The article below will cover the issues of finding the coordinates of the middle of the segment in the presence of the coordinates of its extreme points as initial data. But, before proceeding to the study of the issue, we introduce a number of definitions.

Definition 1

Line segment- a straight line connecting two arbitrary points, called the ends of the segment. As an example, let these be points A and B and, respectively, the segment A B .

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

Definition 2

Cut length is the distance between the ends of the segment at a given scale (segment of unit length). We denote the length of the segment A B as follows: A B .

Definition 3

midpoint A point on a line segment that is equidistant from its ends. If the middle of the segment A B is denoted by the point C, then the equality will be true: A C \u003d C B

Initial data: coordinate line O x and mismatched points on it: A and B . These points correspond real numbers x A and x B . Point C is the midpoint of segment A B: you need to determine the coordinate x C .

Since point C is the midpoint of the segment A B, the equality will be true: | A C | = | C B | . The distance between points is determined by the modulus 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 a formula for the coordinate of the point C: x C \u003d 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, because in the original data - mismatched points. In this way, formula for determining the coordinates of the midpoint of the segment A B with ends A (x A) and B(xB):

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

Initial data: rectangular coordinate system on the plane O x y , two arbitrary non-coinciding points with 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 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).

By construction, the lines A A x , B B x , C C x are parallel; the lines are also parallel to each other. Together with this, according to the Thales theorem, from the equality A C \u003d C B, the equalities follow: A x C x \u003d C x B x and A y C y \u003d C y B y, and they, in turn, indicate that the point C x - the middle of the segment A x B x, and C y is the middle 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 line perpendicular to one of the axes. We will not conduct a detailed analysis of this case, we will consider it only graphically:

Summarizing all of the above, coordinates of the middle 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 middle 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 given points on the axes of the coordinate system.

According to the Thales theorem, the 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 middle of the segment in space, the following formulas are true:

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

The resulting formulas 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 middle of a segment through the coordinates of the radius vectors of its ends

The formula for finding the coordinates of the middle of the 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 geometric definition 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 constructed on the basis of the vectors O A → and O B → , i.e. the point of the middle 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 defined 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 middle of a segment

Among the tasks involving the use of the formulas obtained above, there are both those in which the question is directly to calculate the coordinates of the middle of the segment, 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, as well as 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 middle of the segment A B by the point C . Its coordinates will be determined as half the 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 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 condition of the problem, A M is the median, which means that M is the midpoint of the segment B C . First of all, we find the coordinates of the middle of the segment B C , i.e. M points:

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

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

Solution

The diagonals of a parallelepiped intersect at one point, which is the midpoint of all the diagonals. Based on this statement, we can keep in mind that the point M known by the conditions of the problem is the middle of the segment А С 1 . Based on the formula for finding the coordinates of the middle of the 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) .

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Each point on the planet's surface has a specific position, which corresponds to its own coordinate in latitude and longitude. It is located at the intersection of the spherical arcs of the meridian, which is responsible for longitude, with a parallel, which corresponds to latitude. It is denoted by a pair of angular values ​​expressed in degrees, minutes, seconds, which has the definition of a coordinate system.

Latitude and longitude are the geographical aspect of a plane or sphere, carried over into topographic images. For a more accurate location of any point, its height above sea level is also taken into account, which allows you to find it in three-dimensional space.

Latitude and longitude

The need to find a point by the coordinates of latitude and longitude arises on duty and by occupation among rescuers, geologists, military, sailors, archaeologists, pilots and drivers, but tourists, travelers, seekers, researchers may also need it.

What is latitude and how to find it

Latitude is the distance from an object to the equator line. Measured in angular units (such as degrees, degrees, minutes, seconds, etc.). Latitude on a map or globe is indicated by horizontal parallels - lines that describe a circle parallel to the equator and converge in the form of a series of tapering rings to the poles.

Lines of latitude

Therefore, they distinguish the northern latitude - this is the whole part earth's surface north of the equator, as well as the south - this is the entire part of the planet's surface south of the equator. Equator - zero, the longest parallel.

• The parallels from the equator line to the north pole are considered to be a positive value from 0 ° to 90 °, where 0 ° is the equator itself, and 90 ° is the top of the north pole. They are counted as north latitude (NL).
• Parallels radiating from the equator to the side south pole, denoted by a negative value from 0° to -90°, where -90° is the location of the south pole. They are counted as south latitude (S).
• On a globe, parallels are depicted as circles encircling the ball, which decrease as they approach the poles.
• All points on the same parallel will have the same latitude but different longitudes.
  On the maps, based on their scale, the parallels are in the form of horizontal, curved arc stripes - the smaller the scale, the straighter the parallel strip is shown, and the larger it is, the more curved it is.

