Coordinate plane basics

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 build a coordinate plane, you need to draw $ 2 $ perpendicular lines, at the end of which are indicated with the direction arrows "right" and "up". The lines are marked with divisions, and the point of intersection of the lines is a zero mark for both scales.

Definition 1

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

Two perpendicular axes x and y with divisions make up rectangular, or cartesian, coordinate systemproposed by the French philosopher and mathematician Rene Descartes.

Coordinate plane

  Point coordinates

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

To determine the coordinates of the 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 dashed line in the figure). The intersection of the line with the abscissa axis gives the coordinate $ x $ of the point $ A $, and the intersection with the ordinate axis gives the coordinate of the point $ A $. When recording the coordinates of a point, the coordinate $ x $ is written first, and then the coordinate $ y $.

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

For drawing points on the coordinate plane, they act in the reverse order.

  Building a point by given coordinates

Example 1

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

Decision.

Building a point $ A $:

• put the number $ 2 $ on the axis $ x $ and draw a perpendicular line;
• set the number $ 5 $ on the y axis and draw a line perpendicular to the $ y $ axis. At the intersection of the perpendicular lines we get the point $ A $ with the coordinates $ (2; 5) $.

Building a point $ B $:

• set $ 3 $ on the $ x $ axis and draw a line perpendicular to the x axis;
• on the $ y $ axis, postpone the number $ (- 1) $ and draw a line perpendicular to the $ y $ axis. At the intersection of the perpendicular lines we get the point $ B $ with the coordinates $ (3; –1) $.

Example 2

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

Decision.

Building the point $ C $:

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

Building the point $ D $:

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

Remark 1

Therefore, at the coordinate $ x \u003d 0 $ the point will lie on the $ y $ axis, and at the coordinate $ y \u003d 0 $ the point will lie on the $ x $ axis.

Example 3

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

Decision.

Define the coordinates of the point $ A $. To do this, draw through this point $ 2 $ lines that will be parallel to the coordinate axes. The intersection of the line with the abscissa 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). $

Define the coordinates of the point $ B $. To do this, draw through this point $ 2 $ lines that will be parallel to the coordinate axes. The intersection of the line with the abscissa gives the coordinate $ x $, the intersection of the line with the ordinate gives the coordinate $ y $. We get that the point $ B (–2; 4). $

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

Define the coordinates of the point $ D $. Because it is located on the axis $ x $, then the coordinate $ y $ is equal to zero. The coordinate $ x $ of this point is $ –5 $. Thus, the point $ D (5; 0). $

Example 4

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

Decision.

Building point $ E $:

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

Building point $ F $:

• the coordinate is $ y \u003d 0 $, which means that the point lies on the axis $ x $;
• put the number $ 5 $ on the axis $ x $ and get the point $ F (5; 0). $

Building the point $ G $:

• set the number $ 3 $ on the $ x $ axis and draw a perpendicular line to the $ x $ axis;
• on the $ y $ axis, postpone the number $ 4 $ and draw a perpendicular line to the $ y $ axis;
• at the intersection of perpendicular lines we get the point $ G (3; 4). $

Building point $ H $:

• the coordinate is $ x \u003d 0 $, which means that the point lies on the $ y $ axis;
• put the number $ (- 4) $ on the axis $ y $ and get the point $ H (0; –4). $

Building point $ O $:

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

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Introduction

In the speech of adults, you could hear the phrase: “Leave me your coordinates.” This expression means that the interlocutor must leave his address or phone number by which he can be found. Those of you who played the “sea battle” used the appropriate coordinate system. A similar coordinate system is used in chess. Seats in the auditorium of the cinema are set in two numbers: the first number indicates the number of the row, and the second - the number of the seat in this row. The idea of \u200b\u200bsetting the position of a point on a plane using numbers originated in antiquity. The coordinate system permeates the entire practical life of a person and has a huge practical application. Therefore, we decided to create this project in order to expand our knowledge on the topic “Coordinate plane”

Project objectives:

get acquainted with the history of the emergence of a rectangular coordinate system on the plane;

prominent figures involved in this topic;

find interesting historical facts;

to perceive coordinates well by ear; clearly and accurately carry out the construction;

prepare a presentation.

