Geometric figures are called similar if. Abstract: Similarity of figures

The definition of similarity transformation is the same both on the plane and in space. The transformation of a figure into a figure is called a similarity transformation if during this transformation the distances between points change (increase or decrease) by the same number of times. This means that if arbitrary points A and B of the figure F with this transformation go to the points of the figure where .

The number k is called the similarity coefficient. When the similarity transformation is a movement.

Homothety is a transformation of similarity.

Consider the properties of similarity transformation.

1. During a similarity transformation, three points A, B and C, lying on the same line, are transformed into three Lie points also lying on the same line. Moreover, if point B lies between points A and C, then the point lies between points

2. The similarity transformation transforms lines into straight lines, half-lines into half-lines, segments into segments, planes into planes.

3. The similarity transformation preserves the angles between half-lines.

4. Not every similarity transformation is a homothety.

In Figure 226, the figure is obtained from the figure F by homothety, and the figure is obtained from the figure by symmetry about the line. Converting F to F? is a similarity transformation, since it preserves the relations of distances between corresponding points, but this transformation is not a homothety.

For homothety in space the following theorem is true:

A homothety transformation in space transforms any plane not passing through the homothety center into parallel plane or into yourself.

Figure 227 shows two homothetic cubes with a homothety coefficient equal to 2. The plane ABCD goes into the parallel plane ABCD. The same can be said about the planes of other faces of the cube.

78. Similar figures.

Two figures F are called similar if they are transformed into each other by a similarity transformation. To indicate the similarity of figures, the symbol is used. The entry reads: “The figure is similar to figure F.”

From the properties of the similarity transformation it follows that similar polygons corresponding angles are equal and corresponding sides are proportional.

The notation assumes that the vertices combined by the similarity transformation are in the corresponding places, i.e. A goes to - to

For similar triangles the equalities are true

Two triangles are similar if the corresponding angles are equal and the corresponding sides are proportional. Let us formulate the similarity criteria for triangles.

ABSTRACT

On the topic: “Similarity of figures”

Completed:

student

Checked:

1. Similarity transformation

2. Properties of similarity transformation

3. Similarity of figures

4. Sign of similarity of triangles at two angles

5. Sign of similarity of triangles on two sides and the angle between them

6. Sign of similarity of triangles on three sides

7. Similarity of right triangles

8. Angles inscribed in a circle

9. Proportionality of segments of chords and secants of a circle

10. Problems on the topic “Similarity of figures”

1. SIMILARITY TRANSFORMATION

The transformation of a figure F into a figure F "is called a similarity transformation if, during this transformation, the distances between points change by the same number of times (Fig. 1). This means that if arbitrary points X, Y of the figure F, during a similarity transformation, turn into points X", Y"figure F", then X"Y" = k-XY, and the number k is the same for all points X, Y. The number k is called the similarity coefficient. For k = l, the similarity transformation is obviously a motion.

Let F - this figure and O - fixed point (Fig. 2). Let us draw a ray OX through an arbitrary point X of the figure F and plot on it a segment OX equal to k OX, where k is positive number. The transformation of the figure F, in which each of its points X goes to the point X", constructed in the indicated way, is called homothety with respect to the center O. The number k is called the homothety coefficient, the figures F and F" are called homothetic.

Theorem 1. Homothety is a similarity transformation

Proof. Let O be the homothety center, k be the homothety coefficient, X and Y be two arbitrary points of the figure (Fig. 3)

Fig.3 Fig.4

With homothety, points X and Y go to points X" and Y" on the rays OX and OY, respectively, and OX" = k·OX, OY" = k·OY. This implies the vector equalities OX" = kOX, OY" = kOY.

Subtracting these equalities term by term, we obtain: OY"-OX" = k (OY-OX).

Since OY" - OX"= X"Y", OY -OX=XY, then X"Y" = kХY. This means /X"Y"/=k /XY/, i.e. X"Y" = kXY. Consequently, homothety is a transformation of similarity. The theorem has been proven.

Similarity transformation is widely used in practice when making drawings of machine parts, structures, site plans, etc. These images are similar transformations of imaginary images in full size. The similarity coefficient is called scale. For example, if a section of terrain is depicted on a scale of 1:100, this means that one centimeter on the plan corresponds to 1 m on the ground.

Task. Figure 4 shows a plan of the estate on a scale of 1:1000. Determine the dimensions of the estate (length and width).

Solution. The length and width of the estate on the plan are 4 cm and 2.7 cm. Since the plan is made on a scale of 1:1000, the dimensions of the estate are respectively 2.7 x 1000 cm = 27 m, 4 x 100 cm = 40 m.

