Start in science. Coordinate plane Coordinate plane how to determine coordinates

If you construct two mutually perpendicular numerical axes on a plane: OX and OYthen they will be called coordinate axes... Horizontal axis OX called abscissa (axis x), vertical axis OY - the ordinate (axis y).

Dot Ostanding at the intersection of the axes is called origin... It is the zero point for both axes. Positive numbers are plotted on the abscissa as dots to the right, and on the ordinate as dots up from the zero point. Negative numbers are depicted as dots to the left and down from the origin (dots O). The plane on which the coordinate axes lie is called coordinate plane.

The coordinate axes divide the plane into four parts, called quarters or quadrants... It is customary to number these quarters with Roman numerals in the order in which they are numbered in the drawing.

Point coordinates on a plane

If we take an arbitrary point on the coordinate plane A and draw perpendiculars from it to the coordinate axes, then the bases of the perpendiculars will be two numbers. The number pointed to by the vertical perpendicular is called abscissa point A... The number pointed to by the horizontal perpendicular is ordinate point A.

In the drawing, the abscissa of the point A is 3 and the ordinate is 5.

The abscissa and ordinate are called the coordinates of a given point on the plane.

Point coordinates are written in brackets to the right of the point designation. The abscissa is written first, followed by the ordinate. So record A(3; 5) means that the abscissa of the point A is three, and the ordinate is five.

Point coordinates are numbers that define its position on the plane.

If a point lies on the abscissa axis, then its ordinate is zero (for example, the point B with coordinates -2 and 0). If a point lies on the ordinate axis, then its abscissa is zero (for example, the point C with coordinates 0 and -4).

Origin - point O - has both the abscissa and the ordinate equal to zero: O (0; 0).

This coordinate system is called rectangular or cartesian.

The topic of this video tutorial: Coordinate plane.

Goals and objectives of the lesson:

Acquainted with rectangular coordinate system on the plane
- teach to freely navigate on the coordinate plane
- build points according to its specified coordinates
- determine the coordinates of a point marked on the coordinate plane
- good listening to coordinates
- clearly and accurately perform geometric constructions
- development of creative abilities
- fostering interest in the subject

The term " coordinates"Comes from the Latin word -" ordered "

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 of the plane for which these numbers are coordinates.

Examples are considered:

• by plotting a point by its coordinates
• finding the coordinates of a point located on the coordinate plane

Some additional information:

The idea to set the position of a point on a plane originated in antiquity - 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 a rectangular coordinate system is often called Cartesian.

The words " abscissa», « ordinate», « coordinates"First began to use at the end of the XVII.

For a better understanding of the coordinate plane, let's imagine what we are given: a geographic 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 place.

Additional task.

After studying the video lesson, to consolidate the material, I suggest you take a pen and a leaf 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).
Little 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).

Points are "registered" - "tenants", each point has its own "house number" - its coordinate. If the point is taken in a plane, then for its registration it is necessary to indicate not only the house number, but also the apartment number. Let us recall how this is done.

We will draw two mutually perpendicular coordinate lines and we will consider the point of their intersection - point O. plane to coordinate. Point O is called the origin, the coordinate lines (x-axis and y-axis) are called coordinate axes, and right angles formed by the coordinate axes are called coordinate angles. Coordinate rectangular corners are numbered as shown in Figure 20.

And now let's turn to Figure 21, where a rectangular coordinate system is shown and point M. is marked. Draw a straight line through it, parallel to the y-axis. The straight line intersects the x-axis at some point, this point has a coordinate - on the x-axis. For the point shown in Figure 21, this coordinate is -1.5, it is called the abscissa of point M. Next, draw a straight line through point M parallel to the x axis. The straight line intersects the y-axis at some point, this point has a coordinate - on the y-axis.

For point M, shown in Figure 21, this coordinate is equal to 2, it is called the ordinate of point M. Briefly written as follows: M (-1.5; 2). The abscissa is written in the first place, the ordinate in the second. Use, if necessary, and another form of notation: x \u003d -1.5; y \u003d 2.

Remark 1 ... In practice, to find the coordinates of the point M, instead of straight lines parallel to the coordinate axes and passing through the point M, they construct segments of these straight lines from point M to the coordinate axes (Fig. 22).

Remark 2. In the previous paragraph, we introduced different notation for number intervals. In particular, as we agreed, the notation (3, 5) means that an interval with ends at points 3 and 5 is considered on the coordinate line. In the present section, we consider a pair of numbers as coordinates of a point; for example, (3; 5) is a point on coordinate plane with abscissa 3 and ordinate 5. How is it correct to determine from the symbolic notation what is at stake: the interval or the coordinates of a point? Most often this is clear from the text. And if it's not clear? Pay attention to one detail: we used a comma in the interval designation, and a semicolon in the coordinate designation. This, of course, is not very significant, but still a difference; we will apply it.