Chapter I. Coordinate plane

The idea to set the position of a point on a plane using numbers originated in antiquity - primarily among astronomers and geographers in compiling stellar and geographical maps, calendars.

§1. Origin of coordinates. Coordinate system in geography

200 years before our era, the Greek scientist Hipparchus introduced geographical coordinates. He proposed to draw parallels and meridians on a geographic map and designate numbers as latitude and longitude. Using these two numbers, you can accurately determine the position of the island, village, mountain or well in the desert and put them on a map or globe. Having learned how to determine the latitude and longitude of the ship’s location in the open world, the sailors were able to choose the direction they need.

East longitude and north latitude are denoted by numbers with a plus sign, and west longitude and south latitude are indicated by a minus sign. Thus, a pair of numbers with signs uniquely identifies a point on the globe.

Geographic latitude? - the angle between the vertical line at this point and the plane of the equator, counted from 0 to 90 on both sides of the equator. Geographical longitude? - the angle between the plane of the meridian passing through this point and the plane of the beginning of the meridian (see Greenwich meridian). Longitudes from 0 to 180 east of the beginning of the meridian are called east, west - west.

To find some object in the city, in most cases it is enough to know its address. Difficulties arise if you need to explain where, for example, is a summer cottage, a place in the forest. A universal means of indicating location are geographical coordinates.

In the event of an emergency, a person first must be able to navigate the terrain. Sometimes it is necessary to determine the geographical coordinates of your location, for example, to transmit to the rescue service or for other purposes.

In modern navigation, the WGS-84 world coordinate system is used as standard. All GPS navigators and the main mapping projects on the Internet work in this coordinate system. The coordinates in the WGS-84 system are as common and understood by everyone as universal time. Public accuracy when working with geographic coordinates is 5 - 10 meters on the ground.

Geographic coordinates are signed numbers (latitude from -90 ° to + 90 °, longitude from -180 ° to + 180 °) and can be written in various forms: in degrees (ddd.ddddd °); degrees and minutes (ddd ° mm.mmm "); degrees, minutes and seconds (ddd ° mm" ss.s "). Record forms can be elementary converted to each other (1 degree \u003d 60 minutes, 1 minute \u003d 60 seconds) To indicate the sign of coordinates, letters are often used according to the names of the cardinal points: N and E are north latitude and east longitude are positive numbers, S and W are south latitude and west longitude are negative numbers.

The form for recording coordinates in DEGREES is most convenient for manual input and matches the mathematical record of a number. The form of recording coordinates in DEGREES and MINUTES is preferred in many cases, this format is set by default in most GPS navigators and is standardly used in aviation and at sea. The classical form of recording coordinates in DEGREES, MINUTES, and SECONDS does not really find much practical use.

§2. The coordinate system in astronomy. Constellation Myths

As mentioned above, the idea of \u200b\u200bsetting the position of a point on a plane using numbers was born in ancient times by astronomers when compiling star maps. People needed to count time, predict seasonal phenomena (tides, low tides, seasonal rains, flooding), they had to navigate the area while traveling.

Astronomy is the science of stars, planets, celestial bodies, their structure and development.

Thousands of years have passed, science has stepped far forward, and man still cannot take his admiring gaze from the beauty of the night sky.

Constellations - sections of the starry sky, characteristic figures formed by bright stars. The whole sky is divided into 88 constellations, which facilitate orientation among the stars. Most constellation names come from antiquity.

The most famous constellation is Ursa Major. In ancient Egypt, it was called “Hippopotamus”, and the Kazakhs called “Horse on a leash," although the constellation does not look like one or the other animal. What is it like?

