The coordinates of the midpoint of a line with three points. Midpoint coordinates. Complete Lessons - Knowledge Hypermarket

There is a whole group of tasks (included in the examination problem types) associated with the coordinate plane. These are tasks starting from the most elementary ones, which are solved orally (determining the ordinate or abscissa of a given point, or a symmetric given point, and others), ending with tasks that require high-quality knowledge, understanding and good skills (tasks related to the slope of a straight line).

Gradually, we will consider all of them. In this article, we'll start with the basics. it simple tasks to determine: abscissas and ordinates of a point, length of a segment, midpoint of a segment, sine or cosine of the angle of inclination of a straight line.Most of these tasks will not be interesting. But I consider it necessary to present them.

The fact is that not everyone goes to school. Many people take the Unified State Exam 3-4 or more years after its completion, and they vaguely remember what the abscissa and ordinate are. We will analyze other tasks related to the coordinate plane, do not miss, subscribe, to update the blog. Now n a lot of theory.

Let's build on coordinate plane point A with coordinates x = 6, y = 3.

They say that the abscissa of point A is six, the ordinate of point A is three.

To put it simply, the oh-axis is the abscissa axis, the oh-axis is the ordinate spine.

That is, the abscissa is a point on the x-axis into which a point specified on the coordinate plane is projected; the ordinate is the point on the oy axis to which the specified point is projected.

Segment length on the coordinate plane

The formula for determining the length of a segment, if the coordinates of its ends are known:

As you can see, the length of the segment is the length of the hypotenuse in a right-angled triangle with legs equal

X B - X A and Y B - Y A

* * *

Midpoint of the segment. Her Coordinates.

Formula for finding the coordinates of the midpoint of a segment:

Equation of a straight line passing through two given points

The formula for the equation of a straight line passing through two given points has the form:

where (x 1; y 1) and (x 2; y 2 ) coordinates of the given points.

Substituting the values ​​of the coordinates in the formula, it is reduced to the form:

y = kx + b, where k is the slope of the straight line

We will need this information when solving another group of problems related to the coordinate plane. There will be an article about this, do not miss it!

What else can you add?

The angle of inclination of a straight line (or segment) is the angle between the oX axis and this straight line, ranging from 0 to 180 degrees.

Let's consider the tasks.

From point (6; 8) a perpendicular is dropped onto the ordinate axis. Find the ordinate of the base of the perpendicular.

The base of the perpendicular dropped on the ordinate axis will have coordinates (0; 8). The ordinate is eight.

Answer: 8

Find the distance from the point A with coordinates (6; 8) to the ordinate axis.

The distance from point A to the ordinate is equal to the abscissa of point A.

Answer: 6.

A(6; 8) about the axis Ox.

A point symmetrical to point A relative to the oX axis has coordinates (6; - 8).

The ordinate is minus eight.

Answer: - 8

Find the ordinate of a point symmetric to a point A(6; 8) relative to the origin.

The point symmetrical to point A relative to the origin has coordinates (- 6; - 8).

Its ordinate is - 8.

Answer: –8

Find the abscissa of the midpoint of the line segment connecting the pointsO(0; 0) and A(6;8).

In order to solve the problem, it is necessary to find the coordinates of the midpoint of the segment. The coordinates of the ends of our segment are (0; 0) and (6; 8).

We calculate by the formula:

Got (3; 4). The abscissa is equal to three.

Answer: 3

* The abscissa of the midpoint of a segment can be determined without calculating by the formula, by building this segment on the coordinate plane on a sheet in a cell. The middle of the segment will be easy to determine by the cells.

Find the abscissa of the midpoint of the line segment connecting the points A(6; 8) and B(–2;2).

In order to solve the problem, it is necessary to find the coordinates of the midpoint of the segment. The coordinates of the ends of our segment are (–2; 2) and (6; 8).

We calculate by the formula:

Got (2; 5). The abscissa is two.

