6,5; 0), (4; 7), (2; 9), (3; 11), (3; 13), (0; 10), (- 2; 10), (- 8; 5,5),
(- 8; 3), (- 7; 2), (- 5; 3), (- 5; 4,5), (0; 4), (- 2; 0), (- 2; - 3), (- 5; - 1), (- 7; - 2),
(- 5; - 10), (- 2; - 11), (- 2; - 8,5), (- 4; - 8), (- 4; - 4), (0; - 7,5), (3; - 5).
2) Eye: (- 2; 7).

§ 1 Coordinate system: definition and method of construction

In this lesson we will get acquainted with the concepts of “coordinate system”, “coordinate plane”, “coordinate axes”, and learn how to construct points on a plane using coordinates.

Let's take a coordinate line x with the origin point O, a positive direction and a unit segment.

Through the origin of coordinates, point O of the coordinate line x, we draw another coordinate line y, perpendicular to x, set the positive direction upward, the unit segment is the same. Thus, we have built a coordinate system.

Let's give a definition:

Two mutually perpendicular coordinate lines intersecting at a point, which is the origin of coordinates of each of them, form a coordinate system.

§ 2 Coordinate axis and coordinate plane

The straight lines that form a coordinate system are called coordinate axes, each of which has its own name: the coordinate line x is the abscissa axis, the coordinate line y is the ordinate axis.

The plane on which the coordinate system is selected is called the coordinate plane.

The described coordinate system is called rectangular. It is often called the Cartesian coordinate system in honor of the French philosopher and mathematician René Descartes.

Each point on the coordinate plane has two coordinates, which can be determined by dropping perpendiculars from the point on the coordinate axis. The coordinates of a point on a plane are a pair of numbers, of which the first number is the abscissa, the second number is the ordinate. The abscissa is perpendicular to the x-axis, the ordinate is perpendicular to the y-axis.

Let's mark point A on the coordinate plane and draw perpendiculars from it to the axes of the coordinate system.

Along the perpendicular to the abscissa axis (x-axis), we determine the abscissa of point A, it is equal to 4, the ordinate of point A - along the perpendicular to the ordinate axis (y-axis) is 3. The coordinates of our point are 4 and 3. A (4;3). Thus, coordinates can be found for any point on the coordinate plane.

§ 3 Construction of a point on a plane

How to construct a point on a plane with given coordinates, i.e. Using the coordinates of a point on the plane, determine its position? In this case, we perform the steps in reverse order. On the coordinate axes we find points corresponding to the given coordinates, through which we draw straight lines perpendicular to the x and y axes. The point of intersection of the perpendiculars will be the desired one, i.e. a point with given coordinates.

Let's complete the task: construct point M (2;-3) on the coordinate plane.

To do this, find a point with coordinate 2 on the x-axis and draw a straight line perpendicular to the x-axis through this point. On the ordinate axis we find a point with coordinate -3, through it we draw a straight line perpendicular to the y axis. The point of intersection of perpendicular lines will be the given point M.

Now let's look at a few special cases.

Let us mark points A (0; 2), B (0; -3), C (0; 4) on the coordinate plane.

The abscissas of these points are equal to 0. The figure shows that all points are on the ordinate axis.

Consequently, points whose abscissas are equal to zero lie on the ordinate axis.

Let's swap the coordinates of these points.

The result will be A (2;0), B (-3;0) C (4; 0). In this case, all ordinates are equal to 0 and the points are on the x-axis.

This means that points whose ordinates are equal to zero lie on the abscissa axis.

Let's look at two more cases.

On the coordinate plane, mark the points M (3; 2), N (3; -1), P (3; -4).

It is easy to notice that all the abscissas of the points are the same. If these points are connected, you get a straight line parallel to the ordinate axis and perpendicular to the abscissa axis.

The conclusion suggests itself: points that have the same abscissa lie on the same straight line, which is parallel to the ordinate axis and perpendicular to the abscissa axis.

If you swap the coordinates of the points M, N, P, you get M (2; 3), N (-1; 3), P (-4; 3). The ordinates of the points will be the same. In this case, if you connect these points, you get a straight line parallel to the abscissa axis and perpendicular to the ordinate axis.

Thus, points having the same ordinate lie on the same straight line parallel to the abscissa axis and perpendicular to the ordinate axis.

In this lesson you became acquainted with the concepts of “coordinate system”, “coordinate plane”, “coordinate axes - abscissa axis and ordinate axis”. We learned how to find the coordinates of a point on a coordinate plane and learned how to construct points on the plane using its coordinates.

List of used literature:

Mathematics. Grade 6: lesson plans for I.I.’s textbook. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilina. – Mnemosyne, 2009.
Mathematics. 6th grade: textbook for students of general education institutions. I.I. Zubareva, A.G. Mordkovich. - M.: Mnemosyna, 2013.
Mathematics. 6th grade: textbook for general education institutions/G.V. Dorofeev, I.F. Sharygin, S.B. Suvorov and others/edited by G.V. Dorofeeva, I.F. Sharygina; Russian Academy of Sciences, Russian Academy of Education. - M.: “Enlightenment”, 2010
Handbook of mathematics - http://lyudmilanik.com.ua
Handbook for students in secondary school http://shkolo.ru

Understanding the Coordinate Plane

Each object (for example, a house, a place in the auditorium, a point on the map) has its own ordered address (coordinates), which has a numerical or letter designation.