The ancient Greeks had a legend about the constellations Ursa Major and Ursa Minor. The almighty god Zeus decided to marry the beautiful nymph Calisto, one of the maidservants of the goddess Aphrodite, contrary to the desire of the latter. To save Calisto from the persecution of the goddess, Zeus turned Calisto to Ursa Major, her beloved dog to Ursa Minor and took them to heaven. Transfer the constellations Ursa Major and Ursa Minor from the starry sky to the coordinate plane. . Each of the stars of the “Big Dipper Bucket” has its own name.

THE BIG BEAR

I recognize by the BUCKET I!

Seven stars sparkle here

And here is their name:

DUBHE illuminates the darkness

Next to him is MERAK,

On the side of the FECDA with MEGRETS,

A foolish fellow.

From MEHREC to departure

ALIOT is located,

And after him - MITZAR with ALKOR

(These two shine in chorus).

Our bucket closes

Peerless BENET.

He points to the eye

Path in the constellation VOLOPAS,

Where ARCTUR beautiful shines,

Everyone will notice him now!

No less beautiful legend about the constellations "Cepheus", "Cassiopeia" and "Andromeda".

King Ethiopia once ruled Ethiopia. Once his wife, Queen Cassiopeia, was reckless to brag of her beauty to the inhabitants of the sea - the Nereids. The latter, offended, complained to the god of the sea Poseidon, and angry by the impudence of Cassiopeia, the ruler of the seas let a sea monster - Whale - on the shores of Ethiopia. To save his kingdom from destruction, Cepheus, on the advice of the oracle, decided to make a sacrifice to the monster and give him his beloved daughter Andromeda to eat. He chained Andromeda to the coastal cliff and left it awaiting a decision of his fate.

And at this time, on the other side of the world, the mythical hero Perseus made a bold feat. He entered the secluded island, where the Gorgons lived - amazing monsters in the image of women, whose snakes were teeming with hair instead of hair. The gorgon's eyes were so terrible that everyone they looked at turned into stone instantly.

Taking advantage of the sleep of these monsters, Perseus cut off the head of one of them - Gorgon Medusa. At that moment, the horse Pegasus fluttered out of the severed body of Medusa. Perseus grabbed the head of the jellyfish, jumped on Pegasus and rushed through the air to his homeland. When he flew over Ethiopia, he saw Andromeda chained to a rock. At that moment, Keith had already emerged from the depths of the sea, preparing to swallow his prey. But Perseus, having rushed into a mortal battle with Whale, defeated the monster. He showed Keith the head of the jellyfish that had not yet lost its strength, and the monster turned to stone, turning into an island. As for Perseus, having unchained Andromeda, he returned it to his father, and Cepheus, moved by happiness, gave Andromeda to Perseus as his wife. So this story ended safely, the main characters of which were placed by the ancient Greeks in heaven.

On the star map you can find not only Andromeda with her father, mother and husband, but also the magic horse Pegasus and the culprit of all troubles - the monsters of Whale.

The constellation Ceti is located below Pegasus and Andromeda. Unfortunately, it is not marked by any characteristic bright stars and therefore belongs to the number of minor constellations.

§3. Using the idea of \u200b\u200brectangular coordinates in painting.

Traces of the application of the idea of \u200b\u200brectangular coordinates in the form of a square grid (palette) are depicted on the wall of one of the burial chambers of Ancient Egypt. In the funeral chamber of the pyramid of Father Ramses, there is a network of squares on the wall. With their help, the image was enlarged. Renaissance artists also used a rectangular grid.

The word "perspective" in translation from Latin means "I see clearly." In visual arts, a linear perspective is the image of objects on a plane in accordance with the apparent changes in their magnitude. The foundation of the modern theory of perspective was laid by the great artists of the Renaissance - Leonardo da Vinci, Albrecht Durer and others. One of Durer's engravings (Fig. 3) depicts a method of drawing from life through glass with a square grid applied to it. This process can be described as follows: if you stand in front of a window and, without changing your point of view, circle everything that is visible behind it on the glass, then the resulting pattern will be a promising image of space.

Egyptian design methods that seem to be based on square grid patterns. There are numerous examples in Egyptian art showing that artists and sculptors first drew a grid on the wall, which was to be painted or cut in order to maintain the established proportions. The simple numerical relationships of these nets serve as the core of all the great works of art of the Egyptians.

