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**Axial symmetry**with a 6th grade math lesson involving the perpendicular bisector of a segment as well as the definition and conservation properties of angle measures, lengths and perimeters and areas of figures.

The construction of the symmetrical of a point of a line, of a figure with respect to an axis.

## I. Symmetrical figures

Two figures are symmetrical about a line if they overlap by folding along that line.

This line is called the**axis of symmetry**.

## II. Symmetric of a point.

### 1. definition of the symmetric.

The symmetric of a point A with respect to a line (d) is the point A’, such that the line (d) is the perpendicular bisector of the segment [AA’] (i.e. such that (d) is the perpendicular to the segment [AA’] in its middle).

### 2. Construction of the symmetric with the square and the graduated rule.

### 3. Construction of the symmetric with the compass.

## III. Properties of axial symmetry

The symmetric of a line about an axis is a line.

**Axial symmetry preserves the alignment**.

**Axial symmetry preserves lengths**.

Remark:

The symmetric of the middle of a segment is the middle of the symmetric segment.

The symmetry of a circle about an axis is a circle of the same radius. The centers of the circles are symmetrical about this axis.

Examples:

To construct the symmetric of a complex figure, we break it down into usual figures and construct the symmetric of each of them.

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