The length, as already noted, is indicated by the modulus sign.

If two points of the plane and are given, then the length of the segment can be calculated by the formula

If two points in space and are given, then the length of the segment can be calculated by the formula

Note: The formulas will remain correct if the corresponding coordinates are rearranged: and , but the first option is more standard

Example 3

Solution: according to the corresponding formula:

Answer:

For clarity, I will make a drawing

Section - it's not a vector, and you can't move it anywhere, of course. In addition, if you complete the drawing to scale: 1 unit. \u003d 1 cm (two tetrad cells), then the answer can be checked with a regular ruler by directly measuring the length of the segment.

Yes, the solution is short, but there are a couple of important points in it that I would like to clarify:

First, in the answer we set the dimension: “units”. The condition does not say WHAT it is, millimeters, centimeters, meters or kilometers. Therefore, the general formulation will be a mathematically competent solution: “units” - abbreviated as “units”.

Secondly, let's repeat the school material, which is useful not only for the considered problem:

pay attention to important technical trick – taking the multiplier out from under the root. As a result of the calculations, we got the result and good mathematical style involves taking the factor out from under the root (if possible). The process looks like this in more detail: . Of course, leaving the answer in the form will not be a mistake - but it is definitely a flaw and a weighty argument for nitpicking on the part of the teacher.

Here are other common cases:

Often a sufficiently large number is obtained under the root, for example. How to be in such cases? On the calculator, we check if the number is divisible by 4:. Yes, split completely, thus: . Or maybe the number can be divided by 4 again? . In this way: . The last digit of the number is odd, so dividing by 4 for the third time is clearly not possible. Trying to divide by nine: . As a result:
Ready.

Conclusion: if under the root we get a whole number that cannot be extracted, then we try to take out the factor from under the root - on the calculator we check whether the number is divisible by: 4, 9, 16, 25, 36, 49, etc.

In the course of solving various problems, roots are often found, always try to extract factors from under the root in order to avoid a lower score and unnecessary troubles with finalizing your solutions according to the teacher's remark.

Let's repeat the squaring of the roots and other powers at the same time:

The rules for actions with degrees in a general form can be found in a school textbook on algebra, but I think that everything or almost everything is already clear from the examples given.

Task for an independent solution with a segment in space:

Example 4

Given points and . Find the length of the segment.

Solution and answer at the end of the lesson.

There are three main coordinate systems used in geometry, theoretical mechanics, and other branches of physics: Cartesian, polar, and spherical. In these coordinate systems, each point has three coordinates. Knowing the coordinates of two points, you can determine the distance between these two points.

You will need

• Cartesian, polar and spherical coordinates of the ends of a segment

Instruction

Let's start with a rectangular Cartesian coordinate system. The position of a point in space in this coordinate system is determined by coordinates x,y and z. A radius vector is drawn from the origin of coordinates to the point. The projections of this radius vector onto the coordinate axes will be coordinates this point.
Suppose you now have two points with coordinates x1,y1,z1 and x2,y2 and z2 respectively. Let r1 and r2 be the radius vectors of the first and second points, respectively. Obviously, the distance between these two points will be equal to the modulus of the vector r = r1-r2, where (r1-r2) is the vector difference.
The coordinates of the vector r will obviously be the following: x1-x2, y1-y2, z1-z2. Then the modulus of the vector r or the distance between two points will be: r = sqrt(((x1-x2)^2)+((y1-y2)^2)+((z1-z2)^2)).

Consider now a polar coordinate system in which the coordinate of a point will be given by the radial coordinate r (radius vector in the XY plane), the angular coordinate? (the angle between the r vector and the X axis) and the z coordinate, which is similar to the z coordinate in the Cartesian system. The polar coordinates of a point can be converted to Cartesian as follows: x = r*cos?, y = r*sin?, z = z. Then the distance between two points with coordinates r1, ?1 ,z1 and r2, ?2, z2 will equal R = sqrt(((r1*cos?1-r2*cos?2)^2)+((r1*sin?1-r2*sin?2 )^2)+((z1-z2)^2)) = sqrt((r1^2)+(r2^2)-2r1*r2(cos?1*cos?2+sin?1*sin?2) +((z1-z2)^2))

Now consider a spherical coordinate system. In it, the position of the point is given by three coordinates r, ? and?. r - distance from the origin to the point, ? and? are the azimuth and zenith angles, respectively. Injection? similar to the angle with the same designation in the polar coordinate system, huh? - the angle between the radius vector r and the Z axis, and the 0 coordinates r1, ?1, ?1 and r2, ?2 and ?2 will be equal to R = sqrt(((r1*sin?1*cos?1-r2*sin? 2*cos?2)^2)+((r1*sin?1*sin?1-r2*sin?2*sin?2)^2)+((r1*cos?1-r2*cos?2) ^2)) = (((r1*sin?1)^2)+((r2*sin?2)^2)-2r1*r2*sin?1*sin?2*(cos?1*cos?2 +sin?1*sin?2)+((r1*cos?1-r2*cos?2)^2))

Let the segment be given by two points in the coordinate plane, then you can find its length using the Pythagorean theorem.

