Each point A of the plane is characterized by its coordinates (x, y). They coincide with the coordinates of the vector 0А , coming out of the point 0 - the origin.

Let A and B be arbitrary points of the plane with coordinates (x 1 y 1) and (x 2, y 2), respectively.

Then the vector AB obviously has the coordinates (x 2 - x 1, y 2 - y 1). It is known that the square of the length of a vector is equal to the sum of the squares of its coordinates. Therefore, the distance d between points A and B, or, what is the same, the length of the vector AB, is determined from the condition

d 2 \u003d (x 2 - x 1) 2 + (y 2 - y 1) 2.

$$ d = \sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) $$

The resulting formula allows you to find the distance between any two points of the plane, if only the coordinates of these points are known

Each time, speaking about the coordinates of one or another point of the plane, we have in mind a well-defined coordinate system x0y. In general, the coordinate system on the plane can be chosen in different ways. So, instead of the x0y coordinate system, we can consider the xִy coordinate system, which is obtained as a result of the rotation of the old coordinate axes around the starting point 0 counter-clockwise arrows on the corner α .

If some point of the plane in the x0y coordinate system had coordinates (x, y), then in the new x-y coordinate system it will have other coordinates (x, y).

As an example, consider a point M located on the 0x axis and spaced from the point 0 at a distance equal to 1.

Obviously, in the x0y coordinate system, this point has coordinates (cos α , sin α ), and in the coordinate system хִу the coordinates are (1,0).

The coordinates of any two points of the plane A and B depend on how the coordinate system is set in this plane. But the distance between these points does not depend on how the coordinate system is specified .

Other materials

THEORETICAL QUESTIONS

ANALYTICAL GEOMETRY ON THE PLANE

1. Coordinate method: number line, coordinates on the line; rectangular (Cartesian) coordinate system on the plane; polar coordinates.

Let's take a look at a straight line. Let's choose a direction on it (then it will become an axis) and some point 0 (the origin). A straight line with a chosen direction and origin is called coordinate line(in this case, we assume that the scale unit is selected).

Let M is an arbitrary point on the coordinate line. Let's put in accordance with the point M real number x, equal to the value OM segment : x=OM. Number x called the coordinate of the point M.

Thus, each point of the coordinate line corresponds to a certain real number - its coordinate. The converse is also true, each real number x corresponds to some point on the coordinate line, namely such a point M, whose coordinate is x. This correspondence is called mutually unambiguous.

So, real numbers can be represented by points of the coordinate line, i.e. the coordinate line serves as an image of the set of all real numbers. Therefore, the set of all real numbers is called number line, and any number is a point of this line. Near a point on a number line, a number is often indicated - its coordinate.

Rectangular (or Cartesian) coordinate system on a plane.

Two mutually perpendicular axes About x and About y having common beginning O and the same scale unit, form rectangular (or Cartesian) coordinate system on the plane.

Axis OH called the x-axis, the axis OY- the y-axis. Dot O the intersection of the axes is called the origin. The plane in which the axes are located OH and OY, is called the coordinate plane and is denoted Oh xy.

So, a rectangular coordinate system on a plane establishes a one-to-one correspondence between the set of all points of the plane and the set of pairs of numbers, which makes it possible to apply algebraic methods when solving geometric problems. The coordinate axes divide the plane into 4 parts, they are called quarters, square or coordinate angles.

Polar coordinates.

The polar coordinate system consists of some point O called pole, and the beam emanating from it OE called polar axis. In addition, the scale unit for measuring the lengths of segments is set. Let a polar coordinate system be given and let M is an arbitrary point of the plane. Denote by R– point distance M from the point O, and through φ - the angle by which the beam is rotated counterclockwise the polar axis to coincide with the beam OM.

polar coordinates points M call the numbers R and φ . Number R considered as the first coordinate and called polar radius, number φ - the second coordinate is called polar angle.

Dot M with polar coordinates R and φ are designated as follows: М( ;φ). Let's establish a connection between the polar coordinates of a point and its rectangular coordinates.
In this case, we will assume that the origin of the rectangular coordinate system is at the pole, and the positive semi-axis of the abscissa coincides with the polar axis.

Let the point M have rectangular coordinates X and Y and polar coordinates R and φ .

(1)

Proof.

Drop from the dots M 1 and M 2 perpendiculars M 1 V and M 1 A,. because (x 2 ; y 2). By theory, if M 1 (x 1) and M 2 (x 2) are any two points and α is the distance between them, then α = ‌‌‌‍‌‌|x 2 - x 1 | .