Given the terms and designations introduced, the horizontal coordinate line is called the abscissa, or the x-axis, and the vertical coordinate line, the ordinate, or y-axis. The designations x, y are usually used when specifying a rectangular coordinate system on the plane (see Fig. 20) and often say as follows: given a coordinate system xOy. However, there are other designations: for example, in Figure 23, the coordinate system tOs is specified.
Algorithm for finding the coordinates of a point M, given in a rectangular coordinate system xOy

This is exactly how we acted, finding the coordinates of point M in Figure 21. If point M 1 (x; y) belongs to the first coordinate angle, then x\u003e 0, y\u003e 0; if point М 2 (x; y) belongs to the second coordinate angle, then x< 0, у > 0; if point М 3 (x; y) belongs to the third coordinate angle, then x< О, у < 0; если точка М 4 (х; у) принадлежит четвертому координатному углу, то х > OU< 0 (рис. 24).

And what happens if the point whose coordinates you want to find lies on one of the coordinate axes? Let point A lie on the x-axis and point B on the y-axis (Fig. 25). It makes no sense to draw a straight line through point A parallel to the y-axis and find the point of intersection of this line with the x-axis, since such an intersection point already exists - this is point A, its coordinate (abscissa) is 3. In the same way, it is not necessary to draw through the point And a straight line parallel to the x-axis - this line is the x-axis itself, which intersects the y-axis at point O with the coordinate (ordinate) 0. As a result, for point A we get A (3; 0). Similarly, for point B, we obtain B (0; - 1.5). And for point O we have O (0; 0).

In general, any point on the x-axis has coordinates (x; 0), and any point on the y-axis has coordinates (0; y)

So, we discussed how to find the coordinates of a point in the coordinate plane. And how to solve the inverse problem, that is, how, having given coordinates, construct the corresponding point? To develop an algorithm, we will carry out two auxiliary, but at the same time important reasoning.

First reasoning. Let I be drawn in the xOy coordinate system, parallel to the y-axis and intersecting the x-axis at a point with a coordinate (abscissa) 4

(fig. 26). Any point lying on this straight line has an abscissa 4. So, for points M 1, M 2, M 3 we have M 1 (4; 3), M 2 (4; 6), M 3 (4; - 2). In other words, the abscissa of any point M of the straight line satisfies the condition x \u003d 4. They say that x \u003d 4 - the equation line l or that line I satisfies the equation x \u003d 4.

Figure 27 shows straight lines that satisfy the equations x \u003d - 4 (line I 1), x \u003d - 1
(straight I 2) x \u003d 3.5 (straight I 3). And which line satisfies the equation x \u003d 0? Have you guessed? Y-axis

Second reasoning. Let a straight line I be drawn in the xOy coordinate system, parallel to the x-axis and intersecting the y-axis at a point with coordinate (ordinate) 3 (Fig. 28). Any point lying on this straight line has ordinate 3. So, for points М 1, М 2, М 3 we have: М 1 (0; 3), М 2 (4; 3), М 3 (- 2; 3) ... In other words, the ordinate of any point M of the line I satisfies the condition y \u003d 3. They say that y \u003d 3 is the equation of the line I or that the line I satisfies the equation y \u003d 3.

Figure 29 shows the straight lines that satisfy the equations y \u003d - 4 (straight l 1), y \u003d - 1 (straight I 2), y \u003d 3.5 (straight I 3) - And which straight line satisfies the equation y \u003d 01 Guess? X-axis.

Note that mathematicians, striving for brevity of speech, say "straight line x \u003d 4", and not "straight line satisfying the equation x \u003d 4". Similarly, they say "the line y \u003d 3" and not "the line satisfying the equation y \u003d 3". We will do the same. Let's return now to Figure 21. Note that the point M (- 1.5; 2), which is shown there, is the intersection point of the straight line x \u003d -1.5 and the straight line y \u003d 2. Now, apparently, the algorithm for constructing the point will be clear according to its given coordinates.

Algorithm for constructing a point M (a; b) in a rectangular coordinate system xOy

PRI me r. In the xOy coordinate system, build the points: A (1; 3), B (- 2; 1), C (4; 0), D (0; - 3).

Decision. Point A is the intersection point of the lines x \u003d 1 and y \u003d 3 (see Fig. 30).

