In this lesson we will look at another characteristic of some figures - axial and central symmetry. We encounter axial symmetry every day when we look in the mirror. Central symmetry is very common in living nature. At the same time, figures that have symmetry have a number of properties. In addition, we will later learn that axial and central symmetries are types of movements with the help of which a whole class of problems is solved.
This lesson is devoted to axial and central symmetry.
Definition
The two points are called symmetrical relatively straight if:
In Fig. 1 shows examples of points symmetrical with respect to a straight line and , and .
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Let us also note the fact that any point on a line is symmetrical to itself relative to this line.
Figures can also be symmetrical relative to a straight line.
Let us formulate a strict definition.
Definition
The figure is called symmetrical relative to straight, if for each point of the figure, a point symmetrical to it relative to this straight line also belongs to the figure. In this case the line is called axis of symmetry. The figure has axial symmetry.
Let's look at a few examples of figures that have axial symmetry and their axes of symmetry.
Example 1
The angle has axial symmetry. The axis of symmetry of the angle is the bisector. Indeed: let’s lower a perpendicular to the bisector from any point of the angle and extend it until it intersects with the other side of the angle (see Fig. 2).
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(since - the common side, 
(property of a bisector), and triangles are right-angled). Means, . Therefore, the points are symmetrical with respect to the bisector of the angle.
It follows from this that isosceles triangle has axial symmetry relative to the bisector (height, median) drawn to the base.
Example 2
An equilateral triangle has three axes of symmetry (bisectors/medians/altitudes of each of the three angles (see Fig. 3).
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Example 3
A rectangle has two axes of symmetry, each of which passes through the midpoints of its two opposite sides (see Fig. 4).
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Example 4
A rhombus also has two axes of symmetry: straight lines, which contain its diagonals (see Fig. 5).
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Example 5
A square, which is both a rhombus and a rectangle, has 4 axes of symmetry (see Fig. 6).
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Example 6
For a circle, the axis of symmetry is any straight line passing through its center (that is, containing the diameter of the circle). Therefore, a circle has infinitely many axes of symmetry (see Fig. 7).
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Let us now consider the concept central symmetry.
Definition
The points are called symmetrical relative to the point if: - the middle of the segment.
Let's look at a few examples: in Fig. 8 shows the points and , as well as and , which are symmetrical with respect to the point , and the points and are not symmetrical with respect to this point.
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Some figures are symmetrical about a certain point. Let us formulate a strict definition.
Definition
The figure is called symmetrical about the point, if for any point of the figure the point symmetrical to it also belongs to this figure. The point is called center of symmetry, and the figure has central symmetry.
Let's look at examples of figures with central symmetry.
Example 7
For a circle, the center of symmetry is the center of the circle (this is easy to prove by recalling the properties of the diameter and radius of a circle) (see Fig. 9).
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Example 8
For a parallelogram, the center of symmetry is the point of intersection of the diagonals (see Fig. 10).
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Let's solve several problems on axial and central symmetry.
Task 1.
How many axes of symmetry does the segment have?
A segment has two axes of symmetry. The first of them is a line containing a segment (since any point on a line is symmetrical to itself relative to this line). The second is the perpendicular bisector to the segment, that is, a straight line perpendicular to the segment and passing through its middle.
Answer: 2 axes of symmetry.
Task 2.
How many axes of symmetry does a straight line have?
A straight line has infinitely many axes of symmetry. One of them is the line itself (since any point on the line is symmetrical to itself relative to this line). And also the axes of symmetry are any lines perpendicular to a given line.
Answer: there are infinitely many axes of symmetry.
Task 3.
How many axes of symmetry does the beam have?
The ray has one axis of symmetry, which coincides with the line containing the ray (since any point on the line is symmetrical to itself relative to this line).
Answer: one axis of symmetry.
Task 4.
Prove that the lines containing the diagonals of a rhombus are its axes of symmetry.
Proof:
Consider a rhombus. Let us prove, for example, that the straight line is its axis of symmetry. It is obvious that the points are symmetrical to themselves, since they lie on this line. In addition, the points and are symmetrical with respect to this line, since 
. Let us now choose an arbitrary point and prove that the point symmetric with respect to it also belongs to the rhombus (see Fig. 11).