Remember! The closer a given area is to the equator, the lower its latitude will be.

What is longitude and how to find it

Longitude is the amount by which the position of a given area is removed relative to Greenwich, that is, the zero meridian.

Longitude lines

Longitude is similarly inherent in the measurement in angular units, only from 0 ° to 180 ° and with the prefix - east or west.

• The zero meridian of Greenwich vertically encircles the globe of the Earth, passing through both poles, dividing it into the western and eastern hemispheres.
• Each of the parts to the west of Greenwich (in the Western Hemisphere) will have a West Longitude (WL) designation.
• Each of the parts east of Greenwich and located in the Eastern Hemisphere will bear the designation east longitude(v.p.).
• Finding each point along one meridian has a single longitude, but a different latitude.
• The meridians are plotted on the maps in the form of vertical stripes, curved in the form of an arc. The smaller the scale of the map, the straighter the meridian strip will be.

How to find the coordinates of a given point on the map

Often you have to find out the coordinates of a point that is located on the map in a square between the two nearest parallels and meridians. Approximate data can be obtained by eye by successively estimating the step in degrees between the lines plotted on the map in the area of ​​interest, and then comparing the distance from them to the desired area. For accurate calculations, you will need a pencil with a ruler, or a compass.

• For the initial data, we take the designations of the parallels with the meridian closest to our point.
• Next, we look at the step between their stripes in degrees.
• Then we look at the value of their step on the map in cm.
• Measure with a ruler in cm the distance from given point to the nearest parallel, as well as the distance between this line and the neighboring one, we translate into degrees and take into account the difference - subtracting from the larger one, or adding to the smaller one.
• Thus we get the latitude.

Example! The distance between the parallels 40° and 50°, among which our area is located, is 2 cm or 20 mm, and the step between them is 10°. Accordingly, 1° is equal to 2 mm. Our point is removed from the fortieth parallel by 0.5 cm or 5 mm. We find degrees to our locality 5/2 = 2.5 °, which must be added to the value of the nearest parallel: 40 ° + 2.5 ° = 42.5 ° - this is our northern latitude of the given point. AT southern hemisphere the calculations are similar, but the result has a negative sign.

Similarly, we find the longitude - if the nearest meridian is farther from Greenwich, and the given point is closer, then we subtract the difference, if the meridian is closer to Greenwich, and the point is further, then we add.

If only a compass was found at hand, then each of the segments is fixed with its tips, and the thrust is transferred to the scale.

Similarly, calculations of coordinates on the surface of the globe are made.

The best services for finding a place by coordinates

The easiest way to find out your location is by going to the PC version of the service that works directly with Google Maps. Many utilities simplify the process of entering latitude and longitude in the browser. Let's consider the best of them.

Map & Directions

In addition, Maps & Directions allows you to determine the coordinates of your position on the map for free with just one click. Click on "Find my coordinates", and the service will immediately put a marker and determine the latitude, longitude up to many thousandths, and also the height.

On the same site you can measure the distance between settlements or the area of ​​any given territory, draw a route or calculate the travel time. The service is useful for both travelers and just curious users.

mapcoordinates.net

The useful Mapcoordinates.net utility allows you to find out the coordinates of a point in any region of the world. The service is also integrated with Google Maps, but has a simplified interface, thanks to which even an unprepared user can use it.

In the address bar of the utility, where it says "Search", enter the address of the place, the latitude and longitude of which you want to get. A map with coordinates will appear along with a marker at the desired location. The latitude, longitude, and altitude of the selected point will be displayed above the marker.

Unfortunately, Mapcoordinates.net is not suitable for looking up points by knowing their coordinates. However, for the reverse procedure, this is a very convenient utility. The service supports many languages, including Russian.

Search by coordinates on the map through a browser using the Google Maps service

If for some reason you prefer to work not with simplified services, but directly with Google Maps, then this instruction will be useful for you. The process of searching by coordinates through Google Maps is a little more complicated than in the methods described earlier, but it can be mastered quickly and without much difficulty.

To find out the exact coordinates of a place, follow the following simple instructions:

Open the service on the PC. It is important that the full mode must be enabled, and not the lite (marked with a special lightning icon) mode, otherwise it will not work to get information;

Click on the area of ​​the map where the point or point you need is located with the right mouse button;

Check the option "What's here?" in the menu that appears;

Look at the tab that appears at the bottom of the screen. It will display the latitude, longitude and altitude.

To determine a place by known geographical coordinates, a different procedure will be required:

1. Open Google Maps in full mode on your computer;

   In the search bar at the top of the screen, you can enter the coordinates. This can be done in the following formats: degrees, minutes and seconds; degrees and decimal minutes; decimal degrees;

Press the "Enter" key, and a special marker will appear on the map at the desired location.