Distance from point to point is the length of the segment connecting these points, on a given scale. Thus, when we are talking distance measurement, you need to know the scale (unit of length) in which the measurements will be taken. Therefore, the problem of finding the distance from a point to a point is usually considered either on a coordinate line or in a rectangular Cartesian coordinate system on a plane or in three-dimensional space. In other words, most often you have to calculate the distance between points by their coordinates.

In this article, we, firstly, recall how the distance from a point to a point on a coordinate line is determined. Next, we obtain formulas for calculating the distance between two points of a plane or space according to given coordinates. Finally, let's take a closer look at the solutions characteristic examples and tasks.

Page navigation.

The distance between two points on a coordinate line.

Let's first define the notation. The distance from point A to point B will be denoted as .

From this we can conclude that the distance from point A with coordinate to point B with coordinate is equal to the modulus of the difference in coordinates, that is, for any arrangement of points on the coordinate line.

Distance from a point to a point on a plane, formula.

Let's get a formula for calculating the distance between points and given in a rectangular Cartesian coordinate system on a plane.

Depending on the location of points A and B, the following options are possible.

If points A and B coincide, then the distance between them is zero.

If points A and B lie on a straight line perpendicular to the x-axis, then the points and coincide, and the distance is equal to the distance. In the previous paragraph, we found out that the distance between two points on the coordinate line is equal to the modulus of the difference between their coordinates, therefore, . Consequently, .

Similarly, if points A and B lie on a straight line perpendicular to the y-axis, then the distance from point A to point B is found as .

In this case, the triangle ABC is rectangular in construction, and and . By the Pythagorean theorem we can write the equality , whence .

Let's summarize all the results: the distance from a point to a point on a plane is found through the coordinates of the points by the formula .

The resulting formula for finding the distance between points can be used when points A and B coincide or lie on a straight line perpendicular to one of the coordinate axes. Indeed, if A and B are the same, then . If points A and B lie on a straight line perpendicular to the Ox axis, then . If A and B lie on a straight line perpendicular to the Oy axis, then .

Distance between points in space, formula.

Let us introduce a rectangular coordinate system Оxyz in space. Get the formula for finding the distance from a point to the point .

In general, points A and B do not lie in a plane parallel to one of the coordinate planes. Let's draw through points A and B in the plane perpendicular to the coordinate axes Ox, Oy and Oz. The intersection points of these planes with the coordinate axes will give us the projections of points A and B on these axes. Denote the projections .

The desired distance between points A and B is the diagonal of the rectangular parallelepiped shown in the figure. By construction, the dimensions of this parallelepiped are and . In the course of geometry high school it was proved that the square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions, therefore, . Based on the information of the first section of this article, we can write the following equalities, therefore,

where we get formula for finding the distance between points in space .

This formula is also valid if points A and B

• match;
• belong to one of the coordinate axes or a straight line parallel to one of the coordinate axes;
• belong to one of the coordinate planes or a plane parallel to one of the coordinate planes.

Finding the distance from point to point, examples and solutions.

So, we got the formulas for finding the distance between two points of the coordinate line, plane and three-dimensional space. It's time to consider the solutions of typical examples.

The number of tasks in which the final step is to find the distance between two points according to their coordinates is truly enormous. Full Review such examples are beyond the scope of this article. Here we restrict ourselves to examples in which the coordinates of two points are known and it is required to calculate the distance between them.

Let a rectangular coordinate system be given.

Theorem 1.1. For any two points M 1 (x 1; y 1) and M 2 (x 2; y 2) of the plane, the distance d between them is expressed by the formula

Proof. Let us drop from the points M 1 and M 2 the perpendiculars M 1 B and M 2 A, respectively

on the Oy and Ox axes and denote by K the point of intersection of the lines M 1 B and M 2 A (Fig. 1.4). The following cases are possible:

1) Points M 1, M 2 and K are different. Obviously, the point K has coordinates (x 2; y 1). It is easy to see that M 1 K = ôx 2 – x 1 ô, M 2 K = ôy 2 – y 1 ô. Because ∆M 1 KM 2 is rectangular, then by the Pythagorean theorem d = M 1 M 2 = = .