Instruction

Let the coordinates of the ends of the segment (x1-y1) and (x2-y2) be given. Draw a line segment in a coordinate system.

Lower the perpendiculars from the ends of the segment on the X and Y axes. The segments marked in red in the figure are the projections of the original segment on the coordinate axes.

If you perform a parallel transfer of segments-projections to the ends of the segments, then you get a right triangle. The legs of this triangle will be the transferred projections, and the hypotenuse will be the segment AB itself.

Projection lengths are easy to calculate. The length of the projection on the Y axis will be y2-y1, and the length of the projection on the X axis will be x2-x1. Then by the Pythagorean theorem |AB|²- = (y2 - y1)²- + (x2 - x1)²-, where |AB| - the length of the segment.

Having presented this scheme for finding the length of a segment in the general case, it is easy to calculate the length of a segment without constructing a segment. Let's calculate the length of the segment, the coordinates of the ends of which are (1-3) and (2-5). Then |AB|²- = (2 - 1)²- + (5 - 3)²- = 1 + 4 = 5, so the length of the required segment is 5^1/2.

I will give a detailed example of how you can determine the length of a segment according to given coordinates using the online service on the Test Ru website.

Let's say you need to find the length of a segment on a plane

(in space, you can calculate by analogy, you just need to change the point to the dimension of three)

Segment AB has ends with coordinates A (1, 2) and B (3, 4).

To calculate the length of segment AB use the following steps:

1. Go to the service page for finding the distance between two points online:

We can use this, because the length of the segment along the coordinate. is exactly equal to the distance between points A and B.

To set the correct dimension of point A, drag the lower right edge to the left, as shown in fig.

After entering the coordinates of the first point A(1, 2), then press the button

3. In the second step, you will see a form for entering the second point B, enter its coordinates, as in Fig. below:

Points a and b are entered! Solution:

Find the distance between points (s)

In this article, we will talk about finding the coordinates of the middle of a segment from the coordinates of its ends. First, we will give the necessary concepts, then we will obtain formulas for finding the coordinates of the middle of a segment, and in conclusion, we will consider solutions to typical examples and problems.

Page navigation.

The concept of the middle of a segment.

In order to introduce the concept of the midpoint of a segment, we need definitions of a segment and its length.

The concept of a segment is given in mathematics lessons in the fifth grade of high school as follows: if we take two arbitrary non-coinciding points A and B, attach a ruler to them and draw a line from A to B (or from B to A), then we get segment AB(or segment B A). Points A and B are called the ends of the segment. We should keep in mind that segment AB and segment BA are the same segment.

If the segment AB is infinitely extended in both directions from the ends, then we get straight line AB(or direct VA). Segment AB is the part of the straight line AB enclosed between points A and B. Thus, the segment AB is the union of points A, B and the set of all points of the straight line ABlocated between points A and B. If we take an arbitrary point M of the straight line AB located between points A and B, then they say that the point M lies on segment AB.

Length of segment AB is the distance between points A and B at a given scale (segment of unit length). The length of the segment AB will be denoted as .

Definition.

Dot C is called the middle of the segment AB if it lies on the segment AB and is at the same distance from its ends.

That is, if point C is the midpoint of the segment AB, then it lies on it and.

Further, our task will be to find the coordinates of the middle of the segment ABif the coordinates of points A and B are given on the coordinate line or in a rectangular coordinate system.

The coordinate of the midpoint of the segment on the coordinate line.

Let us be given a coordinate line Ox and two non-coinciding points A and B on it, which correspond to real numbers and . Let point C be the midpoint of segment AB. Let's find the coordinate of the point C.

Since point C is the midpoint of the segment AB, then the equality is true. In the section on the distance from a point to a point on a coordinate line, we showed that the distance between points is equal to the modulus of the difference between their coordinates, therefore, . Then or . From equality find the coordinate of the midpoint of the segment AB on the coordinate line: - it is equal to half the sum of the coordinates of the ends of the segment. From the second equality we get , which is impossible, since we took non-coinciding points A and B.

So, the formula for finding the coordinate of the midpoint of the segment AB with ends and has the form .

Coordinates of the midpoint of a line segment.

Let us introduce a rectangular Cartesian coordinate system Оxyz on the plane. Let us be given two points and and we know that the point C is the midpoint of the segment AB. Let's find the coordinates and points C.

By construction, straight parallel as well as parallel lines , therefore, by Thales theorem from the equality of the segments AC and CB follows the equality of the segments and , as well as the segments and . Therefore, the point is the midpoint of the segment, and the midpoint of the segment. Then, by virtue of the previous paragraph of this article and .

