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Any similarity transformation is a result. Similarity transformation

05.11.2020

Lesson topic: Similarity transformation. Similar figures.

Lesson type: lesson in communication and assimilation of new knowledge.

Lesson objectives:

Educational:

give the concept of transforming the similarity of figures;

similarity transformation properties;

Developing:

1. To develop practical skills in using the similarity of figures in solving problems.

2. To create conditions for real assessment of students' knowledge and capabilities.

Educational:

1. Development of skills of control and mutual control.

2.Education of accuracy in the execution of drawings and notes

During the classes.

1. Organization for the lesson. preparing students for the perception of new knowledge, communicating the topic and objectives of the lesson.

2. Goal setting:

know : definition and properties of similarity transformation, homothety

be able to: build similar and homothetic figures with a given coefficient of similarity

3. Updating previous knowledge

Repetition of the passed material, closely related to the study of new (frontally oral, MD) Working at the blackboard

Card number 1

Construct the figure into which  ABC goes, with parallel transfer to the vector

Card number 2.

Construct the figure into which the segment AB goes when turning about the point O at an angle of 90 o

TO artwork number 3

Construct the figure into which  ABC goes, with symmetry about the point O

Card number 4

Construct the figure into which the figure F goes with symmetry about the straight line y

3) Checking the completion of tasks at the board . To emphasize again that any motion preserves the distance between the points, and therefore when moving, the figures turn into equal figures.

Define the kind of transformations:

What do these transformations have in common?

Motion properties:

When moving, a straight line turns into a straight line, a ray - into a ray, a segment - into a segment.

Distances between points are maintained.

The angles between the beams are preserved.

Corollary: When moving, the figure passes into a figure equal to it !!!

4. Explanation of the new material (lecture with supporting notes, SR with a textbook - design)

First, complete the following assignment: draw in your notebooks, and we on the board, sketch out the class plan.

Why is the table on the plan depicted as a rectangle (and not a circle or

square)?

What is the difference and what do the tables have in common on the plans on the blackboard and in notebooks? (differ in size, but have the same shape).

In life, there are often objects that have the same shape, but different sizes. Such are, for example, photographs of one and the same person, made from one negative in different sizes, plans of a building or an entire city, areas drawn on different scales.

Such figures are usually called like , and a transformation that transforms one figure F into a similar figure F is called a similarity transformation.

On display are posters depicting figures that have the same shape but different sizes. Students are encouraged to give examples of such objects from life.

In order to give a rigorous mathematical definition of a similarity transformation, it is necessary to highlight the properties of this transformation.

Each student has a card (fig. 1)

Similar figures F and F are given. Measure and compare distances AB and AB, BC and B 1 C 1, etc. What can you see the relationship between the distances in such figures? (All distances change by the same number of times, in the drawing by 2 times).

A transform in which the shape retains its appearance but resizes called similarity transformation

those. ХУ "\u003d к · ХУ; АВ \u003d к · AB.

The number k is called the coefficient of similarity.

Similarity transformation has wide practical applications, in particular, when making machine parts, drawing up maps and plans of the area. In this case, the similarity coefficient is called scale.

A special case of similarity transformation is homothety transformation .

Let F be a given figure, O a fixed point, k a positive number. Draw a ray OX through an arbitrary point X of the figure F and put on it the segment OX "equal to k · OX.

Any point X on the plane will correspond to a point X "satisfying the equality OX" \u003d to OX, the transformation is called homothety, relative to the center O with the coefficient to.

The number k is called homothety coefficient, and figures F and F  are called homothetic.

For shapes F and F "specify homothetic points. How are any pair of points and center O located? (On one beam).

What is the peculiarity of the arrangement of homothetic segments? (They are parallel).

Are such figures always homothetic? (Refer to card Fig. 2)

Are homothetic figures always alike?

The theorem answers the last question: Homotetia is a transformation of similarity.