2. PROPERTIES OF SIMILARITY TRANSFORMATION

Just as for motion, it is proved that during a similarity transformation, three points A, B, C, lying on the same line, go into three points A 1, B 1, C 1, also lying on the same line. Moreover, if point B lies between points A and C, then point B 1 lies between points A 1 and C 1. It follows that the similarity transformation transforms lines into straight lines, half-lines into half-lines, and segments into segments.

Let us prove that the similarity transformation preserves the angles between half-lines.

Indeed, let the angle ABC be transformed by a similarity transformation with coefficient k into the angle A 1 B 1 C 1 (Fig. 5). Let us subject angle ABC to a homothety transformation relative to its vertex B with homothety coefficient k. In this case, points A and C will move to points A 2 and C 2. Triangles A 2 BC 2 and A 1 B 1 C 1 are equal according to the third criterion. From the equality of triangles it follows that the angles A 2 BC 2 and A 1 B 1 C 1 are equal. This means that angles ABC and A 1 B 1 C 1 are equal, which is what needed to be proven.

3. SIMILARITY OF FIGURES

Two figures are called similar if they are converted into each other by a similarity transformation. To indicate the similarity of figures, a special icon is used: ∞. The notation F∞F" reads like this: "The figure F is similar to the figure F"."

Let us prove that if the figure F 1 is similar to the figure F 2, and the figure F 2 is similar to the figure F 3, then the figures F 1 and F 3 are similar.

Let X 1 and Y 1 be two arbitrary points of the figure F 1. The similarity transformation that transforms the figure F 1 into F 2 transforms these points into points X 2, Y 2, for which X 2 Y 2 = k 1 X 1 Y 1.

The similarity transformation that transforms the figure F 2 into F 3 transforms the points X 2, Y 2 into the points X 3, Y 3, for which X 3 Y 3 = - k 2 X 2 Y 2.

From equalities

X 2 Y 2= kX 1 Y 1, X 3 Y 3 = k 2 X 2 Y 2

it follows that X 3 Y 3 - k 1 k 2 X 1 Y 1 . This means that the transformation of the figure F 1 into F 3, obtained by sequentially performing two similarity transformations, is similarity. Consequently, the figures F 1 and F 3 are similar, which is what needed to be proven.

In the notation for the similarity of triangles: ΔABC∞ΔA 1 B 1 C 1 - it is assumed that the vertices combined by the similarity transformation are in the corresponding places, i.e. A goes into A 1, B into B 1 and C into C 1.

From the properties of the similarity transformation it follows that for similar figures the corresponding angles are equal, and the corresponding segments are proportional. In particular, for similar triangles ABC and A 1 B 1 C 1

A=A 1, B=B 1, C=C 1

4. SIGNIFICANCE OF SIMILARITY OF TRIANGLES ACCORDING TO TWO ANGLES

Theorem 2. If two angles of one triangle are equal to two angles of another triangle, then such triangles are similar.

Proof. Let triangles ABC and A 1 B 1 C 1 A=A 1, B=B 1. Let us prove that ΔАВС~ΔА 1 В 1 С 1.

Let . Let us subject the triangle A 1 B 1 C 1 to a similarity transformation with a similarity coefficient k, for example, homothety (Fig. 6). In this case we get some triangle A 2 B 2 C 2, equal to a triangle ABC. Indeed, since the similarity transformation preserves angles, then A 2 = A 1, B 2 = B 1. This means that triangles ABC and A have 2 B 2 C 2 A = A 2 , B = B 2 . Next, A 2 B 2 = kA 1 B 1 =AB. Consequently, triangles ABC and A 2 B 2 C 2 are equal according to the second criterion (by side and adjacent angles).

Since triangles A 1 B 1 C 1 and A 2 B 2 C 2 are homothetic and therefore similar, and triangles A 2 B 2 C 2 and ABC are equal and therefore also similar, then triangles A 1 B 1 C 1 and ABC are similar . The theorem has been proven.

Task. Straight, parallel to the side AB of triangle ABC intersects its side AC at point A 1, and its side BC at point B 1. Prove that Δ ABC ~ ΔA 1 B 1 C.

Solution (Fig. 7). Triangles ABC and A 1 B 1 C have a common angle at vertex C, and angles CA 1 B 1 and CAB are equal as the corresponding angles of parallel AB and A 1 B 1 with secant AC. Therefore, ΔАВС~ΔА 1 В 1 С at two angles.

