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 to 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. Thus, 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 the geometric definition of actions on vectors, the following equality will be true: O C → = 1 2 · O A → + O B → . Point C in this case is the intersection point of the diagonals of the parallelogram 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.

Decision

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.

Decision

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.

Decision

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

So, let the segment be given by its ends - points A \u003d (x a; y a; z a) and B \u003d (x b; y b; z b). Then the coordinates of the middle of the segment - we denote it by the point H - can be found by the formula:

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

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

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

Answer: K = (0.5; 0; 1)

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

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

Answer: L = (0.5; 0.5; 1)

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

The tasks that will be considered, it is highly desirable to learn how to solve them fully automatically, and the formulas memorize, don't even remember it on purpose, they will remember it themselves =) This is very important, since other problems of analytical geometry are based on the simplest elementary examples, and it will be annoying to spend extra time eating pawns. You do not need to fasten the top buttons on your shirt, many things are familiar to you from school.

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

After painstaking work, I suddenly noticed that the sizes of web pages are quite large, and if it goes on like this, then you can quietly go wild =) Therefore, I bring to your attention a small essay on a very common geometric problem - on the division of the segment in this respect, and, as a special case, about dividing a segment in half.

For one reason or another, this task did not fit into other lessons, but now there is a great opportunity to consider it in detail and slowly. The good news is that we'll take a break from vectors for a bit and focus on points and line segments.

Section division formulas in this respect

The concept of segment division in this respect

Often you don’t have to wait for what was promised at all, we’ll immediately consider a couple of points and, obviously incredible, a segment:

The problem under consideration is valid both for segments of the plane and for segments of space. That is, the demonstration segment can be placed in any way on a plane or in space. For ease of explanation, I drew it horizontally.

What are we going to do with this segment? Saw this time. Someone is sawing the budget, someone is sawing a spouse, someone is sawing firewood, and we will start sawing a segment into two parts. The segment is divided into two parts using some point, which, of course, is located directly on it:

In this example, the point divides the segment in such a way that the segment is two times shorter than the segment . STILL we can say that the point divides the segment in relation ("one to two"), counting from the top.

In dry mathematical language, this fact is written as follows: , or more often in the form of a familiar proportion: . The ratio of the segments is usually denoted by the Greek letter "lambda", in this case: .

It is easy to make a proportion in a different order: - this record means that the segment is twice as long as the segment, but this does not have any fundamental significance for solving problems. It can be so, and it can be so.

Of course, the segment is easy to divide in some other respect, and as a reinforcement of the concept, the second example:

Here the ratio is valid: . If we make the proportion the other way around, then we get: .

After we figured out what it means to divide the segment in this respect, let's move on to considering practical problems.

If two points of the plane are known, then the coordinates of the point that divides the segment in relation to are expressed by the formulas:

Where did these formulas come from? In the course of analytic geometry, these formulas are strictly derived using vectors (where would we be without them? =)). In addition, they are valid not only for the Cartesian coordinate system, but also for an arbitrary affine coordinate system (see lesson Linear (non) dependence of vectors. Vector basis). Such is the universal task.

Example 1

Find the coordinates of the point that divides the segment in relation to , if the points are known

Decision: In this problem . According to the formulas for dividing the segment in this respect, we find the point:

Answer:

Pay attention to the calculation technique: first you need to separately calculate the numerator and separately the denominator. The result is often (but by no means always) a three- or four-story fraction. After that, we get rid of the multi-storey fraction and carry out final simplifications.

The task does not require a drawing, but it is always useful to complete it on a draft:

Indeed, the relation is satisfied, that is, the segment is three times shorter than the segment . If the proportion is not obvious, then the segments can always be stupidly measured with an ordinary ruler.

Equivalent second way to solve: in it, the countdown starts from a point and the relation is fair: (in human words, the segment is three times longer than the segment). According to the formulas for dividing a segment in this respect:

Answer:

Note that in the formulas it is necessary to move the coordinates of the point to the first place, since the little thriller began with it.

It can also be seen that the second method is more rational due to simpler calculations. But still, this problem is often solved in the "traditional" order. For example, if a segment is given by condition, then it is assumed that you will make up a proportion, if a segment is given, then “tacitly” means proportion.

And I cited the second method for the reason that often they try to deliberately confuse the condition of the problem. That is why it is very important to carry out a draft drawing in order, firstly, to correctly analyze the condition, and, secondly, for verification purposes. It's a shame to make mistakes in such a simple task.

Example 2

Given points . To find:

a) a point dividing the segment with respect to ;
b) a point dividing the segment in relation to .

This is a do-it-yourself example. Full solution and answer at the end of the lesson.

Sometimes there are problems where one of the ends of the segment is unknown:

Example 3

The point belongs to the segment . It is known that the segment is twice as long as the segment . Find a point if .

Decision: It follows from the condition that the point divides the segment in relation to , counting from the top, that is, the proportion is valid: . According to the formulas for dividing a segment in this respect:

Now we do not know the coordinates of the point : , but this is not a particular problem, since they can be easily expressed from the above formulas. In general, it’s not worth expressing anything, it’s much easier to substitute specific numbers and carefully deal with calculations:

Answer:

To check, you can take the ends of the segment and, using the formulas in direct order, make sure that the ratio really turns out to be a point. And, of course, of course, a drawing will not be superfluous. And in order to finally convince you of the benefits of a checkered notebook, a simple pencil and a ruler, I propose a tricky task for an independent solution:

Example 4

Dot . The segment is one and a half times shorter than the segment . Find a point if the coordinates of the points are known .