Answer: 2

* The abscissa of the midpoint of a segment can be determined without calculating by the formula, by building this segment on the coordinate plane on a sheet in a cell.

Find the length of the line segment connecting the points (0; 0) and (6; 8).

The length of a segment at the given coordinates of its ends is calculated by the formula:

in our case, we have O (0; 0) and A (6; 8). Means,

* The order of the coordinates does not matter when subtracted. It is possible to subtract the abscissa and ordinate of point A from the abscissa and ordinate of point O:

Answer: 10

Find the cosine of the slope of the line segment connecting the points O(0; 0) and A(6; 8), with the abscissa.

The angle of inclination of a segment is the angle between this segment and the oX axis.

From point A, let us drop the perpendicular to the oX axis:

That is, the angle of inclination of the segment is the angleSAIv right triangle AVO.

The cosine of an acute angle in a right triangle is

the ratio of the adjacent leg to the hypotenuse

It is necessary to find the hypotenuseOA.

By the Pythagorean theorem:In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

So the cosine of the slope is 0.6

Answer: 0.6

From point (6; 8) a perpendicular is dropped to the abscissa axis. Find the abscissa of the base of the perpendicular.

A straight line is drawn through point (6; 8) parallel to the abscissa axis. Find the ordinate of its point of intersection with the axis OU.

Find the distance from the point A with coordinates (6; 8) to the abscissa axis.

Find the distance from the point A with coordinates (6; 8) to the origin.

Very often in the C2 problem it is required to work with points that divide the segment in half. The coordinates of such points are easily calculated if the coordinates of the ends of the segment are known.

So, let the segment be defined by its ends - points A = (x a; y a; z a) and B = (x b; y b; z b). Then the coordinates of the midpoint of the segment - we denote it by the point H - can be found by the formula:

In other words, the coordinates of the midpoint of a segment are the arithmetic mean of the coordinates of its ends.

· Task ... The unit cube ABCDA 1 B 1 C 1 D 1 is placed in the coordinate system so that the x, y and z axes are directed along the edges AB, AD and AA 1, respectively, and the origin coincides with point A. Point K is the midpoint of the edge A 1 B 1 . Find the coordinates of this point.

Solution... Since point K is the midpoint of the segment A 1 B 1, its coordinates are equal to the arithmetic mean of the coordinates of the ends. Let's write down the coordinates of the ends: A 1 = (0; 0; 1) and B 1 = (1; 0; 1). Now let's find the coordinates of point K:

Answer: K = (0.5; 0; 1)

· Task ... The unit cube ABCDA 1 B 1 C 1 D 1 is placed in the coordinate system so that the x, y and z axes are directed along the edges AB, AD and AA 1, respectively, and the origin coincides with point A. Find the coordinates of the point L at which they intersect diagonals of the square A 1 B 1 C 1 D 1.

Solution... It is known from the planimetry course that the intersection point of the diagonals of a square is equidistant from all its vertices. In particular, A 1 L = C 1 L, i.e. point L is the midpoint of segment A 1 C 1. But A 1 = (0; 0; 1), C 1 = (1; 1; 1), so we have:

Answer: L = (0.5; 0.5; 1)

The simplest problems of analytical geometry.
Actions with vectors in coordinates

It is highly desirable to learn how to solve the tasks that will be considered on a full machine, and the formulas memorize, they will not even specifically memorize, they themselves will be remembered =) This is very important, since other problems of analytical geometry are based on the simplest elementary examples, and it will be annoying to spend additional time eating pawns. There is no need to fasten the top buttons on the shirt, many things are familiar to you from school.

The presentation of the material will go in a parallel course - both for plane and for space. For the reason that all the formulas ... you will see for yourself.

Introduction of Cartesian coordinates in space. Distance between points. Midpoint coordinates.

Lesson objectives:

Educational: Consider the concept of a coordinate system and coordinates of a point in space; derive the formula for the distance in coordinates; derive the formula for the coordinates of the midpoint of the segment.