5. SIGNIFICANCE OF SIMILARITY OF TRIANGLES ON TWO SIDES AND THE ANGLE BETWEEN THEM

Theorem 3. If two sides of one triangle are proportional to two sides of another triangle and the angles formed by these sides are equal, then the triangles are similar.

Proof (similar to the proof of Theorem 2). Let triangles ABC and A 1 B 1 C 1 C=C 1 and AC=kA 1 C 1, BC=kB 1 C 1. Let us prove that ΔАВС~ΔА 1 В 1 С 1.

Let us subject the triangle A 1 B 1 C 1 to a similarity transformation with a similarity coefficient k, for example, homothety (Fig. 8).

In this case, we obtain a certain triangle A 2 B 2 C 2 equal to triangle ABC. Indeed, since the similarity transformation preserves angles, then C 2 = = C 1 . This means that triangles ABC and A have 2 B 2 C 2 C=C 2. Next, A 2 C 2 = kA 1 C 1 =AC, B 2 C 2 = kB 1 C 1 =BC. Consequently, triangles ABC and A 2 B 2 C 2 are equal according to the first criterion (two sides and the angle between them).

Task. In triangle ABC with acute angle C, the altitudes AE and BD are drawn (Fig. 9). Prove that ΔABC~ΔEDC.

Solution. Triangles ABC and EDC have a common vertex angle C. Let us prove the proportionality of the sides of the triangles adjacent to this angle. We have EC = AC cos γ, DC = BC cos γ. That is, the sides adjacent to angle C are proportional for triangles. This means ΔABC~ΔEDC on two sides and the angle between them.

6. SIGNIFICANCE OF SIMILARITY OF TRIANGLES ON THREE SIDES

Theorem 4. If the sides of one triangle are proportional to the sides of another triangle, then such triangles are similar.

Proof (similar to the proof of Theorem 2). Let triangles ABC and A 1 B 1 C 1 AB = kA 1 B 1, AC = kA 1 C 1, BC = kB 1 C 1. Let us prove that ΔАВС~ΔА 1 В 1 С 1.

Let us subject the triangle A 1 B 1 C 1 to a similarity transformation with a similarity coefficient k, for example, homothety (Fig. 10). In this case, we obtain a certain triangle A 2 B 2 C 2 equal to triangle ABC. Indeed, for triangles the corresponding sides are equal:

A 2 B 2 = kA 1 B 1 = AB, A 2 C 2 = kA 1 C 1 = AC, B 2 C 2 = kB 1 C 1 = BC.

Consequently, the triangles are equal according to the third criterion (on three sides).

Task. Prove that the perimeters of similar triangles are related as corresponding sides.

Solution. Let ABC and A 1 B 1 C 1 be similar triangles. Then the sides of triangle A 1 B 1 C 1 are proportional to the sides of triangle ABC, i.e. A 1 B 1 =kAB, B 1 C 1 = kBC, A 1 C 1 =kAC. Adding these equalities term by term, we get:

A 1 B 1 + B 1 C 1 +A 1 C 1 =k(AB+BC+AC).

that is, the perimeters of the triangles are related as corresponding sides.

7. SIMILARITY OF RECTANGULAR TRIANGLES

U right triangle one angle is right. Therefore, according to Theorem 2, for two right triangles to be similar, it is enough that they each have an equal acute angle.

Using this test for the similarity of right triangles, we will prove some relations in triangles.

Let ABC be a right triangle with right angle C. Draw the altitude CD from the vertex right angle(Fig. 11).

Triangles ABC and CBD have common angle at vertex B. Therefore, they are similar: ΔABC~ΔCBD. From the similarity of triangles it follows that the corresponding sides are proportional:

This relationship is usually formulated as follows: a leg of a right triangle is the proportional mean between the hypotenuse and the projection of this leg onto the hypotenuse.

Right triangles ACD and CBD are also similar. They have equal acute angles at vertices A and C. From the similarity of these triangles, the proportionality of their sides follows:

This relationship is usually formulated as follows: the height of a right triangle drawn from the vertex of a right angle is the average proportional between the projections of legs I onto the hypotenuse.

Let's prove next property Triangle bisectors: The bisector of a triangle divides the opposite side into segments proportional to the other two sides.

Let CD be the bisector of triangle ABC (Fig. 12). If the triangle ABC is isosceles with base AB, then the indicated property of the bisector is obvious, since in this case the bisector CD is also the median.

Let's consider general case, when AC≠BC. Let us drop the perpendiculars AF and BE from vertices A and B onto the line CD.