Make a poster: Similarity transformation (properties)

the distance between any two points increases or decreases by the same number of times

the corresponding sides of such figures are parallel

With homothety, only the angles are preserved !!!

center and homothetic points are located on one straight line

5, Checking the understanding of new material :

Construct a point (segment, figure) homothetic to the given one if the homothety coefficient is equal to k.

) k \u003d 2 b) k \u003d 3 c) k \u003d 2

Practical work on cards in 2 versions:

Option 1.

Given a rectangle and a point O. Construct a figure homothetic to this rectangle with respect to the center O with the coefficient k \u003d -2.

Option 2.

Given a square and a point O. Construct a figure homothetic to this square relative to the center O with the coefficient k \u003d 0.5.

Depending on the preparedness of the class, you can organize the exchange of cards between neighbors.

6 ... Lesson summary: (systematization and generalization of knowledge;)

Mark students who were active in the lesson. Report and comment on the grades

7. Homework § №

Examples of

Each homothety is a similarity.
Each movement (including the identical one) can also be considered as a similarity transformation with the coefficient k = 1 .

Similar shapes in the picture have the same colors.

Related definitions

Properties

In metric spaces, the same as in n -dimensional Riemannian, pseudo-Riemannian, and Finsler spaces, similarity is defined as a transformation that transforms the metric of the space into itself up to a constant factor.

The set of all similarities of an n-dimensional Euclidean, pseudo-Euclidean, Riemannian, pseudo-Riemannian, or Finsler space is r -term Lie transformation group called the group of similar (homothetic) transformations of the corresponding space. In each of the spaces of the indicated types r -term group of similar Lie transformations contains ( r - 1) -term normal subgroup of movements.

GEOMETRY
Lesson plans for 10 grades

Lesson 50

Theme. Similarity transformation and its properties

The purpose of the lesson: the formation of students' knowledge about the similarity of spatial figures, the study of the properties of the transformation of similarity and their application to solving problems.

Equipment: cube and tetrahedron models.

During the classes

I. Checking homework

1. Collective discussion of control questions No. 9-11 and solving problems No. 23-25 \u200b\u200b(1).

2. Mathematical dictation.

With a parallel transfer, point A goes to point B: option 1 - A (6; 7; 8), B (8; 2; 6); option 2 - A (2; 1; 3), B (1; 0; 7). Write down:

1) parallel transfer formulas;

2) coordinates of point C, which was formed as a result of parallel transfer of point O (0; 0; 0);

3) coordinates of point D, which was formed as a result of parallel translation of point C;

4) coordinates of point F, into which point M (1; 1; 1) has passed as a result of parallel translation;

5) the formula for parallel transfer, in which point B goes to point A.

Answer. Option 1.1) x1 \u003d x + 2, y1 \u003d y - 5, z1 \u003d z - 2; 2) C (2; -5; -2); 3) D (4; -10; -4); 4) F (-1; 6; 3); 5) x 1 \u003d x - 2, y1 \u003d y + 5, z 1 \u003d z + 2.

Option 2.1) x 1 \u003d x - 1, y 1 \u003d y -1, z 1 \u003d z + 4; 2) C (-1; -1; 4); 3) D (-2; -2, -8); 4) F (2; 2; -3); 5) x 1 \u003d x + 1, y 1 \u003d y + 1, z 1 \u003d z - 4.

II. Perception and awareness of new material

Similarity transformation in space

The transformation of the figure F into the figure F 1 is called similarity transformation if arbitrary points X and Y of the figure F go to points X 1 and Y 1 of the figure F1 such that X1Y 1 \u003d k XY.

Similarity transformation in space, as well as on a plane, transforming straight lines into straight lines, straight into straight, segments into segments and preserves the angles between straight lines.

Two figures in space are called similar if they are translated into each other by a similarity transformation.

The simplest transformation of similarity in space is homothety.

Homotetiya relative to the center O with a coefficient k is a transformation that takes an arbitrary point X to a point X1 of the ray OX, such that OX1 \u003d k OX. (fig. 270).