The same method was used by many Renaissance artists, including Leonardo da Vinci. In ancient Egypt, this was embodied in the Great Pyramid, which is reinforced by its close connection with the pattern on Marlborough Down.

Getting to work, the Egyptian artist drew a wall with a grid of straight lines and then carefully transferred the figures to it. But geometric ordering did not prevent him from recreating nature with detailed accuracy. The appearance of each fish, each bird is transmitted with such veracity that modern zoologists easily determine their species. Figure 4 shows a detail of the composition with the illustration - a tree with birds captured by the Khnumhotep network. The movement of the artist’s hand was guided not only by the reserves of his skills, but also by the eye, sensitive to the outlines of nature.

Fig. 4 Birds on acacia

Chapter II Coordinate Method in Mathematics

§1. The use of coordinates in mathematics. Merits

french mathematician Rene Descartes

For a long time, only the geography of “geography” used this remarkable invention, and it was only in the 14th century that the French mathematician Nicola Orem (1323-1382) tried to apply it to “geo-measurement” - geometry. He proposed covering the plane with a rectangular grid and calling latitude and longitude what we now call the abscissa and ordinate.

Based on this successful innovation, a coordinate method arose that linked geometry to algebra. The main merit in creating this method belongs to the great French mathematician Rene Descartes (1596 - 1650). In his honor, such a coordinate system is called a Cartesian, indicating the place of any point on the plane with distances from this point to "zero latitude" - the abscissa axis "and" zero meridian "- the ordinate axis.

However, this brilliant French scientist and thinker of the XVII century (1596 - 1650) did not immediately find his place in life. Born into a noble family, Descartes received a good education. In 1606, his father sent him to the Jesuit College of La Flèche. Given Descartes’s not very good health, he was given some concessions in the strict regime of this educational institution, for example, he was allowed to get up later than others. Having acquired a lot of knowledge in the college, Descartes at the same time was imbued with antipathy to scholastic philosophy, which he retained throughout his life.

After graduating from college, Descartes continued his education. In 1616 at the University of Poitiers he received a bachelor of law degree. In 1617, Descartes enlisted in the army and traveled extensively throughout Europe.

The year 1619 was scientifically key for Descartes.

It was at this time, as he himself wrote in his diary, that the foundations of a new “amazing science” were revealed to him. Most likely, Descartes had in mind the discovery of a universal scientific method, which he subsequently fruitfully applied in a variety of disciplines.

In the 1620s, Descartes met with the mathematician M. Mersenne, through whom he for many years "kept in touch" with the entire European scientific community.

In 1628, Descartes settled for more than 15 years in the Netherlands, but did not settle in any one place, but changed his place of residence about two dozen times.

In 1633, having learned about the condemnation by the church of Galileo, Descartes refuses to publish the natural philosophical work “The World”, which set forth the ideas of the natural occurrence of the universe according to the mechanical laws of matter.

In 1637, Descartes's work, The Discourse on a Method, was published in French, with which, as many believe, the new European philosophy began.

Great influence on European thought was also exerted by Descartes' last philosophical work, Passion of the Soul, published in 1649. In the same year, at the invitation of the Swedish Queen Christina, Descartes went to Sweden. The harsh climate and the unusual regime (the queen forced Descartes to get up at 5 in the morning to give her lessons and carry out other assignments) undermined Descartes’s health, and, catching a cold, he

died of pneumonia.

According to the tradition introduced by Descartes, the "latitude" of the point is denoted by the letter x, "longitude" by the letter y

This system is based on many ways to indicate a place.

For example, on a ticket to a cinema there are two numbers: a row and a place - they can be considered as the coordinates of a place in the hall.

Similar coordinates are accepted in chess. Instead of one of the numbers, a letter is taken: the vertical rows of cells are indicated by the letters of the Latin alphabet, and the horizontal ones by numbers. Thus, each cell of the chessboard is associated with a pair of letters and numbers, and chess players get the opportunity to record their games. Konstantin Simonov writes about the use of coordinates in his poem "The Son of the Gunner".