Right triangles ACF and VSE are similar, since they have equal acute angles at vertex C. From the similarity of the triangles, the proportionality of the sides follows:

Right triangles ADF and BDE are also similar. Their angles at vertex D are equal to vertical ones. From the similarity of triangles follows the proportionality of the sides:

Comparing this equality with the previous one, we get:

that is, the segments AD and BD are proportional to the sides AC and BC, which was what needed to be proven.

8. ANGLES INCLUDED IN A CIRCLE

An angle breaks a plane into two parts. Each of the parts is called a plane angle. In Figure 13, one of the plane angles with sides a and b is shaded. Plane angles with common sides are called supplementary.

If a plane angle is part of a half-plane, then its degree measure is called degree measure regular angle with the same sides. If a plane angle contains a half-plane, then its degree measure is taken to be 360° - α, where α is the degree measure of an additional plane angle (Fig. 14).

Rice. 13 Fig.14

A central angle in a circle is a plane angle with a vertex at its center. The part of the circle located inside the plane angle is called the arc of the circle corresponding to this central angle (Fig. 15). The degree measure of an arc of a circle is the degree measure of the corresponding central angle.

Rice. 15 Fig. 16

An angle whose vertex lies on a circle and whose sides intersect this circle is called inscribed in a circle. Angle BAC in Figure 16 is inscribed in a circle. Its vertex A lies on the circle, and its sides intersect the circle at points B and C. It is also said that angle A rests on chord BC. Straight line BC divides the circle into two arcs. The central angle corresponding to that of these arcs that does not contain point A is called central angle, corresponding to a given inscribed angle.

Theorem 5. An angle inscribed in a circle is equal to half the corresponding central angle.

Proof. Let's consider first special case, when one of the sides of the angle passes through the center of the circle (Fig. 17, a). Triangle AOB is isosceles because its sides OA and OB are equal in radii. Therefore, angles A and B of the triangle are equal. And since their sum is equal outer corner triangle at vertex O, then angle B of the triangle is equal to half the angle AOC, which is what needed to be proven.

The general case is reduced to the considered special case by drawing the auxiliary diameter BD (Fig. 17, b, c). In the case presented in Figure 17, b, ABC = CBD + ABD = ½ COD + ½ AOD = ½ AOC.

In the case presented in Figure 17, c,

ABC= CBD - ABD = ½ COD - ½ AOD = ½ AOC.

The theorem is completely proven.

9. PROPORTIONALITY OF SEGMENTS OF CHORDS AND SECANTS OF A CIRCLE

If chords AB and CD of a circle intersect at point S

ToAS·BS=CS·DS.

Let us first prove that triangles ASD and CSB are similar (Fig. 19). The inscribed angles DCB and DAB are equal by the corollary of Theorem 5. The angles ASD and BSC are equal as vertical angles. From the equality of the indicated angles it follows that triangles ASZ and CSB are similar.

From the similarity of triangles follows the proportion

AS BS = CS DS, which is what we needed to prove

Fig.19 Fig.20

If two secants are drawn from point P to a circle, intersecting the circle at points A, B and C, D, respectively, then

Let points A and C be the points of intersection of the secants with the circle closest to point P (Fig. 20). Triangles PAD and PCB are similar. They have a common angle at vertex P, and the angles at vertices B and D are equal according to the property of angles inscribed in a circle. From the similarity of triangles follows the proportion

Hence PA·PB=PC·PD, which is what needed to be proven.

10. Problems on the topic “Similarity of figures”

ABSTRACT

On the topic: “Similarity of figures”

Completed:

student

Checked:

1. Similarity transformation

2. Properties of similarity transformation

3. Similarity of figures

4. Sign of similarity of triangles at two angles

5. Sign of similarity of triangles on two sides and the angle between them

6. Sign of similarity of triangles on three sides

7. Similarity of right triangles

8. Angles inscribed in a circle

9. Proportionality of segments of chords and secants of a circle

10. Problems on the topic “Similarity of figures”

1. SIMILARITY TRANSFORMATION

Let F be a given figure and O be a fixed point (Fig. 2). Let us draw a ray OX through an arbitrary point X of the figure F and plot on it a segment OX" equal to k·OX, where k is a positive number. The transformation of the figure F, in which each of its points X goes to the point X", constructed in the indicated way, is called homothety relative to the center O. The number k is called the homothety coefficient, the figures F and F" are called homothetic.

Theorem 1. Homothety is a similarity transformation

Proof. Let O be the homothety center, k be the homothety coefficient, X and Y be two arbitrary points of the figure (Fig. 3)

Fig.3 Fig.4

With homothety, points X and Y go to points X" and Y" on the rays OX and OY, respectively, and OX" = k·OX, OY" = k·OY. This implies the vector equalities OX" = kOX, OY" = kOY. Subtracting these equalities term by term, we obtain: OY"-OX" = k (OY-OX). Since OY" - OX"= X"Y", OY -OX=XY, then X"Y" = kХY. This means /X"Y"/=k /XY/, i.e. X"Y" = kXY. Consequently, homothety is a transformation of similarity. The theorem has been proven.

Task. Figure 4 shows a plan of the estate on a scale of 1:1000. Determine the dimensions of the estate (length and width).

2. PROPERTIES OF SIMILARITY TRANSFORMATION

Let us prove that the similarity transformation preserves the angles between half-lines.

3. SIMILARITY OF FIGURES

Let us prove that if the figure F 1 is similar to the figure F 2, and the figure F 2 is similar to the figure F 3, then the figures F 1 and F 3 are similar.

The similarity transformation that transforms the figure F 2 into F 3 transforms the points X 2, Y 2 into the points X 3, Y 3, for which X 3 Y 3 = - k 2 X 2 Y 2.

From equalities

X 2 Y 2= kX 1 Y 1, X 3 Y 3 = k 2 X 2 Y 2

A=A 1, B=B 1, C=C 1

4. SIGNIFICANCE OF SIMILARITY OF TRIANGLES ACCORDING TO TWO ANGLES

Theorem 2. If two angles of one triangle are equal to two angles of another triangle, then such triangles are similar.

Proof. Let triangles ABC and A have 1 B 1 C 1

Examples

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

Similar figures in the picture have the same colors.

Related definitions

Properties

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

The set of all similarities of n-dimensional Euclidean, pseudo-Euclidean, Riemannian, pseudo-Riemannian or Finsler space is r-member group of Lie transformations, called the group of similar (homothetic) transformations of the corresponding space. In each of the spaces of the specified types r-member group of similar Lie transformations contains ( r− 1) -membered normal subgroup of motions.

Geometry

Similarity of figures

Properties of similar figures

Theorem. When a figure is similar to a figure, and a figure is similar to a figure, then the figures and similar.
From the properties of the similarity transformation it follows that for similar figures the corresponding angles are equal, and the corresponding segments are proportional. For example, in similar triangles ABC And :
; ; ;
.

Signs of similarity of triangles

Theorem 1. If two angles of one triangle are respectively equal to two angles of the second triangle, then such triangles are similar.
Theorem 2. If two sides of one triangle are proportional to two sides of the second triangle and the angles formed by these sides are equal, then the triangles are similar.
Theorem 3. If the sides of one triangle are proportional to the sides of the second triangle, then such triangles are similar.
From these theorems follow facts that are useful for solving problems.
1. A straight line parallel to a side of a triangle and intersecting its other two sides cuts off a triangle similar to this one from it.
In the picture.

2. For similar triangles, the corresponding elements (altitudes, medians, bisectors, etc.) are related as corresponding sides.
3. For similar triangles, the perimeters are related as corresponding sides.
4. If ABOUT- point of intersection of trapezoid diagonals ABCD, That .
In the figure in a trapezoid ABCD:.

5. If the continuation of the sides of the trapezoid ABCD intersect at a point K, then (see figure) .
.

Similarity of right triangles

Theorem 1. If right triangles have equal acute angle, then they are similar.
Theorem 2. If two legs of one right triangle are proportional to two legs of the second right triangle, then these triangles are similar.
Theorem 3. If the leg and hypotenuse of one right triangle are proportional to the leg and hypotenuse of the second right triangle, then such triangles are similar.
Theorem 4. The altitude of a right triangle drawn from the vertex of a right angle splits the triangle into two right triangles similar to this one.
In the picture .

The following follows from the similarity of right triangles.
1. The leg of a right triangle is the mean proportional between the hypotenuse and the projection of this leg onto the hypotenuse:
; ,
or
; .
2. The height of a right triangle drawn from the vertex of a right angle is the average proportional between the projections of the legs onto the hypotenuse:
, or .
3. Property of the bisector of a triangle:
the bisector of a triangle (arbitrary) divides the opposite side triangle into segments proportional to the other two sides.
In the picture in B.P.- bisector.
, or .

The similarity of equilateral and isosceles triangles

1. Everything equilateral triangles similar.
2. If isosceles triangles have equal angles between the sides, then they are similar.
3. If isosceles triangles have proportional bases and side, then they are similar.