Developing: Promote the development of the spatial imagination of students; contribute to the development of problem solving and the development of logical thinking of students.

Educational: Upbringing cognitive activity, sense of responsibility, culture of communication, culture of dialogue.

Equipment: Drawing supplies, presentation, CRC

Lesson type: Lesson in learning new material

Lesson structure:

Organizing time.

Updating basic knowledge.

Learning new material.

Updating new knowledge

Lesson summary.

During the classes

Message from history " Cartesian coordinate system "(Learner)

Solving a geometric, physical, chemical problem, you can use various coordinate systems: rectangular, polar, cylindrical, spherical.

In the general education course, a rectangular coordinate system is studied on a plane and in space. Otherwise, it is called the Cartesian coordinate system after the French scientist philosopher Rene Descartes (1596 - 1650) who first introduced coordinates into geometry.

(A student's story about René Descartes.)

René Descartes was born in 1596 in the city of Lae in the south of France, into a noble family. Father wanted to make Rene an officer. For this, in 1613 he sent Rene to Paris. For many years Descartes had to stay in the army, to participate in military campaigns in Holland, Germany, Hungary, Czech Republic, Italy, in the siege of the Huguenot fortress of La Roshali. But Rene was interested in philosophy, physics and mathematics. Soon after his arrival in Paris, he met a student of Viet, a prominent mathematician of that time - Mersen, and then with other mathematicians of France. While in the army, Descartes devoted all his free time to studying mathematics. He studied German algebra, French and Greek mathematics.

After the capture of La Roschali in 1628, Descartes left the army. He leads a secluded life in order to realize the extensive plans of scientific work.

Descartes was the greatest philosopher and mathematician of his time. The most famous work of Descartes is his "Geometry". Descartes introduced a coordinate system that everyone uses today. He established a correspondence between numbers and line segments and thus introduced the algebraic method to geometry. These discoveries of Descartes gave a huge impetus to the development of both geometry and other branches of mathematics and optics. Now it is possible to depict the dependence of quantities graphically on the coordinate plane, numbers - as segments and perform arithmetic operations on segments and others geometric values as well as various functions. It was a completely new method, distinguished by beauty, grace and simplicity.

Repetition. Rectangular coordinate system on a plane.

Questions:

What is called a coordinate system on a plane?

How are the coordinates of a point on a plane determined?

What are the coordinates of the origin?

What is the formula for the coordinates of the midpoint of a line segment and the distance between points on the plane?

Learning new material:

A rectangular coordinate system in space is a triplet of mutually perpendicular coordinate lines with a common origin. The common origin is indicated by the letterO.

Oh - abscissa axis,

Oy - ordinate axis,

Oz- axis applicate

Three planes passing through the axes of coordinates Ox and Oy, Oy and Oz, Ozand Oh, are called coordinate planes: Oxy, Oyz, OzNS.

In a rectangular coordinate system, each point M in space is associated with a triple of numbers - its coordinates.

M (x, y,z), where x is the abscissa, y is the ordinate,z- applicate.

Coordinate system in space

Point coordinates

Distance between points

1 (x 1 ; y 1 ; z 1 ) and A 2 (x 2 ; y 2 ; z 2 )

Then the distance between points A 1 and A 2 is calculated like this:

The coordinates of the midpoint of the segment in space

There are two arbitrary points A 1 (x 1 ; y 1 ; z 1 ) and A 2 (x 2 ; y 2 ; z 2 ). Then the midpoint of the segment A 1 A 2 will be point C with coordinates x, y, z, where

Gaining solution skills:

1) Find the coordinates of orthogonal projections of pointsA (1, 3, 4) and

B (5, -6, 2) to:

a) planeOxy ; b) planeOyz ; c) axisOx ; d) axisOz .

Answer: a) (1, 3, 0), (5, -6, 0); b) (0, 3, 4), (0, -6, 2); c) (1, 0, 0), (5, 0, 0);

d) (0, 0, 4), (0, 0, 2).