The most important thing when using the Google Maps service is to correctly specify geographic coordinates. Cards only recognize a few data formats, so be sure to keep the following input rules in mind:

When entering degrees, use the special character denoting it "°", not "d";

As a separator between the integer and fractional parts, you must use a dot, not a comma, otherwise the search string will not be able to give a place;

Latitude is listed first, then longitude. The first parameter must be written in the range from -90 to 90, the second - from -180 to 180.

Finding a special character on a PC keyboard is difficult, and in order to adhere to the required list of rules, you need to put in quite a lot of effort. It is much easier to use special utilities - we have listed the best of them in the section above.

Finding a place by latitude and longitude on Android OS

Often you need to find a place by coordinates away from the laptop or personal computer. The mobile application Google Maps, running on the Android platform, will help out. It is usually used to get directions or find out the schedule of vehicles, but the program is also suitable for finding the location of a point or point.

You can download the application for Android on the official page on Google Play. It is available in both Russian and English. After installing the program, follow the instructions below:

Open Google Maps on your device and wait for the map to appear;

Find a place that interests you. Click on it and hold until a special marker is displayed;

A tab will appear at the top of the screen with a search box and full coordinates of the place;

If you need to find a place by coordinates, and not vice versa, then the method on a mobile device is no different from its counterpart on a PC.

The mobile version of the service, as well as working on a PC, will allow you to study the desired place in detail, find out its exact coordinates, or vice versa, recognize the address from known data. This is a convenient way both for home and for the road.

1. To find the coordinates of the intersection point of the graphs of functions, you need to equate both functions to each other, move all terms containing $ x $ to the left side, and the rest to the right side and find the roots of the resulting equation.
2. The second way is to compose a system of equations and solve it by substituting one function into another
3. The third method involves the graphic construction of functions and the visual definition of the intersection point.

Case of two linear functions

Consider two linear functions$ f(x) = k_1 x+m_1 $ and $ g(x) = k_2 x + m_2 $. These functions are called direct. Building them is easy enough, you just need to take any two values ​​$x_1$ and $x_2$ and find $f(x_1)$ and $(x_2)$. Then repeat the same with the $ g(x) $ function. Next, visually find the coordinate of the intersection point of the function graphs.

You should know that linear functions have only one intersection point and only when $ k_1 \neq k_2 $. Otherwise, in the case of $ k_1=k_2 $, the functions are parallel to each other, since $ k $ is the slope factor. If $ k_1 \neq k_2 $, but $ m_1=m_2 $, then the intersection point will be $ M(0;m) $. It is desirable to remember this rule for accelerated problem solving.

Case of two non-linear functions

If a certain point A is given on the coordinate plane and it is required to determine its coordinates, then this is done as follows. Two straight lines are drawn through point A: one parallel to the y axis, the other - x. A line parallel to the y-axis intersects the x-axis (abscissa). The point of intersection of the axis and the line is the x-coordinate of the point A. A straight line parallel to the x-axis intersects the y-axis. The point of intersection of the axis and the line is the y-coordinate of point A . For example, if a line parallel to y intersects the x axis at point -5, and a line parallel to x intersects the y axis at point 2.3, then the coordinates of point A are written as: A (-5; 2.3).

The inverse problem, when it is necessary to draw a point according to the given coordinates, is solved in a similar way. Through points whose values ​​are equal to the given coordinates, lines are drawn on the x and y axes, parallel to each other: through the x coordinate - a straight line parallel to y, through the y coordinate - a straight line parallel to x. The point of intersection of these lines will be the desired point with the given coordinates. For example, given point B (–1.5; –3), it is required to depict it on the coordinate plane. To do this, through the point (–1.5; 0), which lies on the x axis, a straight line is drawn parallel to the y axis. A straight line is drawn through the point (0; -3) parallel to the x axis. Where these lines intersect, point B will be located (–1.5; –3).

In order to find a point on the map by coordinates online using Yandex, Google or OSM technologies, this map uses OSM map technologies: - you need to enter your coordinate data in the fields: latitude and longitude and click the "Find" button, after that the service will calculate place, point on the map, both Russia and the world. This service will help you find out the street, address, city and determine the exact coordinates.

Finding geographic coordinates latitude and longitude by address

To find the coordinates on the map of latitude and longitude of a point at an address online: you need to enter the exact address, city, country in the search field, select the desired one from the list and the service will determine the latitude and longitude of this place, which you can copy from the special field.

You can also show a point on the map and calculate its coordinates, simply by clicking on the map anywhere, the service will calculate: the address of the object and the field will show the coordinate data, which can also be copied.