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 peacefully become brutal =) 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.

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

It is easy to compose the 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 system coordinates (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

Solution: 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: (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 more simple calculations. But anyway this task more often decided 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 . Find:

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

This is an example for independent solution. Complete Solution and the 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 .

Solution: 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 view it doesn’t cost anything to express, 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 .

Solution: The relation follows from the condition: . This example is 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 work the considered examples occur both on their own and as an integral part of larger problems. 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.

Solution: 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:

Initial geometric information

The concept of a segment, like the concept of a point, a straight line, a ray and an angle, refers to the initial geometric information. The study of geometry begins with these concepts.

Under the "initial information" is usually understood as something elementary and simple. In understanding, perhaps this is so. However, such simple concepts are often encountered and are necessary not only in our Everyday life but also in manufacturing, construction and other spheres of our life.

Let's start with definitions.

Definition 1

A segment is a part of a straight line bounded by two points (ends).

If the ends of the segment are points $A$ and $B$, then the formed segment is written as $AB$ or $BA$. Points $A$ and $B$ belong to such a segment, as well as all points of the line lying between these points.

Definition 2

The midpoint of a segment is a point on a segment that bisects it into two equal segments.

If it is a point $C$, then $AC=CB$.

The segment is measured by comparison with a certain segment, taken as a unit of measurement. The most commonly used is the centimeter. If a centimeter fits exactly four times in a given segment, then this means that the length of this segment is equal to $4$ cm.

Let's introduce a simple observation. If a point divides a segment into two segments, then the length of the entire segment is equal to the sum of the lengths of these segments.

The formula for finding the coordinate of the midpoint of a segment

The formula for finding the coordinate of the midpoint of a segment refers to the course of analytical geometry on a plane.

Let's define coordinates.

Definition 3

Coordinates are defined (or ordered) numbers that indicate the position of a point on a plane, on a surface, or in space.

In our case, the coordinates are marked on the plane defined by the coordinate axes.

Figure 3 Coordinate plane. Author24 - online exchange of student papers

Let's describe the picture. A point is chosen on the plane, called the origin of coordinates. It is denoted by the letter $O$. Two straight lines (coordinate axes) are drawn through the origin of coordinates, intersecting at right angles, one of them is strictly horizontal, and the other is vertical. This situation is considered normal. The horizontal line is called the abscissa axis and is denoted $OX$, the vertical line is called the ordinate axis $OY$.

Thus, the axes define the $XOY$ plane.

The coordinates of points in such a system are determined by two numbers.

Exist different formulas(equations) that determine certain coordinates. Usually, in the course of analytical geometry, they study various formulas for lines, angles, lengths of a segment, and others.

Let's go straight to the formula for the coordinate of the middle of the segment.

Definition 4

If the coordinates of the point $E(x,y)$ are the midpoint of the segment $M_1M_2$, then:

Figure 4. The formula for finding the coordinate of the middle of the segment. Author24 - online exchange of student papers

Practical part

Examples from school course geometries are quite simple. Let's look at a few of the main ones.

For a better understanding, let's start with an elementary illustrative example.

Example 1

We have a drawing:

In the figure, the segments $AC, CD, DE, EB$ are equal.

1. The midpoint of which segments is the point $D$?
2. What point is the midpoint of the segment $DB$?

1. the point $D$ is the midpoint of the segments $AB$ and $CE$;
2. point $E$.

Let's consider another simple example in which we need to calculate the length.

Example 2

Point $B$ is the midpoint of segment $AC$. $AB = 9$ cm. What is the length of $AC$?

Since m. $B$ bisects $AC$, then $AB = BC= 9$ cm. So $AC = 9+9=18$ cm.

Answer: 18 cm.

Other similar examples are usually identical and focused on the ability to compare length values ​​​​and their representation with algebraic operations. Often in tasks there are cases when a centimeter does not fit an even number of times into a segment. Then the unit of measurement is divided into equal parts. In our case, a centimeter is divided into 10 millimeters. Separately measure the remainder, comparing with a millimeter. Let us give an example demonstrating such a case.

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 1 . Find the coordinates of this point.

Solution. 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 .

Solution. 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.

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 a point that belongs to this segment and is 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).

Solution.
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).