Using these formulas, one can also calculate the coordinates of the middle of the segment AB in cases where points A and B lie on one of the coordinate axes or on a straight line perpendicular to one of the coordinate axes. Let us leave these cases without comment, and give graphic illustrations.

In this way, the midpoint of segment AB on a plane with ends at points and has coordinates .

Coordinates of the middle of the segment in space.

Let a rectangular coordinate system Oxyz be introduced in three-dimensional space and two points and . We get formulas for finding the coordinates of the point C, which is the midpoint of the segment AB.

Let's consider the general case.

Let and be the projections of points A, B, and C onto the coordinate axes Ox, Oy, and Oz, respectively.

By Thales' theorem, therefore, the points are the midpoints of the segments respectively. Then (see the first paragraph of this article). So we got formulas for calculating the coordinates of the middle of a segment from the coordinates of its ends in space.

These formulas can also be used in cases where points A and B lie on one of the coordinate axes or on a straight line perpendicular to one of the coordinate axes, and also if points A and B lie in one of the coordinate planes or in a plane parallel to one of the coordinate axes. planes.

The coordinates of the middle of the segment through the coordinates of the radius vectors of its ends.

Formulas for finding the coordinates of the middle of a segment are easy to obtain by referring to the algebra of vectors.

Let a rectangular Cartesian coordinate system Oxy be given on the plane and point C be the midpoint of the segment AB, with and .

According to the geometric definition of operations on vectors, the equality (point C is the point of intersection of the diagonals of a parallelogram built on vectors and , that is, point C is the midpoint of the diagonal of the parallelogram). In the article coordinates of a vector in a rectangular coordinate system, we found out that the coordinates of the radius vector of a point are equal to the coordinates of this point, therefore, . Then, after performing the corresponding operations on vectors in coordinates , we have . How can we conclude that point C has coordinates .

Absolutely similarly, the coordinates of the middle of the segment AB can be found through the coordinates of its ends in space. In this case, if C is the midpoint of the segment AB and , then we have .

Finding the coordinates of the middle of the segment, examples, solutions.

In many problems, you have to use formulas to find the coordinates of the midpoint of a segment. Let's consider solutions of the most characteristic examples.

Let's start with an example that only needs to apply a formula.

Example.

The coordinates of two points are given on the plane . Find the coordinates of the midpoint of the segment AB.

Solution.

Let point C be the midpoint of segment AB. Its coordinates are equal to half-sums of the corresponding coordinates of points A and B:

Thus, the midpoint of the segment AB has coordinates.

The length, as already noted, is indicated by the modulus sign.

If two points of the plane and are given, then the length of the segment can be calculated by the formula

If two points in space and are given, then the length of the segment can be calculated by the formula

Note:The formulas will remain correct if the corresponding coordinates are swapped: and , but the first option is more standard

Example 3

Solution: according to the corresponding formula:

Answer:

For clarity, I will make a drawing

Section - it's not a vector, and you can't move it anywhere, of course. In addition, if you complete the drawing to scale: 1 unit. \u003d 1 cm (two tetrad cells), then the answer can be checked with a regular ruler by directly measuring the length of the segment.

Yes, the solution is short, but there are a couple of important points in it that I would like to clarify:

First, in the answer we set the dimension: “units”. The condition does not say WHAT it is, millimeters, centimeters, meters or kilometers. Therefore, the general formulation will be a mathematically competent solution: “units” - abbreviated as “units”.

Secondly, let's repeat the school material, which is useful not only for the considered problem:

pay attention to important technical trick – taking the multiplier out from under the root. As a result of the calculations, we got the result and good mathematical style involves taking the factor out from under the root (if possible). The process looks like this in more detail: Of course, leaving the answer in the form will not be a mistake - but it is definitely a flaw and a weighty argument for nitpicking on the part of the teacher.

Here are other common cases:

Often a sufficiently large number is obtained under the root, for example. How to be in such cases? On the calculator, we check if the number is divisible by 4:. Yes, it was completely divided, thus: . Or maybe the number can be divided by 4 again? . In this way: . The last digit of the number is odd, so dividing by 4 for the third time is clearly not possible. Trying to divide by nine: . As a result:
Ready.

Conclusion: if under the root we get a whole number that cannot be extracted, then we try to take out the factor from under the root - on the calculator we check whether the number is divisible by: 4, 9, 16, 25, 36, 49, etc.

In the course of solving various problems, roots are often found, always try to extract factors from under the root in order to avoid a lower score and unnecessary troubles with finalizing your solutions according to the teacher's remark.

Let's repeat the squaring of the roots and other powers at the same time:

The rules for actions with degrees in a general form can be found in a school textbook on algebra, but I think that everything or almost everything is already clear from the examples given.

Task for an independent solution with a segment in space:

Example 4

Given points and . Find the length of the segment.

Solution and answer at the end of the lesson.