Solving problems in mathematics is often accompanied by many difficulties for students. Helping the student cope with these difficulties, as well as teach them to apply their existing theoretical knowledge when solving specific problems in all sections of the course in the subject “Mathematics” is the main purpose of our site.
When starting to solve problems on the topic, students should be able to construct a point on a plane using its coordinates, as well as find the coordinates of a given point.
Calculation of the distance between two points A(x A; y A) and B(x B; y B) taken on a plane is performed using the formula d = √((x A – x B) 2 + (y A – y B) 2), where d is the length of the segment that connects these points on the plane.
If one of the ends of the segment coincides with the origin of coordinates, and the other has coordinates M(x M; y M), then the formula for calculating d will take the form OM = √(x M 2 + y M 2).
1. Calculation of the distance between two points based on the given coordinates of these points
Example 1.
Find the length of the segment that connects points A(2; -5) and B(-4; 3) on the coordinate plane (Fig. 1).
Solution.
The problem statement states: x A = 2; x B = -4; y A = -5 and y B = 3. Find d.
Applying the formula d = √((x A – x B) 2 + (y A – y B) 2), we get:
d = AB = √((2 – (-4)) 2 + (-5 – 3) 2) = 10.
2. Calculation of the coordinates of a point that is equidistant from three given points
Example 2.
Find the coordinates of point O 1, which is equidistant from three points A(7; -1) and B(-2; 2) and C(-1; -5).
Solution.
From the formulation of the problem conditions it follows that O 1 A = O 1 B = O 1 C. Let the desired point O 1 have coordinates (a; b). Using the formula d = √((x A – x B) 2 + (y A – y B) 2) we find:
O 1 A = √((a – 7) 2 + (b + 1) 2);
O 1 B = √((a + 2) 2 + (b – 2) 2);
O 1 C = √((a + 1) 2 + (b + 5) 2).
Let's create a system of two equations:
(√((a – 7) 2 + (b + 1) 2) = √((a + 2) 2 + (b – 2) 2),
(√((a – 7) 2 + (b + 1) 2) = √((a + 1) 2 + (b + 5) 2).
After squaring the left and right sides of the equations, we write:
((a – 7) 2 + (b + 1) 2 = (a + 2) 2 + (b – 2) 2,
((a – 7) 2 + (b + 1) 2 = (a + 1) 2 + (b + 5) 2.
Simplifying, let's write
(-3a + b + 7 = 0,
(-2a – b + 3 = 0.
Having solved the system, we get: a = 2; b = -1.
Point O 1 (2; -1) is equidistant from the three points specified in the condition that do not lie on the same straight line. This point is the center of a circle passing through three given points (Fig. 2).
3. Calculation of the abscissa (ordinate) of a point that lies on the abscissa (ordinate) axis and is at a given distance from a given point
Example 3.
The distance from point B(-5; 6) to point A lying on the Ox axis is 10. Find point A.
Solution.
From the formulation of the problem conditions it follows that the ordinate of point A is equal to zero and AB = 10.
Denoting the abscissa of point A by a, we write A(a; 0).
AB = √((a + 5) 2 + (0 – 6) 2) = √((a + 5) 2 + 36).
We get the equation √((a + 5) 2 + 36) = 10. Simplifying it, we have
a 2 + 10a – 39 = 0.
The roots of this equation are a 1 = -13; and 2 = 3.
We get two points A 1 (-13; 0) and A 2 (3; 0).
Examination:
A 1 B = √((-13 + 5) 2 + (0 – 6) 2) = 10.
A 2 B = √((3 + 5) 2 + (0 – 6) 2) = 10.
Both obtained points are suitable according to the conditions of the problem (Fig. 3).
4. Calculation of the abscissa (ordinate) of a point that lies on the abscissa (ordinate) axis and is at the same distance from two given points
Example 4.
Find a point on the Oy axis that is at the same distance from points A (6, 12) and B (-8, 10).
Solution.
Let the coordinates of the point required by the conditions of the problem, lying on the Oy axis, be O 1 (0; b) (at the point lying on the Oy axis, the abscissa is zero). It follows from the condition that O 1 A = O 1 B.
Using the formula d = √((x A – x B) 2 + (y A – y B) 2) we find:
O 1 A = √((0 – 6) 2 + (b – 12) 2) = √(36 + (b – 12) 2);
O 1 B = √((a + 8) 2 + (b – 10) 2) = √(64 + (b – 10) 2).
We have the equation √(36 + (b – 12) 2) = √(64 + (b – 10) 2) or 36 + (b – 12) 2 = 64 + (b – 10) 2.
After simplification we get: b – 4 = 0, b = 4.
Point O 1 (0; 4) required by the conditions of the problem (Fig. 4).
5. Calculation of the coordinates of a point that is located at the same distance from the coordinate axes and some given point
Example 5.
Find point M located on the coordinate plane at the same distance from the coordinate axes and from point A(-2; 1).
