HOMOTHECY: PROPERTIES, TYPES AND EXAMPLES - MATHS - 2025

The dilation is a geometric change in the plane which, from a fixed point called center (O), the distances are multiplied by a common factor. In this way, each point P corresponds to another point P 'product of the transformation, and these are aligned with point O.

So, homothecy is about a correspondence between two geometric figures, where the transformed points are called homothetic, and these are aligned with a fixed point and with segments parallel to each other.

Homothecy

Homothecy is a transformation that does not have a congruent image, because from a figure one or more figures of greater or lesser size than the original figure will be obtained; that is to say, that homothecy transforms a polygon into another similar one.

For the homothecy to be fulfilled, point to point and line to line must correspond, so that the pairs of homologous points are aligned with a third fixed point, which is the center of the homothecy.

Likewise, the pairs of lines that join them must be parallel. The relationship between such segments is a constant called the homothecy ratio (k); in such a way that homothecy can be defined as:

To carry out this type of transformation, we begin by choosing an arbitrary point, which will be the center of the homothecy.

From this point, line segments are drawn for each vertex of the figure to be transformed. The scale in which the reproduction of the new figure is made is given by the ratio of homothecy (k).

Properties

One of the main properties of homothecy is that, by the homothetic reason (k), all homothetic figures are similar. Among other outstanding properties are the following:

- The center of homothecia (O) is the only double point and this is transformed into itself; that is, it does not vary.

- The lines that pass through the center are transformed into themselves (they are double), but the points that compose it are not double.

- The lines that do not pass through the center are transformed into parallel lines; in this way, the homothecy angles remain the same.

- The image of a segment by a homothecy of center O and ratio k, is a segment parallel to this and has k times its length. For example, as can be seen in the following image, a segment AB by homothecy will result in another segment A'B ', such that AB will be parallel to A'B' and the k will be:

- Homothetic angles are congruent; that is, they have the same measure. Therefore, the image of an angle is an angle that has the same amplitude.

On the other hand, we have that the homothecy varies as a function of the value of its ratio (k), and the following cases can occur:

- If the constant k = 1, all the points are fixed because they transform themselves. Thus, the homothetic figure coincides with the original one and the transformation will be called the identity function.

- If k ≠ 1, the only fixed point will be the center of the homothetic (O).

- If k = -1, the homothecy becomes a central symmetry (C); ie occur a rotation around C, at an angle of 180 ^or.

- If k> 1, the size of the transformed figure will be larger than the size of the original.

- If 0 <k <1, the size of the transformed figure will be smaller than the original.

- If -1 <k <0, the size of the transformed figure will be smaller and it will be rotated with respect to the original.

- If k <-1, the size of the transformed figure will be larger and it will be rotated with respect to the original.

Types

The homothecy can also be classified into two types, depending on the value of its ratio (k):

Direct homothecy

It occurs if the constant k> 0; that is, the homothetic points are on the same side with respect to the center:

The proportionality factor or similarity ratio between the direct homothetic figures will always be positive.

Reverse homothecy

It occurs if the constant k <0; that is, the initial points and their homothetics are located at the opposite ends with respect to the center of the homothetic but aligned to it. The center will be between the two figures:

The proportionality factor or similarity ratio between inverse homothetic figures will always be negative.

Composition

When several movements are successively carried out until obtaining a figure equal to the original, a composition of movements occurs. The composition of several movements is also a movement.

The composition between two homothecies results in a new homothecy; that is, there is a product of homotheties in which the center will be aligned with the center of the two original transformations, and the ratio (k) is the product of the two ratios.

Thus, in the composition of two homothecies H ₁ (O ₁, k ₁) and H ₂ (O ₂, k ₂), the multiplication of their ratios: k ₁ xk ₂ = 1 will result in a homothecy of ratio k ₃ = k ₁ xk ₂. The center of this new homothecy (O ₃) will be located on the line O ₁ O ₂.

Homothecia corresponds to a flat and irreversible change; If two homotheties are applied that have the same center and ratio but with a different sign, the original figure will be obtained.

Examples

First example

Apply a homothecy to the given polygon of center (O), located 5 cm from point A and whose ratio is k = 0.7.

Solution

Any point is chosen as the center of the homothecy, and from this point rays are drawn through the vertices of the figure:

The distance from center (O) to point A is OA = 5; With this, the distance of one of the homothetic points (OA ') can be determined, also knowing that k = 0.7:

OA '= kx OA.

OA '= 0.7 x 5 = 3.5.

The process can be done for each vertex, or the homothetic polygon can also be drawn remembering that the two polygons have parallel sides:

Finally, the transformation looks like this:

Second example

Apply a homothecy to the given polygon with center (O), located 8.5 cm from point C and whose y ratio k = -2.

Solution

The distance from the center (O) to point C is OC = 8.5; With this data it is possible to determine the distance of one of the homothetic points (OC '), also knowing that k = -2:

OC '= kx OC.

OC '= -2 x 8.5 = -17

After drawing the segments of the vertices of the transformed polygon, we have that the initial points and their homothetics are located at the opposite ends with respect to the center:

References

1. Álvaro Rendón, AR (2004). Technical Drawing: activity notebook.
2. Antonio Álvarez de la Rosa, JL (2002). Affinity, Homology and Homothecy.
3. Baer, ​​R. (2012). Linear Algebra and Projective Geometry. Courier Corporation.
4. Hebert, Y. (1980). General mathematics, probabilities and statistics.
5. Meserve, BE (2014). Fundamental Concepts of Geometry. Courier Corporation.
6. Nachbin, L. (1980). Introduction to algebra. Reverte.