The transformation of the homothety in space takes any plane that does not pass through the center of the homothety into a parallel plane (or into itself, when k \u003d 1).

The proof is carried out as in the textbook.

Solving problems

1. What is a figure similar to a cube with a coefficient of similarity: a) k \u003d 2; b) k \u003d; c) k \u003d 1?

2. Construct a figure homothetic to the given tetrahedron ABCD with respect to the point S (Fig. 271) with the homothetic coefficient: a) k \u003d 2; b) k \u003d; c) k \u003d 1.

3. In what figure does the plane pass during the homothety, if this plane passes through the center of the homothety?

4. Construct a figure into which the cube will go in the case of homothety relative to the point S (Fig. 272) with the homothety coefficient.

5. Triangle ABC is a homothetic triangle A1 B1 C1 relative to the origin with the homothetic coefficient k \u003d 2. Find the coordinates of the vertices of the triangle A1 B1 C1, if A (1; 0; 0), B (0; 3; 0), C (0; 0; - 3).

6. Problem number 29 from the textbook (p. 56).

III. Homework

§4, p. 30; control questions No. 12-13; task number 28 (p. 56).

IV. Lesson summary

Question to the class

1) What is similarity transformation? List its properties.

2) What transformation is called a homothety with center O and coefficient A?

3) In the triangular pyramid SABC, the section MNK is drawn so that SM \u003d 2MA, SK \u003d 2KC, SN \u003d 2NB (Fig. 273). Indicate which of the following statements are correct and which are incorrect:

a) with a homothety with a center S and a coefficient, the point M goes over to the point A;

b) with a homothety with a center S and a coefficient, the ABC plane goes over to the MNK plane;

c) AB \u003d MN;

d) with a homothety with a center S and a coefficient - the SABC pyramid goes into the SMNK pyramid.

4) In the cube ABCDA1 B1 C1 D1, the section BDC 1 and MNK is drawn, where points M, N, K are the midpoints of the edges CC1, BC, DC (Fig. 234). Indicate which of the following statements are correct and which are incorrect:

a) with a homothety with a center C and a coefficient of 0.5, point M goes to point C1;

b) with a homothety with a center C and a coefficient of 2, the MNK plane passes into the BDC1 plane;

c) BD \u003d 2 NK;

d) the sectional area of \u200b\u200bBDC 1 is 4 times larger than the sectional area of \u200b\u200bMNK.

Let us consider some figure and the figure obtained from it by the similarity transformation (center O, coefficient k, see Fig. 263). Let us establish the basic properties of the similarity transformation.

1. Similarity transformation establishes a one-to-one correspondence between the points of the figures.

This means that for a given center O and a coefficient of similarity k, each point of the first figure corresponds to a uniquely defined point of the second figure and that, conversely, every point of the second figure is obtained by transforming the only point of the first figure.

Evidence. The fact that any point A of the original figure corresponds to a certain point A of the transformed figure follows from the definition indicating the exact method of transformation. It is easy to see that, and vice versa, the transformed point A defines the original point A uniquely: both points must lie on one ray at and on opposite rays at and the ratio of their distances to the beginning of the ray O is known: at Therefore, point A, lying at a known distance from the beginning O, is uniquely defined.

The next property can be called the reciprocity property.

2. If some figure is obtained from another figure by a similarity transformation with the center O and the similarity coefficient k, then, and vice versa, the original figure can be obtained by transforming the similarity from the second figure with the same similarity center and similarity coefficient

This property, obviously, follows at least from the reasoning given in the proof of property 1. The reader is left to check that the relation is true for both cases: KO and

The figures obtained from one another by the transformation of similarity are called homothetic or similarly located.

3. Any points lying on one straight line are transformed under homothety into cheeks lying on one straight line parallel to the original (coinciding with it if it passes through O).

Evidence. The case when the straight line passes through O is clear; any points of this line go to points of the same line. Consider the general case: let (Fig. 266) A, B, C - three points of the main figure lying on one straight line; let A be the image of point A under similarity transformation.