2) At what distance is the pointA (1, -2, 3) from the coordinate plane:

a)Oxy ; b)Oxz ; v)Oyz ?

Answer: a) 3; b) 2; in 1

3) Find the coordinates of the midpoint of the segment:

a)AB , ifA (1, 2, 3) andB (-1, 0, 1); b)CD , ifC (3, 3, 0) andD (3, -1, 2).

Answer: a) (1, 1, 2); b) (3, 1, 1).

5. Homework: the textbook of AV Pogorelov "Geometry 10-11" p. 23 - 25, p. 53 answer questions 1 - 3; №7, №10(1)

6. Lesson summary.

table

On surface

In space

Definition. A coordinate system is a set of two intersecting coordinate axes, the point at which these axes intersect - the origin - and unit segments on each of the axes

Definition. A coordinate system is a set of three coordinate axes, the point at which these axes intersect - the origin of coordinates - and unit segments on each of the axes

2 axles,

ОУ - ordinate axis,

ОХ - abscissa axis

3 axles,

ОХ - abscissa axis,

ОУ - ordinate axis,

ОZ - axis of applicator.

OX perpendicular to OA

OX is perpendicular to OA,

OX is perpendicular to OZ,

OA perpendicular to OZ

(Oh; oh)

(OOO)

Direction, unit segment

Distance between points.

Distance between points

Midpoint coordinates.

Midpoint coordinates

Questions:

How is the Cartesian coordinate system introduced? What does it consist of?

How are the coordinates of a point in space determined?

What is the coordinate of the point of intersection of the coordinate axes?

What is the distance from the origin to the given point?

What is the formula for the coordinates of the midpoint of a segment and the distance between points in space?

Assessment of learners

7 reflexion

At the lesson

I found out …

I learned…

I like it…

I was at a loss ...

My mood…

Literature.

A.V. Pogorelov. Tutorial 10-11. M. "Education", 2010

I.S. Petrakov. Math circles in grades 8-10. M, "Education", 1987

The article below will highlight the issues of finding the coordinates of the midpoint of a segment if there are coordinates of its extreme points as the initial data. But, before starting to study the issue, we introduce a number of definitions.

Yandex.RTB R-A-339285-1 Definition 1

Section- a straight line connecting two arbitrary points, called the ends of the segment. As an example, let it be points A and B and, accordingly, segment A B.

If segment A B continues in both directions from points A and B, we get line A B. Then the segment A B is a part of the resulting line bounded by points A and B. Segment A B joins points A and B, which are its ends, as well as a set of points lying between. If, for example, we take any arbitrary point K lying between points A and B, we can say that point K lies on the segment A B.

Definition 2

Segment length- the distance between the ends of the segment at a given scale (segment of unit length). The length of the segment A B is denoted as follows: A B.

Definition 3

Midpoint of the segment- a point lying on a segment and equidistant from its ends. If the midpoint of the segment A B is denoted by the point C, then the equality will be true: A C = C B

Initial data: coordinate line O x and non-coincident points on it: A and B. These points correspond to real numbers x A and x B. Point C - midpoint of segment A B: it is necessary to determine the coordinate x C.

Since point C is the midpoint of the segment A B, the following equality will be true: | A C | = | C B | ... The distance between points is determined by the module of the difference between their coordinates, i.e.

| A C | = | C B | ⇔ x C - x A = x B - x C

Then two equalities are possible: x C - x A = x B - x C and x C - x A = - (x B - x C)

From the first equality we derive the formula for the coordinates of the point C: x C = x A + x B 2 (half the sum of the coordinates of the ends of the segment).

From the second equality we get: x A = x B, which is impossible, since in the original data - mismatched points. Thus, the formula for determining the coordinates of the midpoint of the segment A B with ends A (x A) and B (x B):

The resulting formula will be the basis for determining the coordinates of the midpoint of a segment on a plane or in space.