Walking like a pendulum all night

The eye of the major did not close,

Bye on the radio in the morning

The first signal came:

"It's alright, got it,

The Germans left me

Coordinates (3; 10),

Rather, let's fire!

Guns loaded

The major calculated everything himself.

And with a roar the first volleys

Hit the mountains.

And again a signal on the radio:

"The Germans are right over me,

Coordinates (5; 10),

More like fire!

Earth and rocks flew

A column of smoke rose.

Seemed now from there

No one will leave alive.

The third signal on the radio:

"The Germans are around me,

Coordinates (4; 10),

Do not spare the fire.

The major turned pale upon hearing:

(4; 10) - just

The place where his Lenka

Gotta sit now.

Konstantin Simonov "The son of an artilleryman"

§2. Legends of the invention of the coordinate system

There are several legends about the invention of the coordinate system, which bears the name of Descartes.

Legend 1

This story has survived to our times.

Visiting Parisian theaters, Descartes did not get tired of being surprised at the confusion, the altercation, and sometimes the duel, caused by the lack of an elementary order of distribution of the audience in the auditorium. The numbering system he proposed, in which each place received a row number and a serial number from the edge, immediately removed all the causes for contention and made a real sensation in Parisian high society.

Legend 2. One day, Rene Descartes lay in bed all day thinking about something, and the fly buzzed around and did not allow him to concentrate. He began to ponder how to describe the position of the fly at any given time mathematically in order to be able to slam it without a miss. And ... came up with Cartesian coordinates, one of the greatest inventions in the history of mankind.

Markovtsev Yu.

Once upon a time in an unfamiliar city

Young Descartes arrived.

He was terribly tormented by hunger.

It was a dank month of March.

I decided to turn to a passerby

Descartes, trying to calm the trembling:

Where is the hotel, tell me?

And the lady began to explain:

- Go to the dairy shop

Then to the bakery, after her

Gipsy sells pins

And poison for rats and mice,

You will find in them for sure

Cheeses, biscuits, fruits

And multi-colored silks ...

All these explanations listened

Descartes, trembling from the cold.

He wanted to eat very much,

- Behind the shops - pharmacy

(the pharmacist there is a mustached Swede)

And the church where at the beginning of the century

It seems that my grandfather was married ...

When the lady fell silent for a moment,

Suddenly her servant said:

- Walk three blocks straight

And two to the right. Entrance from the corner.

This is the third fiction about the case that prompted Descartes to the idea of \u200b\u200bcoordinates.

Conclusion

When creating our project, we learned about the use of the coordinate plane in various fields of science and everyday life, some information from the history of the origin of the coordinate plane and mathematicians who made a great contribution to this invention. The material that we collected during the writing of the work can be used in the classroom, as additional material for the lessons. All this may interest students and brighten the learning process.

And we would like to finish with these words:

“Imagine your life as a coordinate plane. The y axis is your position in society. X axis - moving forward, towards the goal, towards your dream. And as we know, it is infinite ... we can fall downward, deeper and deeper minus, we can remain at zero and do nothing, absolutely nothing. We can go up, we can fall, we can go forward or go back, and all because our whole life is a coordinate plane and most importantly here, what is your coordinate ... ”

Bibliography

Glazer G.I. The history of mathematics at school: - M .: Education, 1981. - 239 p., Ill.

Lyatker, Y. A. Descartes. M.: Thought, 1975. - (Thinkers of the Past)

Matvievskaya G.P. Rene Descartes, 1596-1650. M .: Nauka, 1976.

A. Savin. Coordinate Quantum. 1977. No9

Mathematics - supplement to the newspaper "First of September", No. 7, No. 20, No. 17, 2003, No. 11, 2000.

