Distance from point to point, formulas, examples, solutions. How to find the distance on the coordinate plane Finding the distance from point to point, examples and solutions

Lesson #3

TOPIC: Distance between points on a coordinate line

The purpose of the teacher's activity: create conditions for mastering the skills to find the distance between points on the coordinate line, calculating the modulus of the difference, the coordinates of the middle of the segment.

Planned results of the study of the topic:

Personal: show an interest in learning the subject.

Subject: are able to find the distance between points on the coordinate line, calculating the modulus of the difference, the coordinates of the middle of the segment.

Meta-subject results of studying the topic (universal learning activities):

cognitive: focus on a variety of ways to solve problems; are able to summarize and systematize information;

regulatory: take into account the rule in planning and controlling the solution method;

communicative: consider different opinions and strive to coordinate different positions in cooperation.

Lesson script.

I .org moment.
Hello guys. Today we are visiting Let's greet them!

Sit down.

We don't have a typical class. Lesson of generalization of knowledge. We must show what we have learned, what we have learned.

What topic have we been working on lately? (comparison, addition of rational numbers)

The epigraph of the lesson I took these words : We will go on a journey for science today

Let's take fantasy to help

We won't turn off the straight road

And so that we can reach our goals faster

We must go up the stairs!

2. Updating knowledge .

Task "Ladder".

Variant work, verification and self-assessment

3 Well done, we continue to move up for knowledge.Let's check our homework.

1. Find the distance between the points of the coordinate line: D / W

a) A(-4) and B(-6); b) A(5) and B(-7); c) A(3) and B(-18).

DECISION: a) AB = |-6-(-4) |= |-2|=2

b) AB =|-7-5|=12

c) AB = |-18-3 |= 21

2. Find the coordinates of points remote from the point:

a) A(-8) by 5; b) B(6) by -2.7; c) C(4) to -3.2

Decision: a) -8+5=-3 BUT 1 (-3) and -8-5=-13 BUT 2 (-13)

b) 6 + (-2.7) \u003d 3.3 AT 1 (3,3) and 6-(-2.7)=8.7 AT 2 (8,7)

c) 4+(-3.2) = 0.8 With 1 (0,8) 4-(-3,2) = 7,2 With 2 (7,2)

3) Find the coordinate of point C, the middle of the segment, if:

a) A(-12) B (1) b) A(-7) and B(9) c) A(16) and B (-8)

DECISION:

12+1=-11 B) -7+9 =2 C) 16+(-8) =8

11: 2=-5,5 2:2=1 8:2 =4

С(-5.5) С(1) С(4)

You have a sample of homework on your desks. Check and rate the self-assessment sheet.

4 . Blitz - Poll :

1. What is a coordinate line?

2. What rules for comparing rational numbers do you know?

3.What is the modulus of a number?

4. How to add two numbers with the same signs?

5. How to add two numbers with different signs?

6. How to determine the distance between the points of the coordinate line?

Well, now we will show how we can apply our knowledge in practice.

5.Fix bugs

    12+4 =-16 -12+(-18) =6 9-14=5

    16 +(-10)=6 30 +(-10) =-20 5 –(-3)=2

    6 –(-5) =11 -20 -14 =-34 -2 +7=9

    11-28 =-39 -34 -5 =-29 9 -13=22

Perform a self test.

    12+4 =--8 -12+(-18) =30 9-14= -5

    16 +(-10)=-26 30 +(-10) =20 5 –(-3)=8

    26 –(-5) =-21 -20 -14 =-34 -2 +7=5

    11-28 =--17 -34 -5 =-41 9 -13=-4

6. Determine the distance between the points: and find the middle of the segment (according to options)

(exchange of notebooks and mutual check.)



7. Well, now we will rest. Our eyes must rest

8. Independent work (in a notebook) grading.

    1st option 2nd option

    1,5-4,6 0,8 -1,2

    -2,8 +3,8 4-9,4

    0,45 -1 -4,3 +(-1,2) (Slide 9)

    Target: check the ability to apply the laws of addition to transform expressions; develop cognitive interest, independence; cultivate perseverance and perseverance in achieving the goal.


    Find the value of the expression and, according to the result, color the gnome in accordance with the table. (the card with the gnome remains with the students as a talisman)

    Well done boys!

    You completed the tasks

    And shined with knowledge.

    And the magic key to learning -

    Your perseverance and patience!