Solution at the end of the lesson. By the way, it is not the only one, if you go a different way from the sample, then this will not be a mistake, the main thing is that the answers match.

For spatial segments, everything will be exactly the same, only one more coordinate will be added.

If two points in space are known, then the coordinates of the point that divides the segment in relation to are expressed by the formulas:
.

Example 5

Points are given. Find the coordinates of a point belonging to the segment if it is known that .

Decision: The relation follows from the condition: . This example was taken from a real test, and its author allowed himself a little prank (suddenly someone stumbles) - it would be more rational to write the proportion in the condition like this: .

According to the formulas for the coordinates of the middle of the segment:

Answer:

Three-dimensional drawings for verification purposes are much more difficult to perform. However, you can always make a schematic drawing to understand at least the condition - which segments need to be correlated.

As for the fractions in the answer, don't be surprised, it's common. I said it many times, but I repeat: in higher mathematics it is customary to wield ordinary regular and improper fractions. Answer in the form will do, but the variant with improper fractions is more standard.

Warm-up task for independent solution:

Example 6

Points are given. Find the coordinates of the point if it is known that it divides the segment with respect to .

Solution and answer at the end of the lesson. If it is difficult to orient in proportions, make a schematic drawing.

In independent and control works, the considered examples are found both on their own and as an integral part of larger tasks. In this sense, the problem of finding the center of gravity of a triangle is typical.

I don’t see much point in analyzing a kind of task where one of the ends of the segment is unknown, since everything will look like a flat case, except that there are a little more calculations. Better remember the school years:

Formulas for the coordinates of the middle of the segment

Even unprepared readers can remember how to cut a segment in half. The task of dividing a segment into two equal parts is a special case of dividing a segment in this respect. The two-handed saw works in the most democratic way, and each neighbor at the desk gets the same stick:

At this solemn hour, the drums beat, saluting the significant proportion. And general formulas miraculously transformed into something familiar and simple:

A convenient moment is the fact that the coordinates of the ends of the segment can be painlessly rearranged:

In general formulas, such a luxurious number, as you understand, does not work. Yes, and here there is no special need for it, so, a pleasant trifle.

For the spatial case, an obvious analogy is valid. If the ends of the segment are given, then the coordinates of its middle are expressed by the formulas:

Example 7

The parallelogram is given by the coordinates of its vertices. Find the point of intersection of its diagonals.

Decision: Those who wish can complete the drawing. I especially recommend graffiti to those who have completely forgotten the school geometry course.

According to a well-known property, the diagonals of a parallelogram are divided in half by their intersection point, so the problem can be solved in two ways.

Method one: Consider opposite vertices . Using the formulas for dividing a segment in half, we find the midpoint of the diagonal:

How to find the coordinates of the midpoint of a segment
First, let's figure out what the middle of the segment is.
The midpoint of a segment is considered to be a point that belongs to this segment and is at the same distance from its ends.

The coordinates of such a point are easy to find if the coordinates of the ends of this segment are known. In this case, the coordinates of the middle of the segment will be equal to half the sum of the corresponding coordinates of the ends of the segment.
The coordinates of the midpoint of a segment are often found by solving problems on the median, midline, etc.
Consider the calculation of the coordinates of the middle of the segment for two cases: when the segment is given on the plane and given in space.
Let the segment on the plane be given by two points with coordinates and . Then the coordinates of the middle of the PH segment are calculated by the formula:

Let the segment be given in space by two points with coordinates and . Then the coordinates of the middle of the PH segment are calculated by the formula:

Example.
Find the coordinates of the point K - the middle of the MO, if M (-1; 6) and O (8; 5).

Decision.
Since the points have two coordinates, it means that the segment is given on the plane. We use the corresponding formulas:

Consequently, the middle of the MO will have coordinates K (3.5; 5.5).

Answer. K (3.5; 5.5).

Doesn't make any work. To calculate them, there is a simple expression that is easy to remember. For example, if the coordinates of the ends of a segment are respectively (x1; y1) and (x2; y2), respectively, then the coordinates of its middle are calculated as the arithmetic mean of these coordinates, that is:

That's the whole difficulty.
Consider the calculation of the coordinates of the center of one of the segments on a specific example, as you asked.

Task.
Find the coordinates of a certain point M if it is the midpoint (center) of the segment KR, the ends of which have the following coordinates: (-3; 7) and (13; 21), respectively.

Decision.
We use the above formula:

Answer. M (5; 14).

Using this formula, you can also find not only the coordinates of the middle of a segment, but also its ends. Consider an example.

Task.
The coordinates of two points (7; 19) and (8; 27) are given. Find the coordinates of one of the ends of the segment if the previous two points are its end and middle.

Decision.
Let's denote the ends of the segment as K and P, and its middle as S. Let's rewrite the formula taking into account the new names:

Substitute the known coordinates and calculate the individual coordinates: