Question

a dilation is a nonrigid transformation that can produce enlarged or reduced images from a given pre-image. how do you know that a dilation will produce similar figures? explain.

Explanation:

A dilation scales all the linear dimensions (sides, radii, etc.) of a figure by a constant scale factor \( k \) (where \( k>0 \)). For two figures to be similar, their corresponding angles must be congruent, and their corresponding sides must be in proportion.

• Angle Preservation: Dilation does not change the measure of angles. This is because dilation is a type of similarity transformation that preserves the shape's angular relationships. For example, in a triangle, the angles are determined by the relative slopes of the sides, and scaling the sides by a constant factor does not alter these slopes or the angle measures.
• Proportional Sides: When we apply a dilation with scale factor \( k \) to a figure, every side length of the image is \( k \) times the corresponding side length of the pre - image. So, the ratio of corresponding side lengths between the image and the pre - image is \( k \) (a constant).

Since the corresponding angles are congruent and the corresponding sides are proportional (with the scale factor as the constant of proportionality), by the definition of similar figures, the pre - image and the image after dilation are similar.

Answer:

A dilation produces similar figures because it preserves angle measures (corresponding angles are congruent) and scales all side lengths by a constant factor (corresponding sides are proportional). By the definition of similar figures (congruent corresponding angles and proportional corresponding sides), the pre - image and the dilated image are similar.