In this article, we will consider ways to determine the distance from point to point theoretically and using the example of specific tasks. And to begin with, let's introduce some definitions.
Definition 1
Distance between points Is the length of the segment connecting them, on the available 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 a coordinate line, in a coordinate plane or three-dimensional space.
Initial data: coordinate line O x and an arbitrary point A lying on it. Any point of the line has one real number: let it be a certain number for point A x A,it is also 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 in a given scale.
If point A corresponds to an integer real number, sequentially postponing from point O to point along the straight line OA segments - units of length, we can determine the length of the segment O A by the total number of pending unit segments.
For example, point A corresponds to the number 3 - to get to it from point O, you will need to postpone three unit segments. If point A has a coordinate - 4 - unit segments are plotted in the same way, but in a different, negative direction. Thus, in the first case, the distance O And is equal to 3; in the second case, О А \u003d 4.
If point A has a rational number as a coordinate, then from the origin (point O) we postpone an integer number of unit segments, and then its necessary part. But it is not always geometrically possible to make a measurement. For example, it seems difficult to postpone the fraction 4 111 on the coordinate straight line.
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 the distance); if the coordinate is less than zero, then O A \u003d - x A. In general, these statements are true for any real number x A.
To summarize: the distance from the origin to the point corresponding to a real number on the coordinate line is:
- 0 if the point coincides with the origin;
- x A if x A\u003e 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 down the distance from point O to point A with the coordinate x A: O A \u003d x A
The statement will be true: the distance from one point to another will be equal to the modulus of the coordinate difference.Those. for points A and B lying on the same coordinate line for any of their locations and having coordinates, respectively x A and x B: A B \u003d 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 us draw perpendiculars to the coordinate axes O x and O y through points A and B and obtain 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 points and coincide, and | A B | \u003d | А 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 \u003d y B - y A, and therefore A B \u003d A y B y \u003d y B - y A.
If points A and B lie on a straight line perpendicular to the O y axis (ordinate axis) - by analogy with the previous paragraph: A B \u003d A x B x \u003d 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, deriving the calculation formula:
We see that triangle ABC is rectangular in construction. Moreover, A C \u003d A x B x and B C \u003d A y B y. Using the Pythagorean theorem, we compose the equality: AB 2 \u003d AC 2 + BC 2 ⇔ AB 2 \u003d A x B x 2 + A y B y 2, and then we transform it: AB \u003d A x B x 2 + A y B y 2 \u003d x B - x A 2 + y B - y A 2 \u003d (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 calculation using the formula using the coordinates of these points
A B \u003d (x B - x A) 2 + (y B - y A) 2
The resulting formula also confirms the previously formed statements for cases of coincidence of points or situations when the points lie on straight lines perpendicular to the axes. So, for the case of coincidence of points A and B, the equality will be true: A B \u003d (x B - x A) 2 + (y B - y A) 2 \u003d 0 2 + 0 2 \u003d 0
For a situation where points A and B lie on a straight line perpendicular to the abscissa axis:
A B \u003d (x B - x A) 2 + (y B - y A) 2 \u003d 0 2 + (y B - y A) 2 \u003d y B - y A
For the case when points A and B lie on a straight line perpendicular to the ordinate axis:
A B \u003d (x B - x A) 2 + (y B - y A) 2 \u003d (x B - x A) 2 + 0 2 \u003d 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 obtain 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 parallelepiped. According to the construction of the measurement of this parallelepiped: A x B x, A y B y and A z B z
It is known from the geometry course that the square of the diagonal of a parallelepiped is equal to the sum of the squares of its measurements. 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 \u003d x B - x A, A y B y \u003d y B - y A, A z B z \u003d z B - z A
Let's transform the expression:
AB 2 \u003d A x B x 2 + A y B y 2 + A z B z 2 \u003d x B - x A 2 + y B - y A 2 + z B - z A 2 \u003d \u003d (x B - x A) 2 + (y B - y A) 2 + z B - z A 2
The final formula for determining the distance between points in space will look like this:
A B \u003d x B - x A 2 + y B - y A 2 + (z B - z A) 2
The resulting formula is also valid for cases when:
The points are the same;
They lie on the same coordinate axis or a straight line parallel to one of the coordinate axes.