Solution.
The required point M, like point A(-2; 1), is located in the second coordinate angle, since it is equidistant from points A, P 1 and P 2 (Fig. 5). The distances of point M from the coordinate axes are the same, therefore, its coordinates will be (-a; a), where a > 0.
From the conditions of the problem it follows that MA = MR 1 = MR 2, MR 1 = a; MP 2 = |-a|,
those. |-a| = a.
Using the formula d = √((x A – x B) 2 + (y A – y B) 2) we find:
MA = √((-a + 2) 2 + (a – 1) 2).
Let's make an equation:
√((-а + 2) 2 + (а – 1) 2) = а.
After squaring and simplification we have: a 2 – 6a + 5 = 0. Solve the equation, find a 1 = 1; and 2 = 5.
We obtain two points M 1 (-1; 1) and M 2 (-5; 5) that satisfy the conditions of the problem. 
6. Calculation of the coordinates of a point that is located at the same specified distance from the abscissa (ordinate) axis and from the given point
Example 6.
Find a point M such that its distance from the ordinate axis and from point A(8; 6) is equal to 5.
Solution.
From the conditions of the problem it follows that MA = 5 and the abscissa of point M is equal to 5. Let the ordinate of point M be equal to b, then M(5; b) (Fig. 6).
According to the formula d = √((x A – x B) 2 + (y A – y B) 2) we have:
MA = √((5 – 8) 2 + (b – 6) 2).
Let's make an equation:
√((5 – 8) 2 + (b – 6) 2) = 5. Simplifying it, we get: b 2 – 12b + 20 = 0. The roots of this equation are b 1 = 2; b 2 = 10. Consequently, there are two points that satisfy the conditions of the problem: M 1 (5; 2) and M 2 (5; 10).
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Distance from point to point is the length of the segment connecting these points on 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 taken. 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 you have to calculate the distance between points using their coordinates.
In this article, we will firstly recall how the distance from point to 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. In conclusion, we will consider in detail the solutions to typical examples and problems.
Page navigation.
The distance between two points on a coordinate line.
Let's first define the notation. We will denote the distance from point A to point B 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 location of points on the coordinate line.
Distance from point to point on a plane, formula.
We obtain 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 abscissa axis, then the points coincide, and the distance is equal to the distance . In the previous paragraph, we found out that the distance between two points on a 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 axis, 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 down the equality, whence .
Let us summarize all the results obtained: the distance from a point to a point on a plane is found through the coordinates of the points using 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 us introduce a rectangular coordinate system Oxyz in space. Let's get a 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 us draw through points A and B planes perpendicular to the coordinate axes Ox, Oy and Oz. The intersection points of these planes with the coordinate axes will give us projections of points A and B onto these axes. We denote the projections 
.
The required 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 equal 
And . In a high school geometry course, it was proven that the square of the diagonal of a cuboid 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,
where do we get it from formula for finding the distance between points in space .
This formula is also valid if points A and B
- match up;
- belong to one of the coordinate axes or a 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 have obtained formulas for finding the distance between two points on a coordinate line, plane and three-dimensional space. It's time to look at solutions to typical examples.
The number of problems in which the final step is to find the distance between two points according to their coordinates is truly enormous. A full review of such examples is beyond the scope of this article. Here we will limit ourselves to examples in which the coordinates of two points are known and it is necessary to calculate the distance between them.
Using coordinates, the location of an object on the globe is determined. Coordinates are indicated by latitude and longitude. Latitudes are measured from the equator line on both sides. In the Northern Hemisphere the latitudes are positive, in the Southern Hemisphere they are negative. Longitude is measured from the prime meridian either east or west, respectively, either eastern or western longitude is obtained.According to the generally accepted position, the prime meridian is taken to be the one that passes through the old Greenwich Observatory in Greenwich. Geographic coordinates of the location can be obtained using a GPS navigator. This device receives satellite positioning system signals in the WGS-84 coordinate system, uniform for the whole world.
Navigator models differ in manufacturer, functionality and interface. Currently, built-in GPS navigators are also available in some cell phone models. But any model can record and save the coordinates of a 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 you can do this. The canonical form of representing geographic coordinates: degrees, minutes, seconds.For example, you can determine the distance between the following coordinates: point No. 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 length 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 are entered into the query fields for the first and second coordinates. When calculating, the online calculator gave the result - 3,800,619 m.
The next method is more labor-intensive, but also more visual. You must use any available mapping or navigation program. Programs in which you can create points using coordinates and measure 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 labels indicating 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 length of the distance between point No. 1 and point No. 2 is 3,817,353 m.
