Homothecia . The study of the equality of geometric figures was closely related to the concept of movement and if a movement is applied to a geometric figure, another figure is obtained that is equal to it.
Summary
[ hide ]
- 1 Determination
- 2 Properties
- 3 Composition
- 1 Movement
- 4 Similar Transformations
- 1 Similar Figures
- 5 See also
- 6 Sources
Determination
A homothecia is a transformation of the plane itself that is defined as follows:
- A PointO is determined as the center of the homotecia.
- A real number is determined as the ratio of the homotecia.
- The imageP ‘of a point P is located on the ray OP
- O is his own image (O ‘and O coincide).
A homotecia of center O and ratio k is denoted H (O; k)
Properties
For all H (O; k) the following is true: 1. The image of a line is a line parallel to it. 2. The image of a segment is a segment parallel to it and that has k times its length. 3. The image of an angle is an angle that has the same width.
The properties of homotecia can be demonstrated by applying the reciprocal theorem of the Transverse Theorem and the Fundamental Theorem of Triangle Similarity .
Composition
Movement
When several movements are carried out successively (for example if a translation is made to a figure, then a central symmetry is applied to the obtained image and a reflection is finally applied to the new image), a figure equal to the original is obtained . This successive realization of movements is called composition of movements.
The composition of various movements is also a movement. Analogously, when several homogeneities are carried out successively, one then speaks of composition of homothecies.
The composition of two homothecies H (O1, k1) and H (O2, k2) where k1 • k2 = 1 is again a homotecia, its ratio k3 = k1 • k2 and its center O3 is located on the line O1O2. You can also make the composition of a homotecia with one movement.
Generalizing, a H (O; -k) homotecia with k> 0 can be defined as the composition of a H (O; k) homotecia with a center symmetry of center O.
Similar Transformations
Similar Figures
Any composition of a homotecia with a movement is called a similar transformation.
All movement is in fact a similar transformation, since it is enough to consider it as the composition of a movement with a homotecia of ratio k = 1. It is always true that the image of any geometric figure by a similar transformation is similar to the original figure.
Two geometric figures F1 and F2 are similar, if there is a similar transformation by which one is transformed into the other. Then F1 ~ F2 is written.