Siegel F.Yu. Star ABC: Student Manual. - M .: Education, 1981. - 191 p., Silt

Steve Parker, Nicholas Harris. Illustrated Encyclopedia for children. Secrets of the universe. Kharkov Belgorod. 2008

Materials from the site http://istina.rin.ru/

On surface. Let one be x and the other be y. And let these lines be mutually perpendicular (that is, they intersect at right angles). Moreover, the point of their intersection will be the origin for both lines, and the unit segment is the same (Fig. 1).

So we got rectangular coordinate system, and our plane became coordinate. Lines x and y are called coordinate axes. Moreover, the x axis is the abscissa axis, and the y axis is the ordinate axis. A similar plane is usually indicated by the name of the axes and the reference point - xOy. The rectangular coordinate system is also called cartesian coordinate systemsince the first time the French mathematician and philosopher Rene Descartes began to actively use it.

The rectangular angles formed by the straight lines x and y are called coordinate angles. Each corner has its own number as shown in fig. 2.

So, when we talked about the coordinate line, every point of this line had one coordinate. Now, when it comes to the coordinate plane, then every point on this plane will already have two coordinates. One corresponds to the line x (this coordinate is called abscissa), the other corresponds to the straight line y (this coordinate is called ordinate) It is written this way: M (x; y), where x is the abscissa and y is the ordinate. It reads as: "Point M with x, y coordinates."

How to determine the coordinates of a point on a plane?
Now we know that each point on the plane has two coordinates. In order to find out its coordinates, it is enough for us to draw two straight lines perpendicular to the coordinate axes through this point. The intersection points of these lines with the coordinate axes will be the desired coordinates. So, for example, in fig. 3 we determined that the coordinates of point M are 5 and 3.

How to build a point on a plane by its coordinates?
It also happens that we already know the coordinates of a point on a plane. And we need to find her location. Suppose we have the coordinates of the point (-2; 5). That is, the abscissa is -2, and the ordinate is 5. Take the point with the coordinate -2 on the x-axis (abscissa axis) and draw a line a parallel to the y axis through it. Note that any point on this line will have an abscissa equal to -2. Now we find on the y line (ordinate axis) a point with coordinate 5 and draw a straight line b parallel to the x axis through it. Note that any point on this line will have an ordinate equal to 5. At the intersection of lines a and b, there will be a point with coordinates (-2; 5). Denote it by the letter P (Fig. 4).

We also add that the line a, all of whose points have an abscissa of -2, is given by the equation
x \u003d -2 or that x \u003d -2 is the equation of the line a. For convenience, we can say not “line, which is given by the equation x \u003d -2”, but simply “line x \u003d -2”. Indeed, for any point of the line a, the equality x \u003d -2 is valid. And line b, all of whose points have ordinate 5, is in turn given by the equation y \u003d 5 or that y \u003d 5 is the equation of line b.

In everyday life, you might hear the phrase: “ Leave me your coordinates!».

How do you understand this phrase?

This expression means that the interlocutor must leave his address  or phone number, i.e. data by which it can be found.

Definition

The numbers by which indicate where some object is located, call it coordinates.

You have already met with coordinates more than once in mathematics. You can perform two operations: mark a point on a coordinate line with a given coordinate and, conversely, determine the coordinate of a given point. To do this, on a straight line, select the reference point, the positive direction and the unit segment. After that, any point on the line gets its own coordinate.

Point coordinate   indicates, therefore, its place on the coordinate line.

The question arises: is it possible to determine the location of a point on a plane?

Surely, at least once in your life you played a game like " Sea battle».

The field of this game consists of a square measuring 10 by 10 cells. In this field, ships are represented: 1 four-cell, 2 three-cell, 3 two-cell and 4 single-cell. In this case, between any two neighboring ships there must be a gap of at least one cell.

The screen shows one of the options for the location of the ships. Each square cell is indicated by a pair: (letter - number) indicated along the bottom and left sides of the square. For example, a ship is located in a cage (F; 4). The essence of this game is to find all the ships of the opponent first. When designating the position of the cell, the first indicates its horizontal coordinate, and the second - vertical.

This is precisely the essence of the coordinates, or, as they usually say, coordinate systems : this is the rule by which the position of an object is determined.