Initial data: rectangular coordinate system on the O x y plane, two arbitrary non-coinciding points with the given coordinates A x A, y A and B x B, y B. Point C is the midpoint of segment A B. It is necessary to determine the coordinates x C and y C for point C.

Let us take for analysis the case when points A and B do not coincide and do not lie on the same coordinate line or a straight line perpendicular to one of the axes. A x, A y; B x, B y and C x, C y - projections of points A, B and C on the coordinate axes (straight lines O x and O y).

According to the construction, the lines A A x, B B x, C C x are parallel; straight lines are also parallel to each other. Together with this, according to Thales' theorem, the equality A C = C B implies the equalities: A x C x = C x B x and A y C y = C y B y, and they in turn indicate that the point C x is the middle of the segment A x B x, and C y is the midpoint of the segment A y B y. And then, based on the formula obtained earlier, we get:

x C = x A + x B 2 and y C = y A + y B 2

The same formulas can be used in the case when points A and B lie on the same coordinate line or a straight line perpendicular to one of the axes. We will not carry out a detailed analysis of this case, we will consider it only graphically:

Summarizing all of the above, coordinates of the midpoint of the segment A B on the plane with the coordinates of the ends A (x A, y A) and B (x B, y B) defined as:

(x A + x B 2, y A + y B 2)

Initial data: coordinate system О x y z and two arbitrary points with given coordinates A (x A, y A, z A) and B (x B, y B, z B). It is necessary to determine the coordinates of the point C, which is the midpoint of the segment A B.

A x, A y, A z; B x, B y, B z and C x, C y, C z - projections of all specified points on the axis of the coordinate system.

According to Thales' theorem, the following equalities are true: A x C x = C x B x, A y C y = C y B y, A z C z = C z B z

Therefore, the points C x, C y, C z are the midpoints of the segments A x B x, A y B y, A z B z, respectively. Then, to determine the coordinates of the midpoint of a segment in space, the following formulas are valid:

x C = x A + x B 2, y c = y A + y B 2, z c = z A + Z B 2

The formulas obtained are also applicable in cases where points A and B lie on one of the coordinate lines; on a straight line perpendicular to one of the axes; in one coordinate plane or a plane perpendicular to one of the coordinate planes.

Determining the coordinates of the midpoint of a segment through the coordinates of the radius vectors of its ends

The formula for finding the coordinates of the midpoint of a segment can also be derived according to the algebraic interpretation of vectors.

Initial data: rectangular Cartesian coordinate system O x y, points with given coordinates A (x A, y A) and B (x B, x B). Point C is the midpoint of segment A B.

According to the geometric definition of actions on vectors, the following equality will be true: O C → = 1 2 · O A → + O B →. Point C in this case is the intersection point of the diagonals of the parallelogram built on the basis of the vectors O A → and O B →, i.e. point of the midpoint of the diagonals. The coordinates of the radius vector of the point are equal to the coordinates of the point, then the equalities are true: O A → = (x A, y A), O B → = (x B, y B). Let's perform some operations on vectors in coordinates and get:

O C → = 1 2 O A → + O B → = x A + x B 2, y A + y B 2

Therefore, point C has coordinates:

x A + x B 2, y A + y B 2

By analogy, a formula is determined for finding the coordinates of the midpoint of a segment in space:

C (x A + x B 2, y A + y B 2, z A + z B 2)

Examples of solving problems for finding the coordinates of the midpoint of a segment

Among the tasks involving the use of the formulas obtained above, there are both those in which the question of calculating the coordinates of the midpoint of a segment is directly involved, and those that involve bringing the given conditions to this question: the term "median" is often used, the goal is to find the coordinates of one from the ends of the segment, and also common problems on symmetry, the solution of which, in general, should also not cause difficulties after studying this topic. Let's consider typical examples.

Example 1

Initial data: on the plane - points with given coordinates A (- 7, 3) and B (2, 4). It is necessary to find the coordinates of the midpoint of the segment A B.