2) Point K coincides with point M 2, but is different from point M 1 (Fig. 1.5). In this case y 2 = y 1

and d \u003d M 1 M 2 \u003d M 1 K \u003d ôx 2 - x 1 ô \u003d =

3) The point K coincides with the point M 1, but is different from the point M 2. In this case x 2 = x 1 and d =

M 1 M 2 \u003d KM 2 \u003d ôy 2 - y 1 ô \u003d = .

4) Point M 2 coincides with point M 1. Then x 1 \u003d x 2, y 1 \u003d y 2 and

d \u003d M 1 M 2 \u003d O \u003d.

The division of the segment in this respect.

Let an arbitrary segment M 1 M 2 be given on the plane and let M be any point of this

segment other than the point M 2 (Fig. 1.6). The number l defined by the equality l = , is called attitude, in which the point M divides the segment M 1 M 2.

Theorem 1.2. If the point M (x; y) divides the segment M 1 M 2 in relation to l, then the coordinates of this are determined by the formulas

x = , y = , (4)

where (x 1; y 1) are the coordinates of the point M 1, (x 2; y 2) are the coordinates of the point M 2.

Proof. Let us prove the first of formulas (4). The second formula is proved similarly. Two cases are possible.

x = x 1 = = = .

2) The straight line M 1 M 2 is not perpendicular to the Ox axis (Fig. 1.6). Let's drop the perpendiculars from the points M 1 , M, M 2 to the axis Ox and denote the points of their intersection with the axis Ox respectively P 1 , P, P 2 . According to the proportional segments theorem =l.

Because P 1 P \u003d ôx - x 1 ô, PP 2 \u003d ôx 2 - xô and the numbers (x - x 1) and (x 2 - x) have the same sign (for x 1< х 2 они положительны, а при х 1 >x 2 are negative), then

l == ,

x - x 1 \u003d l (x 2 - x), x + lx \u003d x 1 + lx 2,

x = .

Corollary 1.2.1. If M 1 (x 1; y 1) and M 2 (x 2; y 2) are two arbitrary points and the point M (x; y) is the midpoint of the segment M 1 M 2, then

x = , y = (5)

Proof. Since M 1 M = M 2 M, then l = 1 and by formulas (4) we obtain formulas (5).

Area of ​​a triangle.

Theorem 1.3. For any points A (x 1; y 1), B (x 2; y 2) and C (x 3; y 3) that do not lie on the same

straight line, the area S of triangle ABC is expressed by the formula

S \u003d ô (x 2 - x 1) (y 3 - y 1) - (x 3 - x 1) (y 2 - y 1) ô (6)

Proof. The area ∆ ABC shown in fig. 1.7, we calculate as follows

S ABC \u003d S ADEC + S BCEF - S ABFD.

Calculate the area of ​​the trapezoid:

S-ADEC=
,

SBCEF=

S ABFD =

Now we have

S ABC \u003d ((x 3 - x 1) (y 3 + y 1) + (x 3 - x 2) (y 3 + y 2) - (x 2 - -x 1) (y 1 + y 2)) \u003d (x 3 y 3 - x 1 y 3 + x 3 y 1 - x 1 y 1 + + x 2 y 3 - -x 3 y 3 + x 2 y 2 - x 3 y 2 - x 2 y 1 + x 1 y 1 - x 2 y 2 + x 1 y 2) \u003d (x 3 y 1 - x 3 y 2 + x 1 y 2 - x 2 y 1 + x 2 y 3 -

X 1 y 3) \u003d (x 3 (y 1 - y 2) + x 1 y 2 - x 1 y 1 + x 1 y 1 - x 2 y 1 + y 3 (x 2 - x 1)) \u003d (x 1 (y 2 - y 1) - x 3 (y 2 - y 1) + + y 1 (x 1 - x 2) - y 3 (x 1 - x 2)) \u003d ((x 1 - x 3) ( y 2 - y 1) + (x 1 - x 2) (y 1 - y 3)) \u003d ((x 2 - x 1) (y 3 - y 1) -

- (x 3 - x 1) (y 2 - y 1)).

For another location ∆ ABC, formula (6) is proved similarly, but it can be obtained with the “-” sign. Therefore, in the formula (6) put the sign of the modulus.

Lecture 2

The equation of a straight line on a plane: the equation of a straight line with the main coefficient, the general equation of a straight line, the equation of a straight line in segments, the equation of a straight line passing through two points. Angle between lines, conditions of parallelism and perpendicularity of lines on a plane.