Coordinate systems   meet in our life constantly.

You are familiar with the coordinate system in the cinema's auditorium (row number and seat number), in the train (car number and seat number), with the geographical coordinate system (longitude and latitude).

What you need to know in order to find your place in the cinema? Seats in the cinema auditorium two  by numbers: the first number indicates the number of the row, and the second - the number of the chair in this row. So, in order to take its place in the audience correctly, you need to know two coordinates : row and place.

for instance , the ticket indicates: 3 row 2 place. See where this place is located.

Please note that when determining the location we need to know two characteristics   or two meanings .

In a similar way, the position of a point on a plane can be indicated.

Rene Descartes - French mathematician introduced in 1637 coordinate system , which is used all over the world and is known to every student. She is also called " Cartesian coordinate system ».

To define a Cartesian rectangular coordinate system on a plane, two mutually perpendicular coordinate lines are drawn x  and atcalled coordinate axes .

The intersection point of the axes is “ OIs called origin .

On each axis OX  and Oy  set a positive direction and select a single segment.

Each of the coordinate axes has its own name: the horizontal axis is called abscissa axis   (or x axis ), the vertical axis is called ordinate axis   (or axis y ) These straight lines make up coordinate system on the plane .

Definition

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

The axes break the coordinate plane into four parts, which are called coordinate quarters . They are numbered in Roman numerals and counterclockwise.

They say: first quarter, second quarter, third quarter and fourth quarter.

Each point of such a plane has two coordinates.

Consider how the position of a point on the coordinate plane is determined.

for instance we have a point M. And you need to determine its coordinates. To do this, draw a perpendicular from this point to the horizontal axis or the abscissa axis.

Axis intersection point x  called the abscissa of the point M .

In our case, the abscissa points M 3.

Axis intersection point at  called ordinate point M .

In our case, the ordinate of the point M 5.

Abscissa and ordinate points M  are called coordinates   this points . They are usually written next to the letter denoting a point, in parentheses. Moreover, the abscissa is always written in the first place, and the ordinate in the second.

Read this entry like this: "Point M with abscissa 3 and ordinate 5", or "Point M with coordinates 3 and 5". Please note, if you rearrange the coordinates in places, you get a completely different point. For example, point N (5; 3).

Definition

The coordinates of the point (x; y) on the plane   Is a pair of numbers in which the abscissa (x) is in the first place, and the ordinate (y) of this point is in the second.

Will do conclusion: coordinates can be specified for any point on the coordinate plane: to do this, draw perpendiculars from the point to the coordinate axes and determine to what number of the coordinate axis the base of the perpendicular corresponds.

Points of any straight line perpendicular to the abscissa axis have the same abscissa .

for instance all points of the line a  have an abscissa 4. All points on the ordinate axis have an abscissa of 0, i.e. the coordinates of any point on the ordinate axis have the form (0; y).

Points of any straight line perpendicular to the ordinate axis have the same ordinate .

for instance all points of the line b  have an ordinate of -3. All points of the abscissa axis have an ordinate of 0, i.e. the coordinates of any point on the abscissa axis have the form (x; 0).

Origin   - point O - lies on both the abscissa and the ordinate. So its coordinates are (0; 0).

There are several ways to construct a point by its coordinates.

for instance , construct a point A (-5; 7).

First way: on axis x  find the abscissa of the point A. We have it equal to -5. Draw a perpendicular from this point about the axis OH. Next, on the y axis, we find the ordinate of the point. It is equal to 7. Draw a perpendicular from this point relative to the axis OU. The point where both perpendiculars intersect is the desired point A.

Second way  plot points on given coordinates. Can be shifted along the axis OH  5 units to the left, because the abscissa of a point is a negative number. And then, parallel to the axis ABOUTX  up by 7 units, as the ordinate of a point is a positive number. The point where both perpendiculars intersect is the desired point A.

Let's make another very important conclusion:

Each point 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 the coordinates .

The task

Draw points on the coordinate plane, and then connect them in succession with line segments.

What figure did we get in the end? Right! This is a cat !!!