Solution

Let us denote the midpoint of the segment A B by point C. Its coordinates will be defined as the half-sum of the coordinates of the ends of the segment, i.e. points A and B.

x C = x A + x B 2 = - 7 + 2 2 = - 5 2 y C = y A + y B 2 = 3 + 4 2 = 7 2

Answer: coordinates of the middle of the segment A B - 5 2, 7 2.

Example 2

Initial data: the coordinates of the triangle A B C are known: A (- 1, 0), B (3, 2), C (9, - 8). It is necessary to find the length of the median A M.

Solution

1. By the hypothesis of the problem, M is the median, and therefore M is the midpoint of the segment B C. First of all, we find the coordinates of the midpoint of the segment B C, i.e. point M:

x M = x B + x C 2 = 3 + 9 2 = 6 y M = y B + y C 2 = 2 + (- 8) 2 = - 3

1. Since now we know the coordinates of both ends of the median (points A and M), we can use the formula to determine the distance between the points and calculate the length of the median A M:

A M = (6 - (- 1)) 2 + (- 3 - 0) 2 = 58

Answer: 58

Example 3

Initial data: in a rectangular coordinate system of three-dimensional space, a parallelepiped A B C D A 1 B 1 C 1 D 1 is given. The coordinates of the point C 1 (1, 1, 0) are given, and also the point M is defined, which is the midpoint of the diagonal B D 1 and has coordinates M (4, 2, - 4). It is necessary to calculate the coordinates of point A.

Solution

The diagonals of a parallelepiped have an intersection at one point, which is the midpoint of all diagonals. Based on this statement, it can be borne in mind that the point M, known from the conditions of the problem, is the midpoint of the segment A C 1. Based on the formula for finding the coordinates of the midpoint of a segment in space, we find the coordinates of point A: x M = x A + x C 1 2 ⇒ x A = 2 x M - x C 1 = 2 4 - 1 + 7 y M = y A + y C 1 2 ⇒ y A = 2 y M - y C 1 = 2 2 - 1 = 3 z M = z A + z C 1 2 ⇒ z A = 2 z M - z C 1 = 2 (- 4) - 0 = - 8

Answer: coordinates of point A (7, 3, - 8).

If you notice an error in the text, please select it and press Ctrl + Enter

• Midpoint coordinates.

Lesson objectives

• Expand the horizons of concepts.
• Get acquainted with new definitions and recall some already studied.
• Learn to apply the properties of shapes when solving problems.
• Developing - to develop students' attention, perseverance, perseverance, logical thinking, math speech.
• Educational - through a lesson to bring up an attentive attitude to each other, instill the ability to listen to comrades, mutual assistance, independence.

Lesson Objectives

• Test the students' ability to solve problems.

Lesson plan

1. Introduction.
2. Repetition of previously studied material.
3. Midpoint coordinates.
4. Logical tasks.

introduction

Before moving on to the material itself on the topic, I would like to talk a little about the segment, not only as a mathematical definition. Many scientists have tried look at the segment differently, saw something unusual in him. Some talented artists made geometric shapes convey mood and emotions.

There are many theories about how color affects our mood and why.

Color can be felt, it is closely related to our emotions. The color of nature, architecture, plants, clothes that surrounds us gradually influences our mood.

According to experts, the color scale can affect a person.

• Red color can cheer up, give strength.
• Pink the color symbolizes peace and quiet.
• Orange is a warm, restless color that gives energy and uplifting.
• In imperial China yellow was considered such a sacred color that only the emperor could wear yellow clothes. The Egyptians and Mayans considered the sun to be yellow and revered its life-sustaining power. Yellow flowers can invigorate and delight when you are not feeling well.
• Green- healing color. Provides a sense of balance and harmony.
• Blue enhances creativity.
• Purple- the color of thoughtfulness, spirituality and peace. It is associated with intuition and concern for others.
• White usually considered the color of purity and innocence. It is also associated with inspiration, illumination, spirituality, and love.