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Draw a perpendicular to the line through the point and extend it until it intersects with . Consider triangles and . These triangles are right-angled (by construction), in addition, they have: - a common leg, and 
(since the diagonals of a rhombus are its bisectors). So these triangles are equal: 
. This means that all their corresponding elements are equal, therefore: . From the equality of these segments it follows that the points and are symmetrical with respect to the straight line. This means that it is the axis of symmetry of the rhombus. This fact can be proven similarly for the second diagonal.
Proven.
Task 5.
Prove that the point of intersection of the diagonals of a parallelogram is its center of symmetry.
Proof:
Consider a parallelogram. Let us prove that the point is its center of symmetry. It is obvious that the points and , and are pairwise symmetrical with respect to the point , since the diagonals of a parallelogram are divided in half by the point of intersection. Let us now choose an arbitrary point and prove that the point symmetric with respect to it also belongs to the parallelogram (see Fig. 12).
Movement concept
Let us first examine the concept of movement.
Definition 1
A mapping of a plane is called a motion of the plane if distances are preserved during this mapping.
There are several theorems related to this concept.
Theorem 2
The triangle, when moving, turns into an equal triangle.
Theorem 3
Any figure, when moving, transforms into a figure equal to it.
Axial and central symmetry are examples of motion. Let's look at them in more detail.
Axial symmetry
Definition 2
Points $A$ and $A_1$ are called symmetrical with respect to the line $a$ if this line is perpendicular to the segment $(AA)_1$ and passes through its center (Fig. 1).
Figure 1.
Let's consider axial symmetry using an example problem.
Example 1
Construct a symmetrical triangle for a given triangle relative to any of its sides.
Solution.
Let us be given a triangle $ABC$. We will construct its symmetry with respect to the side $BC$. The side $BC$ with axial symmetry will transform into itself (follows from the definition). Point $A$ will go to point $A_1$ as follows: $(AA)_1\bot BC$, $(AH=HA)_1$. Triangle $ABC$ will transform into triangle $A_1BC$ (Fig. 2).
Figure 2.
Definition 3
A figure is called symmetrical with respect to straight line $a$ if every symmetrical point of this figure is contained in the same figure (Fig. 3).
Figure 3.
Figure $3$ shows a rectangle. It has axial symmetry with respect to each of its diameters, as well as with respect to two straight lines that pass through the centers of opposite sides of a given rectangle.
Central symmetry
Definition 4
Points $X$ and $X_1$ are called symmetrical with respect to point $O$ if point $O$ is the center of the segment $(XX)_1$ (Fig. 4).
Figure 4.
Let's consider central symmetry using an example problem.
Example 2
Construct a symmetrical triangle for a given triangle at any of its vertices.
Solution.
Let us be given a triangle $ABC$. We will construct its symmetry relative to the vertex $A$. The vertex $A$ with central symmetry will transform into itself (follows from the definition). Point $B$ will go to point $B_1$ as follows: $(BA=AB)_1$, and point $C$ will go to point $C_1$ as follows: $(CA=AC)_1$. Triangle $ABC$ will transform into triangle $(AB)_1C_1$ (Fig. 5).
Figure 5.
Definition 5
A figure is symmetrical with respect to point $O$ if every symmetrical point of this figure is contained in the same figure (Fig. 6).
Figure 6.
Figure $6$ shows a parallelogram. It has central symmetry about the point of intersection of its diagonals.
Example task.
Example 3
Let us be given a segment $AB$. Construct its symmetry with respect to the line $l$, which does not intersect the given segment, and with respect to the point $C$ lying on the line $l$.
Solution.
Let us schematically depict the condition of the problem.
Figure 7.
Let us first depict axial symmetry with respect to straight line $l$. Since axial symmetry is a movement, then by Theorem $1$, the segment $AB$ will be mapped onto the segment $A"B"$ equal to it. To construct it, we will do the following: draw straight lines $m\ and\n$ through points $A\ and\B$, perpendicular to straight line $l$. Let $m\cap l=X,\ n\cap l=Y$. Next we draw the segments $A"X=AX$ and $B"Y=BY$.
