In similar polygons, how do the lengths of corresponding sides compare?

In similar polygons, the lengths of corresponding sides have a constant ratio. This means that if you take any pair of corresponding sides from two similar polygons, the length of one side will always be a specific multiple of the length of the other side, depending on the scale factor of the similarity. This ratio remains consistent across all sets of corresponding sides in those polygons.

This constant ratio reflects the fundamental property of similarity in geometry, where all proportions between corresponding lengths are maintained. For example, if one side of a smaller polygon is 3 units long and the corresponding side of a larger polygon is 6 units, the ratio of the lengths of these two sides is 3:6, which simplifies to 1:2. This ratio will hold true for all corresponding sides in the two polygons, demonstrating that their shapes are the same but their sizes differ.

In contrast, equal lengths would only occur if the polygons are congruent rather than similar. The lengths do not depend on the area, as similar polygons with different areas can still share the same ratio of side lengths. Saying that the lengths can be any length ignores the defined relationship that must exist between corresponding sides of similar figures.