A fifth grade math class on this chapter is always necessary for the student’s progress.

This lesson involves the following concepts:

– The definition

– property of conservation of lengths;

– conservation property of angle measures;

– conservation of the alignment;

– preservation of parallelism;

– transformation of a line into another parallel line;

– conservation of the perimeter of a figure;

– conservation of the area of a geometric figure.

The student should know how to construct the image of a figure by a central symmetry of center O but also how to use the various conservation properties to carry out demonstrations in geometry for the fifth level and throughout his schooling.

## I. Definition of central symmetry

### 1. central symmetry and half-turn

Definition:

Two figures and are symmetrical with respect to a point O when one can pass from one to the other by a half-turn of center O, that is to say a rotation of an angle of 180° and center O.

The figure is called the image of by the central symmetry of center O.

Example:

• The figure is the symmetrical of the figure with respect to the point O.
• Similarly, the figure is the symmetric of the figure with respect to the point O.
• The figures and are symmetrical with respect to the point O.
• We also say that the point O is the center of the symmetry that transforms the figure into the figure .

### 2.symmetric of a point

Definition:

Consider a central symmetry of center O.

Point M’ is the image of point M by the central symmetry of center O

if and only if the point O is the middle of the segment [MM’].

Example:

• The symmetric of A with respect to O is A’.
• The symmetric of A’ with respect to O is A.
• A and A’ are symmetrical with respect to O.

Remark:

The symmetric of O about O is the point O itself.

## II. properties of central symmetry

### 1.symmetric of a segment

Ownership:
• The symmetric of a segment by a central symmetry is a segment of the same length.
• Central symmetry preserves the lengths of segments, perimeters and areas of geometric figures.
Example:
• To construct the symmetry of the segment [CD] with respect to the point O, we construct the symmetry of the points C and D, noted C’ and D’, with respect to the point O.
• By the symmetry of center O, the symmetrical of the segment [CD] is then the segment [C’D’].
• The symmetric of the middle of a segment is the middle of the symmetric segment.

### 2.symmetry of a line

Ownership:
• The image of a line by a central symmetry is a line that is parallel to it.
• Central symmetry transforms a line into another line that is parallel to it.

### 3.symmetric of a polygon

Ownership:

Central symmetry preserves everything, mainly :

• the lengths;
• the perimeters of figures;
• areas of figures;
• angle measurements;
• parallelism;
• orthogonality.
Ownership:

The symmetric of a polygon is a polygon with the same number of sides and the same shape.

To construct the symmetric of a polygon, we construct the symmetric of each side then,

we connect the vertices in the right order.

### 4.symmetrical of a circle

Ownership:

The symmetric of a circle is a circle with the same radius and having for center the symmetric of the center of the first circle.

Remark:

To construct the symmetric of a circular arc with respect to a point, we construct the symmetries of the center and the ends of the symmetric circular arc.

## III.center of symmetry of a figure

Ownership:

A point O is the center of symmetry of a figure if the image of the figure is coincident with .

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