## similarity transformation

### In mathematics, a similarity transformation is a type of transformation that preserves the shape but not necessarily the size of an object

In mathematics, a similarity transformation is a type of transformation that preserves the shape but not necessarily the size of an object. It involves a combination of translations, rotations, reflections, and dilations.

More formally, a similarity transformation refers to a composition of rigid motions followed by a dilation. A rigid motion is a transformation that preserves distances and angles. Examples of rigid motions include translations (sliding), rotations (turning), and reflections (flipping). A dilation, on the other hand, is a transformation that changes the size of the object by either enlarging or reducing it. It involves multiplying the coordinates of each point by a constant factor.

The similarity transformation can be defined as follows: Let A and B be two figures in the plane. A similarity transformation is a transformation that maps A onto B, such that for every pair of corresponding points A and B, the ratio of the distances between the images of these points is constant. This constant is called the scale factor.

The scale factor, denoted by k, is the ratio of the lengths of corresponding sides or segments of the two figures. It can be calculated by dividing the length of a corresponding side in the image by the length of the corresponding side in the original figure.

The properties of similarity transformations include:

1. Shape preservation: The transformed figure is similar to the original figure, meaning that their corresponding angles are congruent and their corresponding sides are proportional.

2. Orientation preservation: If the original figure is oriented clockwise or counterclockwise, the transformed figure will have the same orientation.

3. Collinearity preservation: Collinear points in the original figure remain collinear in the transformed figure.

Similarity transformations are commonly used in geometry to compare and analyze different figures. They can be used to determine if two figures are similar, to find missing side lengths or angles in similar figures, or to create scaled-up or scaled-down versions of a figure.

Overall, similarity transformations provide a way to study the relationships and properties between geometric figures without changing their overall shape.

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