Mathematicians have developed a model that allows you to determine the position of an object and is called coordinate plane.

To construct a coordinate plane, you need to draw $2$ perpendicular straight lines, at the end of which the directions “to the right” and “up” are indicated using arrows. Divisions are applied to the lines, and the point of intersection of the lines is the zero mark for both scales.

Definition 1

The horizontal line is called x-axis and is denoted by x, and the vertical line is called y-axis and is denoted by y.

Two perpendicular x and y axes with divisions make up rectangular, or Cartesian, coordinate system, which was proposed by the French philosopher and mathematician Rene Descartes.

Coordinate plane

Point coordinates

A point on a coordinate plane is defined by two coordinates.

To determine the coordinates of point $A$ on the coordinate plane, you need to draw straight lines through it that will be parallel to the coordinate axes (indicated by a dotted line in the figure). The intersection of the line with the x-axis gives the $x$ coordinate of point $A$, and the intersection with the y-axis gives the y-coordinate of point $A$. When writing the coordinates of a point, the $x$ coordinate is first written, and then the $y$ coordinate.

Point $A$ in the figure has coordinates $(3; 2)$, and point $B (–1; 4)$.

To plot a point on the coordinate plane, proceed in the reverse order.

Constructing a point at specified coordinates

Example 1

On the coordinate plane, construct points $A(2;5)$ and $B(3; –1).$

Solution.

Construction of point $A$:

put the number $2$ on the $x$ axis and draw a perpendicular line;
On the y-axis we plot the number $5$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $A$ with coordinates $(2; 5)$.

Construction of point $B$:

Let us plot the number $3$ on the $x$ axis and draw a straight line perpendicular to the x axis;
On the $y$ axis we plot the number $(–1)$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $B$ with coordinates $(3; –1)$.

Example 2

Construct points on the coordinate plane with given coordinates $C (3; 0)$ and $D(0; 2)$.

Solution.

Construction of point $C$:

put the number $3$ on the $x$ axis;
coordinate $y$ is equal to zero, which means point $C$ will lie on the $x$ axis.

Construction of point $D$:

put the number $2$ on the $y$ axis;
coordinate $x$ is equal to zero, which means point $D$ will lie on the $y$ axis.

Note 1

Therefore, at coordinate $x=0$ the point will lie on the $y$ axis, and at coordinate $y=0$ the point will lie on the $x$ axis.

Example 3

Determine the coordinates of points A, B, C, D.$

Solution.

Let's determine the coordinates of point $A$. To do this, we draw straight lines through this point $2$ that will be parallel to the coordinate axes. The intersection of the line with the x-axis gives the coordinate $x$, the intersection of the line with the ordinate gives the coordinate $y$. Thus, we obtain that the point $A (1; 3).$

Let's determine the coordinates of point $B$. To do this, we draw straight lines through this point $2$ that will be parallel to the coordinate axes. The intersection of the line with the x-axis gives the coordinate $x$, the intersection of the line with the ordinate gives the coordinate $y$. We find that point $B (–2; 4).$

Let's determine the coordinates of point $C$. Because it is located on the $y$ axis, then the $x$ coordinate of this point is zero. The y coordinate is $–2$. Thus, point $C (0; –2)$.

Let's determine the coordinates of point $D$. Because it is on the $x$ axis, then the $y$ coordinate is zero. The $x$ coordinate of this point is $–5$. Thus, point $D (5; 0).$

Example 4

Construct points $E(–3; –2), F(5; 0), G(3; 4), H(0; –4), O(0; 0).$

Solution.

Construction of point $E$:

put the number $(–3)$ on the $x$ axis and draw a perpendicular line;
on the $y$ axis we plot the number $(–2)$ and draw a perpendicular line to the $y$ axis;
at the intersection of perpendicular lines we obtain the point $E (–3; –2).$

Construction of point $F$:

coordinate $y=0$, which means the point lies on the $x$ axis;
Let us plot the number $5$ on the $x$ axis and obtain the point $F(5; 0).$

Construction of point $G$:

put the number $3$ on the $x$ axis and draw a perpendicular line to the $x$ axis;
on the $y$ axis we plot the number $4$ and draw a perpendicular line to the $y$ axis;
at the intersection of perpendicular lines we obtain the point $G(3; 4).$

Construction of point $H$:

coordinate $x=0$, which means the point lies on the $y$ axis;
Let us plot the number $(–4)$ on the $y$ axis and obtain the point $H(0;–4).$

Construction of point $O$:

both coordinates of the point are equal to zero, which means that the point lies simultaneously on both the $y$ axis and the $x$ axis, therefore it is the intersection point of both axes (the origin of coordinates).