Examples of solving problems on finding the distance between points
Example 1Initial data: given a coordinate line and points lying on it with the given coordinates A (1 - 2) and B (11 + 2). It is necessary to find the distance from the origin point O to point A and between points A and B.
Decision
- The distance from the origin to the point is equal to the modulus of the coordinate of this point, respectively O A \u003d 1 - 2 \u003d 2 - 1
- The distance between points A and B is defined as the modulus of the difference between the coordinates of these points: A B \u003d 11 + 2 - (1 - 2) \u003d 10 + 2 2
Answer: O A \u003d 2 - 1, A B \u003d 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 the values \u200b\u200bof this number at which the distance AB will be equal to 5.
Decision
To find the distance between points A and B, use the formula A B \u003d (x B - x A) 2 + y B - y A 2
Substituting the real values \u200b\u200bof the coordinates, we get: A B \u003d (λ + 1 - 1) 2 + (3 - (- 1)) 2 \u003d λ 2 + 16
And we also use the existing condition that AB \u003d 5 and then the equality will be true:
λ 2 + 16 \u003d 5 λ 2 + 16 \u003d 25 λ \u003d ± 3
Answer: AB \u003d 5, if λ \u003d ± 3.
Example 3
Initial data: given 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.
Decision
To solve the problem, we use the formula A B \u003d x B - x A 2 + y B - y A 2 + (z B - z A) 2
Substituting real values, we get: A B \u003d (- 7 - 1) 2 + (- 2 - 2) 2 + (4 - 3) 2 \u003d 81 \u003d 9
Answer: | A B | \u003d 9
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Using coordinates, they determine the location of an object on the globe. Coordinates are indicated by latitude and longitude. Latitudes are measured from the equatorial line on both sides. Latitudes are positive in the Northern Hemisphere and negative in the Southern Hemisphere. Longitude is measured from the initial meridian either to the east or to the west, respectively, it turns out either east longitude or west.According to the generally accepted position, the meridian, which passes through the old Greenwich Observatory in Greenwich, is taken as the initial one. The geographic coordinates of the location can be obtained using a GPS navigator. This device receives signals from the satellite positioning system in the WGS-84 coordinate system, the same for the whole world.
Navigator models differ in manufacturers, functionality and interface. Currently, some models of cell phones have built-in GPS navigators. But any model can record and store the coordinates of the point.
Distance between GPS coordinates
To solve practical and theoretical problems in some industries, it is necessary to be able to determine the distances between points by their coordinates. There are several ways to do this. The canonical form of representation of geographic coordinates: degrees, minutes, seconds.For example, you can determine the distance between the following coordinates: point # 1 - latitude 55 ° 45′07 ″ N, longitude 37 ° 36′56 ″ E; point No. 2 - latitude 58 ° 00′02 ″ N, longitude 102 ° 39′42 ″ E
The easiest way is to use a calculator to calculate the distance between two points. In the browser search engine, you must set the following search parameters: online - to calculate the distance between two coordinates. In the online calculator, latitude and longitude values \u200b\u200bare entered into the query fields for the first and second coordinates. When calculating the online calculator, the result was 3,800,619 m.
The next method is more time consuming, but also more visual. You must use any available mapping or navigation software. The programs in which you can create points by coordinates and measure the distances between them include the following applications: BaseCamp (a modern analogue of the MapSource program), Google Earth, SAS.Planet.
All of the above programs are available to any network user. For example, to calculate the distance between two coordinates in Google Earth, you need to create two placemarks with the coordinates of the first point and the second point. Then, using the Ruler tool, you need to connect the first and second marks with a line, the program will automatically display the measurement result and show the path on the satellite image of the Earth.
In the case of the example given above, the Google Earth program returned the result - the distance between point number 1 and point number 2 is 3,817,353 m.
Why there is an error in determining the distance
All distance calculations between coordinates are based on arc length calculations. The radius of the Earth is involved in calculating the arc length. But since the shape of the Earth is close to an oblate ellipsoid, the radius of the Earth at certain points is different. To calculate the distance between the coordinates, the average value of the Earth's radius is taken, which gives an error in measurement. The greater the measured distance, the greater the error.Theorem 1. For any two points and a plane, the distance between them is expressed by the formula:
For example, if points and are given, then the distance between them:
2. Area of \u200b\u200ba triangle.
Theorem 2.
