Geometric transformations
Problem — How to manipulate and study geometric figures by changing their position or size while keeping their essential properties?
- Understand what a geometric transformation in the plane is.
- Discover the main types: translation, rotation, axial and central symmetry, homothety.
- Learn to recognize the properties preserved by each transformation.
- Know how to construct and describe a geometric transformation on a figure.
- Develop the ability to solve geometric problems using these transformations.
Part 1: Introduction to geometric transformations
A geometric transformation is an operation that associates each point of the plane with another point of the plane, modifying the initial figure. This operation can move, rotate, reflect, or resize the figure.
In geometry, studying transformations helps better understand the structure and properties of figures. They also help solve problems by reproducing or modifying figures while preserving certain characteristics.
Key concepts
- Image of a point by a transformation: the point obtained after applying the transformation.
- Image figure: the set of image points of the initial figure.
- Invertible transformation: a transformation for which we can find back the initial figure by applying an inverse transformation.
We introduced the notion of geometric transformation as an operation that modifies the position or size of figures in the plane. Understanding this idea is fundamental to studying the different types of transformations and their essential properties. This prepares us to explore the most important transformations in geometry.
Part 2: Isometric transformations - preserving distances
A transformation is called an isometry if it preserves distances between all points. In other words, the image figures are exactly superimposable on the initial figures by sliding, without deformation.
There are three main isometric transformations: translation, rotation, and axial symmetry. Each modifies the figure without changing its shape or size.
Translation
Translation moves all points of the figure in the same direction and by the same vector.
- Example: move a triangle 3 cm to the right and 2 cm up.
- Properties: preserves angles, lengths, and the figure's orientation.
Rotation
Rotation turns the figure around a fixed point called the center of rotation, by a given angle and direction (clockwise or counterclockwise).
- Example: rotate a square 90° around its center.
- Properties: preserves distances and angles, but may change the figure's orientation.
Axial symmetry
Axial symmetry reflects the figure relative to a given axis called the axis of symmetry.
- Example: symmetry of a polygon relative to a vertical line.
- Properties: preserves distances and angles, but reverses orientation (creates a "mirror image").
Isometries are key transformations because they preserve the shape and size of figures. Translation, rotation, and axial symmetry allow moving, rotating, or reflecting a figure without deforming it. These transformations are used to test properties of figures or solve problems where shape must stay unchanged.
Part 3: Central symmetry
Central symmetry is a transformation that associates each point with an image point such that the center of symmetry is the midpoint of the segment joining the point and its image.
We can see central symmetry as a 180° rotation around a fixed point called the center of symmetry.
Characteristics
- Each point and its image are aligned with the center of symmetry, which is the midpoint of this segment.
- Preserves distances and angles, so it is an isometry.
- Reverses the figure's orientation.
Concrete example
Given a triangle ABC and a point O chosen as the center of symmetry. By performing central symmetry with center O, triangle ABC is transformed into a triangle A'B'C' where each point satisfies that O is the midpoint of [AA'], [BB'], and [CC'].
Central symmetry is a simple and very useful transformation, especially for regular figures. It allows creating symmetric figures relative to a point and is part of isometries. Understanding how it works helps solve many geometric problems.
Part 4: Homothety - enlargement and reduction
Homothety is a transformation that enlarges or reduces a figure from a fixed point called the center of homothety, according to a ratio k, called the homothety coefficient.
This transformation changes the size of the figure but preserves its shape and angles.
Properties of homothety
- If k > 1, the figure is enlarged.
- If 0 < k < 1, the figure is reduced.
- The points of the figure and their images are aligned with the center of homothety.
- Lengths are multiplied by |k|.
- Angles are preserved.
Concrete example
Given a square with side length 4 cm, we perform a homothety with center O and ratio k = 2. The image figure is a square with sides measuring 8 cm, each image point lies on the line connecting center O to the original point and is twice as far from O as that point.
Homothety allows changing the size of a figure while preserving its shape. It is an essential transformation to understand enlargement and reduction concepts in geometry. It is often used in modeling and technical drawing.
Part 5: Using geometric transformations
Geometric transformations are powerful tools in mathematics. They allow:
- Solving figure construction problems (e.g., reproducing a triangle after a translation).
- Demonstrating remarkable properties of figures using symmetry or rotation.
- Changing coordinate systems or position in the plane.
- Better visualizing and understanding geometry in space and in the plane.
Application example
To prove two segments have the same length, we can use a translation to move one onto the other and verify the superposition, showing that translation is an isometry that preserves distances.
Geometric transformations are not just theoretical tools: they are very practical for constructing, comparing, and analyzing figures. A deep understanding is essential to progress in geometry, especially in 8th grade where they play a central role in the curriculum.
This lesson presented the main types of geometric transformations: translation, rotation, axial and central symmetry, and homothety. We saw how each changes figures in the plane while preserving important properties, especially distances, angles, or overall shape. Mastering these transformations allows confident exploration of plane geometry, analyzing complex figures, and solving various geometric problems. Understanding and using these transformations is a fundamental skill in mathematics for 8th grade and beyond.