# Symmetry. Symmetry of plane figures

*Mirror symmetry. Symmetry plane. Symmetrical figures.*

Mirror equal figures. Central symmetry. Symmetry center.

Rotation symmetry. Symmetry axis. Axial symmetry.

Examples of a symmetry kinds. Symmetry of plane figures.

Examples of symmetry of plane figures.

Mirror equal figures. Central symmetry. Symmetry center.

Rotation symmetry. Symmetry axis. Axial symmetry.

Examples of a symmetry kinds. Symmetry of plane figures.

Examples of symmetry of plane figures.

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Mirror symmetry.
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A geometric figure is called a

*symmetrical*

*relatively a plane*S ( Fig.104 ), if for each point E of this figure a point E’of the same figure can be found, so that a segment EE’ is perpendicular to a plane S and is divided by this plane into two ( EA = AE’ ). A plane S is called a

*symmetry plane*. Symmetrical figures, subjects and bodies are nor equal one to another in a narrow meaning ( for instance, a left glove is not suitable for a right hand and vice versa ). They are called

*mirror equal*.

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Central symmetry.
**
A geometric figure (or a body) is called a

*symmetrical*

*relatively a center*C ( Fig.105 ), if for each point A of this figure the point E of the same figure can be found, so that a segment AE goes through the center C and is divided in this point into two (AC = CE). A point C is called a

*symmetry center*.

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Rotation symmetry.
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A body (a figure) has a

*rotation symmetry*( Fig.106 ), if at turning by an angle 360 °/

*n*( here

*n*– integer ) around some straight line AB (

*symmetry axis*) it coincides completely with its initial position. At

*n*= 2 we have an

*axial symmetry.*Triangles ( Fig.105 ) have also an axial symmetry.

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Examples of above mentioned kinds of symmetry.
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*A sphere ( ball )*has a central, a mirror and a rotation symmetry. A symmetry center is a center of a ball; a symmetry plane is a plane of any large circle; a symmetry axis is a diameter of a ball.

*A round cone*has an axial symmetry; a symmetry axis is an axis of a cone.

*A right prism*has

*a mirror symmetry. A symmetry plane is parallel to its bases and placed by equal distance between them.*

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Symmetry of plane figures.
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*Mirror-axial symmetry.*If the plane figure ABCDE ( Fig.107 ) is symmetrical relatively a plane S ( that is possible only if the figure plane is perpendicular to the plane S ), then the straight line KL, along which these planes are intersected, is a

*symmetry axis*of the 2-nd order of the figure ABCDE. In this case the figure ABCDE is called a

*mirror symmetrical*one.

*
Central symmetry.
*
If the plane figure ( ABCDEF, Fig.108 ) has a symmetry axis of the 2-nd order, perpendicular to the figure plane
(a straight line MN, Fig.108 ), then a point O, in which MN and the figure plane ABCDEF intersect, is a

*symmetry*

*center*.

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Examples of symmetry of plane figures.
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*A parallelogram*has only a central symmetry. Its symmetry center is a point of ntersection of diagonals.

*An isosceles trapezoid*has only an axis symmetry. Its symmetry axis is a perpendicular, drawn through middles of its bases.

*A rhombus*has both a central and an axial symmetry. Its symmetry axis is any of its diagonals; a symmetry center is a point of their intersection.

*A circle*has … What can you say about a circle’s kinds of symmetry ?