Point B is the point of intersection of lines x \u003d - 2 and y \u003d 1 (Fig. 30). Point C belongs to the x-axis, and point D to the y-axis (see Fig. 30).

In conclusion of the section, we note that for the first time a rectangular coordinate system on a plane began to be actively used to replace algebraic models geometric French philosopher Rene Descartes (1596-1650). Therefore, sometimes they say "Cartesian coordinate system", "Cartesian coordinates".

A complete list of topics by grade, a calendar plan according to the school curriculum in mathematics online, video material in mathematics for grade 7 download

A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions

§ 1 Coordinate system: definition and construction method

In this lesson, we will get acquainted with the concepts of "coordinate system", "coordinate plane", "coordinate axes", we will learn how to build points on a plane by coordinates.

Take the 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, 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 define:

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

§ 2 Coordinate axis and coordinate plane

The straight lines that form the coordinate system are called coordinate axes, each of which has its own name: the x coordinate line is the abscissa axis, the y coordinate line 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 after the French philosopher and mathematician René Descartes.

Each point of 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 the plane are a pair of numbers, of which the first number is the abscissa, the second number is the ordinate. The abscissa is shown by the perpendicular to the x-axis, the ordinate is the perpendicular to the y-axis.

Let's mark point A on the coordinate plane, 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 4, the ordinate of point A - along the perpendicular to the ordinate (y-axis) is 3. The coordinates of our point are 4 and 3. A (4; 3). Thus, coordinates can be found for any point in the coordinate plane.

§ 3 Construction of a point on the plane

How to build a point on a plane with given coordinates, i.e. determine its position by the coordinates of a point on the plane? In this case, we perform the actions in the reverse order. On the coordinate axes, we find the 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. point with given coordinates.

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

To do this, on the abscissa axis, we find a point with coordinate 2, draw a straight line through this point perpendicular to the x axis. On the ordinate we find a point with a coordinate of -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's mark on the coordinate plane the points A (0; 2), B (0; -3), C (0; 4).

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

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

Let's change the coordinates of these points in places.

It turns out A (2; 0), B (-3; 0) C (4; 0). In this case, all ordinates are 0 and the points are on the abscissa axis.

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

Let us examine two more cases.

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

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

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

If you change the coordinates of points M, N, P in places, you get M (2; 3), N (-1; 3), P (-4; 3). The ordinates of the points will become 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, the points having the same ordinate lie on one straight line parallel to the abscissa axis and perpendicular to the ordinate axis.

In this lesson, you got acquainted with the concepts of "coordinate system", "coordinate plane", "coordinate axes - abscissa axis and ordinate axis". Learned how to find the coordinates of a point on the coordinate plane and learned how to build points on the plane by its coordinates.

List of used literature:

1. Mathematics. Grade 6: lesson plans for the textbook by I.I. Zubareva, A.G. Mordkovich // compiled by L.A. Topilin. - Mnemosyne, 2009.
2. Mathematics. Grade 6: a textbook for students of educational institutions. I.I.Zubareva, A.G. Mordkovich. - M .: Mnemosina, 2013.
3. Mathematics. Grade 6: textbook for educational institutions / G.V. Dorofeev, I.F. Sharygin, S.B. Suvorov and others / edited by G.V. Dorofeeva, I.F. Sharygin; Russian Academy of Sciences, Russian Academy of Education. - M .: "Education", 2010
4. Mathematics reference - http://lyudmilanik.com.ua
5. Handbook for high school students http://shkolo.ru

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

Definition 1

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

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

Coordinate plane

Point coordinates

A point on a 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, which will be parallel to the coordinate axes (in the figure, they are highlighted by a dotted line). The intersection of the straight line with the abscissa gives the $ x $ coordinate of the point $ A $, and the intersection with the ordinate gives the coordinate at the point $ A $. When writing the coordinates of a point, the $ x $ coordinate is written first, and then the $ y $ coordinate.

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

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

Drawing a point by specified coordinates

Example 1

Draw points $ A (2; 5) $ and $ B (3; –1) on the coordinate plane. $

Decision.

Plotting point $ A $:

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

Plotting point $ B $:

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

Example 2

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

Decision.

Plotting point $ C $:

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

Plotting point $ D $:

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

Remark 1

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

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

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

Let's define the coordinates of the 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, the point is $ C (0; –2) $.

Let's define the coordinates of the point $ D $. Because it is located on the $ x $ axis, then the $ y $ coordinate is zero. The $ x $ coordinate 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.

Plotting point $ E $:

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

Plotting point $ F $:

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

Plotting point $ G $:

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

Plotting point $ H $:

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

Plotting point $ O $:

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