2.1. Let a rectangular coordinate system and some line L be given on the plane.

Definition 2.1. An equation of the form F(x;y) = 0 relating the variables x and y is called line equation L(in a given coordinate system) if this equation is satisfied by the coordinates of any point lying on the line L, and not by the coordinates of any point not lying on this line.

Examples of equations of lines on a plane.

1) Consider a straight line parallel to the axis Oy of a rectangular coordinate system (Fig. 2.1). Let us denote by the letter A the point of intersection of this line with the axis Ox, (a; o) ─ its or-

dinats. The equation x = a is the equation of the given line. Indeed, this equation is satisfied by the coordinates of any point M(a;y) of this line and the coordinates of any point not lying on the line are satisfied. If a = 0, then the line coincides with the Oy axis, which has the equation x = 0.

2) The equation x - y \u003d 0 defines the set of points in the plane that make up the bisectors of I and III coordinate angles.

3) The equation x 2 - y 2 \u003d 0 is the equation of two bisectors of coordinate angles.

4) The equation x 2 + y 2 = 0 defines a single point O(0;0) on the plane.

5) The equation x 2 + y 2 \u003d 25 is the equation of a circle of radius 5 centered at the origin.

In this article, we will consider ways to determine the distance from a point to a point theoretically and on the example of specific tasks. Let's start with some definitions.

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

Distance between points- this is the length of the segment connecting them, in the existing scale. It is necessary to set the scale in order to have a unit of length for measurement. Therefore, basically the problem of finding the distance between points is solved by using their coordinates on the coordinate line, in the coordinate plane or three-dimensional space.

Initial data: the coordinate line O x and an arbitrary point A lying on it. One real number is inherent in any point of the line: let this be a certain number for point A xA, it is the coordinate of point A.

In general, we can say that the estimation of the length of a certain segment occurs in comparison with the segment taken as a unit of length on a given scale.

If point A corresponds to an integer real number, having set aside successively from point O to a point along a straight line O A segments - units of length, we can determine the length of segment O A by the total number of pending single segments.

For example, point A corresponds to the number 3 - in order to get to it from point O, it will be necessary to set aside three unit segments. If point A has a coordinate of - 4, single segments are plotted in a similar way, but in a different, negative direction. Thus, in the first case, the distance O A is 3; in the second case, O A \u003d 4.

If point A has a rational number as a coordinate, then from the origin (point O) we set aside an integer number of unit segments, and then its necessary part. But geometrically it is not always possible to make a measurement. For example, it seems difficult to put aside the coordinate direct fraction 4 111 .

In the above way, it is completely impossible to postpone an irrational number on a straight line. For example, when the coordinate of point A is 11 . In this case, it is possible to turn to abstraction: if the given coordinate of point A is greater than zero, then O A \u003d x A (the number is taken as a distance); if the coordinate is less than zero, then O A = - x A . In general, these statements are true for any real number x A .

Summarizing: the distance from the origin to the point, which corresponds to a real number on the coordinate line, is equal to:

• 0 if the point is the same as the origin;

• x A if x A > 0 ;

• - x A if x A< 0 .

In this case, it is obvious that the length of the segment itself cannot be negative, therefore, using the modulus sign, we write the distance from the point O to the point A with the coordinate x A: O A = x A

The correct statement would be: the distance from one point to another will be equal to the modulus of the difference in coordinates. Those. for points A and B lying on the same coordinate line at any location and having, respectively, the coordinates x A and x B: A B = x B - x A .

Initial data: points A and B lying on a plane in a rectangular coordinate system O x y with given coordinates: A (x A , y A) and B (x B , y B) .

Let's draw perpendiculars to the coordinate axes O x and O y through points A and B and get the projection points as a result: A x , A y , B x , B y . Based on the location of points A and B, the following options are further possible:

If points A and B coincide, then the distance between them is zero;

If points A and B lie on a straight line perpendicular to the O x axis (abscissa axis), then the points and coincide, and | A B | = | A y B y | . Since the distance between the points is equal to the modulus of the difference between their coordinates, then A y B y = y B - y A , and, therefore, A B = A y B y = y B - y A .