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, it is required to know the scale (unit of length) in which the measurements will be taken. Therefore, the problem of finding the distance from a point to a 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 by their coordinates.

In this article, we, firstly, recall how the distance from a point to a 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 consider in detail the solutions of typical examples and problems.

Page navigation.

The distance between two points on a coordinate line.

Let's first define the notation. The distance from point A to point B will be denoted 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, i.e, for any arrangement of points on the coordinate line.

Distance from a point to a point on a plane, formula.

Let's get 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 x-axis, then the points and coincide, and the distance is equal to the distance. In the previous paragraph, we found out that the distance between two points on the 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 y-axis, then the distance from point A to point B is found as .

In this case, the triangle ABC is rectangular in construction, and and . By the Pythagorean theorem we can write the equality , whence .

Let's summarize all the results: the distance from a point to a point on a plane is found through the coordinates of the points by 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 are the same, 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 Оxyz in space. Get the 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's draw through points A and B in the plane perpendicular to the coordinate axes Ox, Oy and Oz. The intersection points of these planes with the coordinate axes will give us the projections of points A and B on these axes. Denote the projections .


The desired 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 and . In the high school geometry course, it was proved that the square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions, therefore,. Based on the information of the first section of this article, we can write the following equalities, therefore,

where we get formula for finding the distance between points in space .

This formula is also valid if points A and B

  • match;
  • belong to one of the coordinate axes or a straight 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 got the formulas for finding the distance between two points of the coordinate line, plane and three-dimensional space. It's time to consider the solutions of typical examples.

The number of tasks in which the final step is to find the distance between two points according to their coordinates is truly enormous. A complete review of such examples is beyond the scope of this article. Here we restrict ourselves to examples in which the coordinates of two points are known and it is required to calculate the distance between them.

In mathematics, both algebra and geometry pose problems of finding the distance to a point or line from a given object. It is found in completely different ways, the choice of which depends on the initial data. Consider how to find the distance between given objects in different conditions.

Using Measuring Tools

At the initial stage of mastering mathematical science, they teach how to use elementary tools (such as a ruler, protractor, compass, triangle, and others). Finding the distance between points or lines with their help is not difficult at all. It is enough to attach the scale of divisions and write down the answer. One has only to know that the distance will be equal to the length of the straight line that can be drawn between the points, and in the case of parallel lines, the perpendicular between them.

Using theorems and axioms of geometry

In learning to measure distance without the help of special devices or For this, numerous theorems, axioms and their proofs are needed. Often the tasks of how to find the distance come down to the formation and search for its sides. To solve such problems, it is enough to know the Pythagorean theorem, the properties of triangles and how to transform them.

Points on the coordinate plane

If there are two points and their position on the coordinate axis is given, then how to find the distance from one to the other? The solution will include several steps:

  1. We connect the points with a straight line, the length of which will be the distance between them.
  2. Find the difference between the values ​​of the coordinates of the points (k; p) of each axis: |k 1 - k 2 |= d 1 and | p 1 - p 2 |= d 2 (we take the values ​​modulo, because the distance cannot be negative) .
  3. After that, we square the resulting numbers and find their sum: d 1 2 + d 2 2
  4. The final step is to extract from the resulting number. This will be the distance between the points: d \u003d V (d 1 2 + d 2 2).

As a result, the whole solution is carried out according to one formula, where the distance is equal to the square root of the sum of the squares of the difference in coordinates:

d \u003d V (| k 1 - k 2 | 2 + | p 1 - p 2 | 2)

If the question arises of how to find the distance from one point to another, then the search for an answer to it will not differ much from the above. The decision will be made according to the following formula:

d \u003d V ( | k 1 - k 2 | 2 + | p 1 - p 2 | 2 + | e 1 - e 2 | 2)

Parallel lines

The perpendicular drawn from any point lying on one straight line to the parallel will be the distance. When solving problems in a plane, it is necessary to find the coordinates of any point of one of the lines. And then calculate the distance from it to the second straight line. To do this, we bring them to the general form Ax + Vy + C = 0. It is known from the properties of parallel lines that their coefficients A and B will be equal. In this case, you can find by the formula:

d \u003d | C 1 - C 2 | / V (A 2 + B 2)

Thus, when answering the question of how to find the distance from a given object, it is necessary to be guided by the condition of the problem and the tools provided for its solution. They can be both measuring devices, and theorems and formulas.

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