Figure 8.
Let us now depict the central symmetry with respect to the point $C$. Since central symmetry is a motion, then by Theorem $1$, the segment $AB$ will be mapped onto the segment $A""B""$ equal to it. To construct it, we will do the following: draw the lines $AC\ and\ BC$. Next we draw the segments $A^("")C=AC$ and $B^("")C=BC$.
Figure 9.
So, as for geometry: there are three main types of symmetry.
Firstly, central symmetry (or symmetry about a point) - this is a transformation of the plane (or space), in which a single point (point O - the center of symmetry) remains in place, while the remaining points change their position: instead of point A, we get point A1 such that point O is the middle of the segment AA1. To construct a figure Ф1, symmetrical to the figure Ф relative to point O, you need to draw a ray through each point of the figure Ф, passing through point O (center of symmetry), and on this ray lay a point symmetrical to the chosen one relative to point O. The set of points constructed in this way will give the figure F1.
Of great interest are figures that have a center of symmetry: with symmetry about the point O, any point in the figure Φ is again transformed into a certain point in the figure Φ. There are many such figures in geometry. For example: a segment (the middle of the segment is the center of symmetry), a straight line (any point of it is the center of its symmetry), a circle (the center of the circle is the center of symmetry), a rectangle (the point of intersection of its diagonals is the center of symmetry). There are many centrally symmetrical objects in living and inanimate nature (student message). Often people themselves create objects that have a center symmetryries (examples from handicrafts, examples from mechanical engineering, examples from architecture and many other examples).
Secondly, axial symmetry (or symmetry about a straight line) - this is a transformation of a plane (or space), in which only the points of the straight line p remain in place (this straight line is the axis of symmetry), while the remaining points change their position: instead of point B we obtain a point B1 such that the straight line p is the perpendicular bisector to the segment BB1 . To construct a figure Ф1, symmetrical to the figure Ф, relative to the straight line р, it is necessary for each point of the figure Ф to construct a point symmetrical to it relative to the straight line р. The set of all these constructed points gives the desired figure F1. There are many geometric shapes having an axis of symmetry.
A rectangle has two, a square has four, a circle has any straight line passing through its center. If you look closely at the letters of the alphabet, you can find among them those that have horizontal or vertical, and sometimes both, axes of symmetry. Objects with axes of symmetry are quite often found in living and inanimate nature (student reports). In his activity, a person creates many objects (for example, ornaments) that have several axes of symmetry.
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Thirdly, plane (mirror) symmetry (or symmetry about a plane)
- this is a transformation of space in which only points of one plane retain their location (α-symmetry plane), the remaining points of space change their position: instead of point C, a point C1 is obtained such that the plane α passes through the middle of the segment CC1, perpendicular to it.
To construct a figure Ф1, symmetrical to the figure Ф relative to the plane α, it is necessary for each point of the figure Ф to build points symmetrical relative to α; they, in their set, form the figure Ф1.
Most often, in the world of things and objects around us, we encounter three-dimensional bodies. And some of these bodies have planes of symmetry, sometimes even several. And man himself, in his activities (construction, handicrafts, modeling, ...) creates objects with planes of symmetry.
It is worth noting that along with the three listed types of symmetry, they distinguish (in architecture)portable and rotating, which in geometry are compositions of several movements.
Axial symmetry. With axial symmetry, each point of the figure goes to a point that is symmetrical to it relative to a fixed straight line.
Picture 35 from the presentation “Ornament” for geometry lessons on the topic “Symmetry”Dimensions: 360 x 260 pixels, format: jpg. To download a free image for a geometry lesson, right-click on the image and click “Save image as...”. To display pictures in the lesson, you can also download the entire presentation “Ornament.ppt” with all the pictures in a zip archive for free. The archive size is 3324 KB.
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Picture 9 from the presentation “Types of symmetry” for geometry lessons on the topic “Symmetry”
Dimensions: 1503 x 939 pixels, format: jpg. To download a free image for a geometry lesson, right-click on the image and click “Save image as...”. To display pictures in the lesson, you can also download for free the entire presentation “Types of symmetry.ppt” with all the pictures in a zip archive. Archive size - 1936 KB.
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