This course on translation and rotation in the fourth grade (4e) with the definition and properties of these two transformations of the plane studied in college must be well learned. Thus, this lesson is written by a team of national education teachers and it is in accordance with the current programs.

The student will have to know how to construct the image of a figure but also, to use the various properties of conservation of these two transformations of the plane in order to carry out a demonstration in geometry.

## I. The translation

### 1. definition of translation

Definition:

When the figure is dragged so that A arrives at B, it overlaps with the figure .

The figure is the image of the figure by the translation that transforms A into B.

### 2.image of a point and a segment by translation

Ownership:

The image of point M by the translation that transforms A into B is point M’, such that the segments

[MB] and [AM’] have the same environment.

If the points are not aligned, then ABM’M is a parallelogram.

Ownership:

The image of a segment by a translation is a segment of the same length.

Translation preserves angle measures, perimeters, areas and parallelism.

Example:

In the translation that transforms A and B, the segment [MN] has as image [M’N’] are parallel [M’N’] are parallel and of the same length.

## II. The rotation

### 1. Definition of rotation

Definition:

When the figure is rotated about point O by an angle of measure in a counterclockwise direction, it is superimposed with the figure .

is the image of the figure by the rotation of center O and angle .

Remark:

• Throughout this lesson, the direction of rotation will always be trigonometric (counterclockwise).
• The rotation of center O and angle 180° is the central symmetry of center O.

### 2. Image of a point by a rotation

Ownership:

Consider two distinct points O and M.

The image of the point M by the rotation of center O and angle is the point M’ such that OM’=OM and .

## III. Properties of translation and rotation

Ownership:

Translation and rotation preserve lengths, perimeters, areas of figures as well as alignment and parallelism and measures of angles.

Example:

Quadrilateral A’B’C’D’ is the image of ABCD by the rotation of center O and angle 60°.

The quadrilateral is the image of ABCD by the translation that transforms A into .

• The areas and perimeters of the three quadrilaterals are equal.
• Points A, B and K are aligned, so their images are also aligned.
• The point J is the middle of the segment [BC], so its image J’ by the rotation is the middle of the segment [B’C’].
• The angle is the image of the angle by the translation, so they have the same measure.
• The angle is the image of the angle by the rotation, so they have the same measure.

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