For any points
not lying on one straight line, the area of \u200b\u200ba triangle is expressed by the formula:
For example, let's find the area of \u200b\u200ba triangle formed by points, and.
Comment. If the area of \u200b\u200bthe triangle is zero, it means that the points are collinear.
3. Division of a segment in a given ratio.
Let an arbitrary segment be given on the plane and let
–Any point of this segment other than the end points. The number defined by equality is called attitude,in which the point divides the segment.
The problem of dividing a segment in a given relation is that for a given ratio and given coordinates of points
and find the coordinates of the point.
Theorem 3.
If a point divides a segment
in a relationship
,
then the coordinates of this point are determined by the formulas: 
(1.3), where are the coordinates of the point, are the coordinates of the point.
Corollary: If is the middle of the segment
, where and, then (1.4) (since).
For example. Points and are given. Find the coordinates of a point that is two times closer to than to
Solution: The desired point divides the segment
in regards as 
then 
,
, got
Polar coordinates
The most important after the rectangular coordinate system is the polar coordinate system. It consists of some point called pole, and the ray emanating from it - polar axis... In addition, the scale unit is set for measuring the lengths of the segments. 
Let a polar coordinate system be given and let be an arbitrary point of the plane. Let be the distance from the point
to point; is the angle by which the polar axis must be rotated to align with the beam.
Polar coordinates of a point are called numbers. In this case, the number is considered the first coordinate and is called polar radius, the number is the second coordinate and is called polar angle.
It is indicated. The polar radius can be any non-negative value :. The polar angle is generally considered to vary within the following ranges: However, in some cases it is necessary to determine the angles counted clockwise from the polar axis.
The relationship between the polar coordinates of a point and its rectangular coordinates.
We will assume that the origin of the rectangular coordinate system is at the pole, and the positive semiaxis of the abscissa coincides with the polar axis. 
Let - in a rectangular coordinate system and - in a polar coordinate system. Defined - right-angled triangle c. Then (1.5). These formulas express rectangular coordinates in terms of polar coordinates.
On the other hand, by the Pythagorean theorem and
(1.6) - these formulas express polar coordinates through rectangular ones.
Note that the formula defines two polar angle values, since. Of these two values \u200b\u200bof the angle, the one at which the equalities are satisfied is selected.
For example, let's find the polar coordinates of a point .. or, since I quarter.
Example 1: Find a point symmetrical to a point 
With respect to the bisector of the first coordinate angle.
Decision:
Let's draw through the point AND straight l 1 perpendicular to the bisector l the first coordinate angle. Let be . On a straight line l 1 postpone the segment CA 1 , equal to segment AC. Rectangular triangles ASO and AND 1 COequal to each other (on two legs). Hence it follows that | OA| = |OA 1 |. Triangles ADO and OEA 1 are also equal to each other (hypotenuse and acute angle). We conclude that | AD| \u003d | OE | = 4, | OD | \u003d | EA 1 | = 2, i.e. point has coordinates x \u003d 4, y \u003d -2, those. AND 1 (4;-2).
Note that the general statement holds: the point A 1, symmetrical to the point relative to the bisector of the first and third coordinate angles, has coordinates, that is .
Example 2: Find the point at which the straight line passing through the points and , will cross the axis Oh.
Decision:
The coordinates of the desired point FROM there is ( x; 0). And since the points AND, INand FROM lie on one straight line, then the condition (x 2 -x 1 ) (y 3 -y 1 ) - (x 3 -x 1 ) (y 2 -y 1 ) = 0 (formula (1.2), the area of \u200b\u200ba triangle ABC is zero!), where are the coordinates of the point AND, - points IN, - points FROM... We get, i.e., ,. Hence the point FROM has coordinates ,, i.e.
Example 3:Points, are specified in the polar coordinate system. To find: and) distance between points and ; b) the area of \u200b\u200bthe triangle OM 1 M 2 (ABOUT - pole).
Decision:
a) We use formulas (1.1) and (1.5):
i.e, .
b) using the formula for the area of \u200b\u200ba triangle with sides and and b and the angle between them (), we find the area of \u200b\u200bthe triangle OM 1 M 2 . .