If points A and B lie on a straight line perpendicular to the O y axis (y-axis) - by analogy with the previous paragraph: A B = A x B x = x B - x A

If points A and B do not lie on a straight line perpendicular to one of the coordinate axes, we find the distance between them by deriving the calculation formula:

We see that the triangle A B C is right-angled by construction. In this case, A C = A x B x and B C = A y B y . Using the Pythagorean theorem, we compose the equality: A B 2 = A C 2 + B C 2 ⇔ A B 2 = A x B x 2 + A y B y 2 , and then transform it: A B = A x B x 2 + A y B y 2 = x B - x A 2 + y B - y A 2 = (x B - x A) 2 + (y B - y A) 2

Let's form a conclusion from the result obtained: the distance from point A to point B on the plane is determined by the calculation by the formula using the coordinates of these points

A B = (x B - x A) 2 + (y B - y A) 2

The resulting formula also confirms the previously formed statements for the cases of coincidence of points or situations when the points lie on straight lines perpendicular to the axes. So, for the case of the coincidence of points A and B, the equality will be true: A B = (x B - x A) 2 + (y B - y A) 2 = 0 2 + 0 2 = 0

For the situation when points A and B lie on a straight line perpendicular to the x-axis:

A B = (x B - x A) 2 + (y B - y A) 2 = 0 2 + (y B - y A) 2 = y B - y A

For the case when points A and B lie on a straight line perpendicular to the y-axis:

A B = (x B - x A) 2 + (y B - y A) 2 = (x B - x A) 2 + 0 2 = x B - x A

Initial data: rectangular coordinate system O x y z with arbitrary points lying on it with given coordinates A (x A , y A , z A) and B (x B , y B , z B) . It is necessary to determine the distance between these points.

Consider the general case when points A and B do not lie in a plane parallel to one of the coordinate planes. Draw through points A and B planes perpendicular to the coordinate axes, and get the corresponding projection points: A x , A y , A z , B x , B y , B z

The distance between points A and B is the diagonal of the resulting box. According to the construction of the measurement of this box: A x B x , A y B y and A z B z

From the course of geometry it is known that the square of the diagonal of a parallelepiped is equal to the sum of the squares of its dimensions. Based on this statement, we obtain the equality: A B 2 \u003d A x B x 2 + A y B y 2 + A z B z 2

Using the conclusions obtained earlier, we write the following:

A x B x = x B - x A , A y B y = y B - y A , A z B z = z B - z A

Let's transform the expression:

A B 2 = A x B x 2 + A y B y 2 + A z B z 2 = x B - x A 2 + y B - y A 2 + z B - z A 2 = = (x B - x A) 2 + (y B - y A) 2 + z B - z A 2

Final formula for determining the distance between points in space will look like this:

A B = x B - x A 2 + y B - y A 2 + (z B - z A) 2

The resulting formula is also valid for cases where:

The dots match;

They lie on the same coordinate axis or on a straight line parallel to one of the coordinate axes.

Examples of solving problems for finding the distance between points

Example 1

Initial data: a coordinate line and points lying on it with given coordinates A (1 - 2) and B (11 + 2) are given. It is necessary to find the distance from the reference point O to point A and between points A and B.

Solution

1. The distance from the reference point to the point is equal to the module of the coordinate of this point, respectively O A \u003d 1 - 2 \u003d 2 - 1
2. The distance between points A and B is defined as the modulus of the difference between the coordinates of these points: A B = 11 + 2 - (1 - 2) = 10 + 2 2

Answer: O A = 2 - 1, A B = 10 + 2 2

Example 2

Initial data: given a rectangular coordinate system and two points lying on it A (1 , - 1) and B (λ + 1 , 3) ​​. λ is some real number. It is necessary to find all values ​​of this number for which the distance A B will be equal to 5.

Solution

To find the distance between points A and B, you must use the formula A B = (x B - x A) 2 + y B - y A 2

Substituting real values coordinates, we get: A B = (λ + 1 - 1) 2 + (3 - (- 1)) 2 = λ 2 + 16

And also we use the existing condition that A B = 5 and then the equality will be true:

λ 2 + 16 = 5 λ 2 + 16 = 25 λ = ± 3

Answer: A B \u003d 5 if λ \u003d ± 3.

Example 3

Initial data: a three-dimensional space in a rectangular coordinate system O x y z and the points A (1 , 2 , 3) ​​and B - 7 , - 2 , 4 lying in it are given.

Solution

To solve the problem, we use the formula A B = x B - x A 2 + y B - y A 2 + (z B - z A) 2

Substituting the real values, we get: A B = (- 7 - 1) 2 + (- 2 - 2) 2 + (4 - 3) 2 = 81 = 9

Answer: | A B | = 9

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