Final answer:
A dilation that results in a congruent figure has a scale factor of 1. Scale factors define the ratio of corresponding lengths between two geometric figures, and a factor of 1 means the figures are identical in size. Example problems often involve using proportions to find missing dimensions using clear ratios.
Step-by-step explanation:
A dilation in geometry that produces a congruent figure must have a scale factor of 1, because only identical scale transformations will result in congruent figures, which are figures with the same shape and size. The scale factor essentially is the ratio of any two corresponding lengths in the images. When the scale factor is 1, the figure's size does not change, indicating it remains congruent to the original.
For instance, if we consider a problem that utilizes scale factors, such as the proportion 1:2 = 4:x, we're working to find the value of x that maintains the ratio. In this case, we're matching up the actual dimensions to the scale dimensions. To solve for x, we cross-multiply to get 1 * x = 2 * 4, which simplifies to x = 8. This means the actual dimension corresponding to a scale dimension of 4, when using a 1:2 scale factor, is 8.
When working with different units, we sometimes use a process that seems like we're multiplying by a series of 1's because the numeric value of the conversion factor may not be 1, but the concept of multiplication by 1 is applied in the form of unity fractions that represent equivalent amounts in different measurement units. This allows for the conversion of dimensions while preserving the proportionality dictated by the scale factor.