Let us show that the images B and C also lie on AK. Indeed, the drawn direct and direct AC cut off the proportional parts on OA, OB, OS: Thus, it can be seen that the points lying on the rays OB and OS and on the straight line AK (similarly it will turn out and at are respectively for B and C. You can say that during the transformation of the similarity, any straight line that does not pass through the center of the similarity is transformed into a straight line parallel to itself.

It is already clear from what has been said that any segment is also transformed into a segment.

4. When transforming the similarity, the ratio of any pair of corresponding segments is equal to the same number - the coefficient of similarity.

Evidence. A distinction should be made between two cases.

1) Let the given segment AB not lie on the ray passing through the center of the similarity (Fig. 266). In this case, these two segments - the original AB and, like it, the corresponding AB - are the segments of parallel straight lines enclosed between the sides of the angle AOB. Applying the property of item 203, we find what was required to prove.

2) Let the given segment, and hence the corresponding one, lie on one straight line passing through the center of the similarity (segments AB and AB in Fig. 267). From the definition of such a transformation, we get whence, forming a derivative proportion, we find what was required to prove.

5. The angles between the corresponding straight lines (segments) of similarly located figures are equal.

Evidence. Let the given angle and the angle corresponding to it when transforming the similarity with the center O and some coefficient k. In fig. 263, 264 presents two options:. In any of these cases, by property 3, the sides of the corners are pairwise parallel. Moreover, in one case, both pairs of sides are equally directed, in the second, both are oppositely directed. Thus, by the property of angles with parallel sides, the angles are equal.

So it's proven

Theorem 1. For similarly located figures, any corresponding pairs of line segments are in the same constant ratio, equal to the coefficient of similarity; any pairs of corresponding angles are equal.

Thus, of two similarly located figures, either one can be considered an image of the other at some chosen scale.

Example 1. Construct a figure similarly located with a square ABCD (Fig. 268) at a given center of similarity O and a coefficient of similarity

Decision. We connect one of the vertices of the square (for example, A) with the center O and build a point A such that this point will correspond to A in the similarity transformation. Further construction is convenient to carry out as follows: connect the remaining vertices of the square with O and draw straight lines through A, parallel to the corresponding sides AB and AD. At the points of their intersection with O B and and, the vertices B and D will be placed. We also draw the BC parallel to the BC and find the fourth vertex C. Why is ABCD also a square? Justify yourself!

Example 2. In fig. 269 \u200b\u200bshows a pair of similarly spaced triangular plates. One of them shows point K. Construct the corresponding point on the second.

Decision. Let us connect K with one of the vertices of the triangle, for example with A. The resulting line will intersect the side BC at point L. Find the corresponding point L as the intersection of BC and construct the required point K on the segment, intersecting it with the line OK.

Theorem 2. A figure homothetic to a circle (circle) is again a circle (circle). The centers of the circles correspond similarly.

Evidence. Let C be the center of a circle Φ of radius R (Fig. 270), O - the center of similarity. The similarity coefficient is denoted by k. Let C be a point similar to the center C of a circle. (We do not yet know whether it will retain the role of the center!) Consider all possible radii of the circle, all of them, when transforming the similarity, will turn into segments parallel to themselves and having equal lengths

Thus, all ends of the transformed radii will be located again on the same circle with center C and radius R, which is what we had to prove.

Conversely, any two circles are in homothetic correspondence (in the general case, even in two ways, with two different centers).

Indeed, we draw any radius of the first circle (the radius of the CM in Fig. 271) and both parallel to it the radius of the second circle. The points of intersection of the line of the centers of the SS and the straight lines connecting the end of the radius CM with the ends of the radii parallel to it, that is, points O and O "in Fig. 271, can be taken as centers of homothety (of the first and second kind).

In the case of concentric circles, there is a single center of homothety — the common center of the circles; equal circles are in accordance with the homothety with the center in the middle of the segment.

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