Why there is an error when determining the distance
All calculations of the extent between coordinates are based on the calculation of the arc length. The radius of the Earth is involved in calculating the length of the arc. But since the shape of the Earth is close to an oblate ellipsoid, the radius of the Earth varies at certain points. To calculate the distance between coordinates, the average value of the Earth's radius is taken, which gives an error in the measurement. The greater the distance being measured, the greater the error.Mathematics
§2. Coordinates of a point on the 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 transmit a drawing made up of dots by telegraph, communicate it to a computer, which does not understand drawings at all, but understands numbers well.
In the previous paragraph, we defined some sets of points on the plane using relationships between numbers. Now let's try to consistently translate other geometric concepts and facts into the language of numbers.
We will start with a simple and common task.
Find the distance between two points on the plane.
Solution:
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 points, knowing their coordinates. When deriving this rule, of course, it is allowed to resort to a 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 on the given numbers - the coordinates of the points - in order to obtain the desired number - the distance between dots.
Perhaps some readers will find this approach to solving the problem strange and far-fetched. What is simpler, they will say, the points are given, even by 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 doesn’t have a ruler, and she doesn’t draw, but she can count so quickly that it’s not a problem for her at all. Note that our problem is formulated so that the rule for calculating the distance between two points consists of commands that can be executed by a machine.
It is better to first solve the problem posed for the special case when one of these points lies at the origin of coordinates. Start with a few numerical examples: find the distance from the origin of the points; And .
Note. Use the Pythagorean theorem.
Now write a general formula to calculate 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 conditions stated above. In particular, it can be used in calculations on machines that can multiply numbers, add them, and extract square roots.
Now let's solve the general problem
Given two points on a plane, find the distance between them.
Solution:
Let us denote by , , , the projections of points and on the coordinate axes.
Let us denote the point of intersection of the lines with the letter . From a right triangle using the Pythagorean theorem we obtain:
But the length of the segment is equal to the length of the segment. The 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 to .
Arguing similarly, we find that the length of the segment is equal to . Substituting the found values and into the formula we get.
In this article we will look at ways to determine the distance from point to point theoretically and using the example of specific tasks. To begin with, let's introduce some definitions.
Definition 1
Distance between points is the length of the segment connecting them, on the existing scale. It is necessary to set a 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 on 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 length of a certain segment is assessed in comparison with a segment taken as a unit of length on a given scale.
If point A corresponds to an integer real number, by laying off sequentially from point O to point along the straight line O A segments - units of length, we can determine the length of the segment O A from the total number of set aside unit segments.
For example, point A corresponds to the number 3 - to get to it from point O, you will need to lay off three unit segments. If point A has coordinate - 4, unit segments are laid out in a similar way, but in a different, negative direction. Thus, in the first case, the distance O A is equal to 3; in the second case O A = 4.
If point A has a rational number as a coordinate, then from the origin (point O) we plot 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 plot the fraction 4 111 on the coordinate line.
Using the above method, it is completely impossible to plot 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 = x A (the number is taken as the 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.
To summarize: the distance from the origin to the point that corresponds to a real number on the coordinate line is equal to:
- 0 if the point coincides with 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 point O to point A with the coordinate xA: O A = x A
The following 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 location and having corresponding coordinates xA 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 us draw perpendiculars through points A and B to the coordinate axes O x and O y and obtain as a result the projection points: A x, A y, B x, B y. Based on the location of points A and B, the following options are then 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 coincide, and | A B | = | A y B y | . Since the distance between the points is equal to the modulus of the difference of 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 (ordinate 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 will find the distance between them by deriving the calculation formula:
We see that triangle A B C is rectangular in construction. In this case, A C = A x B x and B C = A y B y. Using the Pythagorean theorem, we create 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 draw 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 = (x B - x A) 2 + (y B - y A) 2
The resulting formula also confirms previously formed statements for cases of coincidence of points or situations when the points lie on straight lines perpendicular to the axes. So, if points A and B coincide, the following equality will be true: A B = (x B - x A) 2 + (y B - y A) 2 = 0 2 + 0 2 = 0
For a situation where 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 ordinate 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: a 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.
Let's consider the general case when points A and B do not lie in a plane parallel to one of the coordinate planes. Let us draw planes perpendicular to the coordinate axes through points A and B 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 measurements of this parallelepiped: A x B x , A y B y and A z B z
From the geometry course we know 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 = 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 when:
The points coincide;
They lie on one 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: 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 origin point O to point A and between points A and B.
Solution
- The distance from the reference point to the point is equal to the modulus of the coordinate of this point, respectively O A = 1 - 2 = 2 - 1
- We define the distance between points A and B 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: a rectangular coordinate system and two points lying on it A (1, - 1) and B (λ + 1, 3) are given. λ is some real number. It is necessary to find all values of this number at 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 the real coordinate values, we get: A B = (λ + 1 - 1) 2 + (3 - (- 1)) 2 = λ 2 + 16
We also 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 = 5 if λ = ± 3.
Example 3
Initial data: a three-dimensional space is specified in the rectangular coordinate system O x y z and the points A (1, 2, 3) and B - 7, - 2, 4 lying in it.
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 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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