Point to point distance is the length of the line segment connecting these points, at a given scale. Thus, when it comes to measuring distance, you need to know the scale (unit of length) in which the measurements will be carried out. Therefore, the problem of finding the distance from point to 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 it is necessary to calculate the distance between points by their coordinates.
In this article, we, firstly, recall how the distance from point to point on the coordinate line is determined. Next, we will obtain formulas for calculating the distance between two points of a plane or space according to given coordinates. In conclusion, let us consider in detail the solutions of typical examples and tasks.
Page navigation.
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.
Hence we can conclude that distance from point A with coordinate to point B with coordinate is equal to the modulus of the difference of coordinates, i.e, 
at any location of points on the coordinate line.
Distance from point to 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 the 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 abscissa 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, 
... Hence, .
Similarly, if points A and B lie on a straight line perpendicular to the ordinate, then the distance from point A to point B is found as.
In this case, triangle ABC is rectangular in construction, and 
and. By pythagorean theorem we can write equality, whence.
Let's summarize all the results obtained: the distance from a point to a point on the 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 coincide, 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's introduce a rectangular coordinate system Oxyz in space. Let's get the formula for finding the distance from the point 
to the point 
.
In general, points A and B do not lie in a plane parallel to one of the coordinate planes. Let us draw through points A and B planes perpendicular to the coordinate axes Ox, Oy and Oz. The points of intersection of these planes with the coordinate axes will give us the projection of points A and B on these axes. We denote 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 a high school geometry course, 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 in the first section of this article, we can write the following equalities, therefore,
whence 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 formulas for finding the distance between two points of the coordinate line, plane and three-dimensional space. It's time to consider solutions to typical examples.
The number of problems in the solution of which the final step is to find the distance between two points by their coordinates is truly enormous. A complete overview of such examples is beyond the scope of this article. Here we will restrict ourselves to examples in which the coordinates of two points are known and it is required to calculate the distance between them.
Maths
§2. Point coordinates on a plane
3. Distance between two points.
You and I can now talk about points in the language of numbers. For example, we no longer need to explain: take a point that is three units to the right of the axis and five units below the axis. Suffice it to say simply: take the point.
We have already said that this creates certain advantages. So, we can send a drawing made of dots by telegraph, tell it to a computer, which does not understand the drawings at all, but understands numbers well.
In the previous section, we specified some sets of points on the plane with the help of relations between numbers. Now let's try to consistently translate other geometric concepts and facts into the language of numbers.
We'll start with a simple and common task.
Find the distance between two points of the plane.
Decision:
As always, we assume that the points are given by their coordinates, and then our task is to find a rule by which we can calculate the distance between the points, knowing their coordinates. When deriving this rule, of course, it is allowed to resort to the drawing, but the rule itself should not contain any references to the drawing, but should only show what actions and in what order must be performed with these numbers - the coordinates of the points in order to get the required number - the distance between dots.
Perhaps some of the readers will find this approach to solving the problem strange and far-fetched. What is easier, they say, the points are given, even if they are coordinates. Draw these points, take a ruler and measure the distance between them.
This method is sometimes not so bad. However, imagine again that you are dealing with a computer. She does not have a ruler, and she does not draw, but she knows how to count so quickly that it does not pose any problem for her at all. Note that our task is set so that the rule for calculating the distance between two points consists of commands that the machine can execute.
It is better to solve the problem posed first for a particular case when one of these points lies at the origin. Start with a few numerical examples: find the distance from the origin of the points; and.
Indication. Use the Pythagorean theorem.
Now write a general formula for calculating the distance of a point from the origin.
The distance of a point from the origin is determined by the formula:
Obviously, the rule expressed by this formula satisfies the above conditions. In particular, it can be used for calculations on machines that can multiply numbers, add them, and extract square roots.
Now let's solve the general problem
Given two points of the plane and find the distance between them.
Decision:
Let us denote by,,, projections of points and on the coordinate axis.
The point of intersection of the lines and will be designated by a letter. From a right-angled triangle, according to the Pythagorean theorem, we obtain:
But the length of the segment is equal to the length of the segment. Points and, lie on the axis and have coordinates and, respectively. According to the formula obtained in paragraph 3 of paragraph 2, the distance between them is equal.
Arguing similarly, we find that the length of the segment is equal to. Substituting the found values \u200b\u200band into the formula we get.