But how many people have so many opinions. Each has its own truth.

There is also an interesting theory of how it is connected the shape of a line or segment with its character.

Form, like color, is a property of an object. The form- these are the external outlines of a visible object, reflecting its spatial aspects (forma, translated from Latin - external view). Everything that surrounds us has a certain shape. It is the artist's task to understand and depict its constructive structure and semantic content. And we, as spectators, need to be able to read the image, decipher the character and meaning different forms... On a sheet of paper and a computer screen, a shape is formed when a line is closed. Therefore, the nature of the form depends on the nature of the line with which it is formed.

Which of these lines can you express calmness, anger, indifference, excitement, joy?

There can be no definite answer in this case. For example, a barbed line can express anger, schadenfreude, or violent joy bordering on recklessness.

What mood or emotion corresponds to each of these lines?

How does the shape depend on the nature of the line by which it is formed?

Repetition of previously learned material

In space

There are two arbitrary points A1 (x 1; y 1; z 1) and A2 (x 2; y 2; z 2). Then the midpoint of the segment A1A2 will be the point WITH with coordinates x, y, z, where

Division of a segment in a given ratio

If x 1 and y 1 are coordinates of point A, and x 2 and y 2 are coordinates of point B, then the x and y coordinates of point C, dividing segment AB in relation, are determined by the formulas

The area of ​​a triangle according to the known coordinates of its vertices A (x 1, y 1), B (x 2, y 2), C (x 3, y 3) is calculated by the formula.

The number obtained using this formula should be taken in absolute value.

Example # 1

Find the midpoint of line segment AB.

Answer: The coordinates of the midpoint of the segment are (1.5; 2)

Example # 2.

Find the midpoint of line segment AB.

Answer: The coordinates of the midpoint of the segment are (21; 0)

Example No. 3.

Find the coordinates of point C if AC = 5.5 and CB = 19.5.

A (1; 7), B (43; -4)

Answer: Point C coordinates (10.24; 4.58)

Tasks

Problem number 1

Find the midpoint of the line segment DB.

Problem number 2.

Find the midpoint of the CD segment.

How statues are made.

It is said about many famous sculptors that when asked how such wonderful statues can be made, the answer was: "I take a block of marble and cut off everything unnecessary from it." V different books this can be read about Michelangelo, about Thorvaldsen, about Rodin.

In the same way, you can get any bounded flat geometric shape: you need to take some square in which it lies, and then cut off all unnecessary. However, it is not necessary to cut off immediately, but gradually, discarding a piece in the shape of a circle at every step. In this case, the circle itself is thrown out, and its border - the circle - remains in the figure.

At first glance, it seems that only a certain type of figure can be obtained this way. But the whole point is that not one or not two circles are discarded, but an infinite, more precisely, countable set of circles. Any shape can be obtained in this way. To be convinced of this, it is enough to take into account that the set of circles for which both the radius and both coordinates of the center are rational is countable.

And now, to get any shape, it is enough to take a square containing it (a block of marble) and overgrow all circles of the above type, which do not contain a single point of the shape we need. If you throw out circles not from the square, but from the entire plane, then with the described technique you can get unlimited figures.

Questions

1. What is a line segment?
   
2. What does the segment consist of?
3. How can you find the midpoint of a line segment?

List of sources used

1. Kuznetsov A.V., teacher of mathematics (grades 5-9), Kiev
2. "Single State exam 2006. Mathematics. Educational and training materials for training students / Rosobrnadzor, ISOP - M .: Intellect-Center, 2006 "
3. K. Mazur "The solution of the main competition problems in mathematics of the collection edited by M. I. Skanavi"
4. L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev, E. G. Poznyak, I. I. Yudina "Geometry, 7 - 9: a textbook for educational institutions"

Worked on the lesson

A. V. Kuznetsov

S.A. Poturnak

Tatiana Prosnyakova