When a shape, such as a pentagon, is dilated by a scale factor \( k \), the lengths of the sides are multiplied by \( k \). However, the area of the shape is affected by the square of the scale factor. Specifically, if a shape is dilated by a scale factor \( k \), the area \( A \) of the shape transforms according to the equation:
\[ A' = k^2 \times A \]
Where \( A' \) is the new area after dilation, and \( A \) is the original area.
If the scale factor \( k \) is less than 1, the following occurs:
- The lengths of the sides shrink, as they are multiplied by a fraction (less than 1).
- The area shrinks even more, as it is multiplied by \( k^2 \) (which is less than \( k \) when \( k < 1 \)).
Since \( k^2 < k \) for \( k < 1 \), this means that the area decreases more rapidly than the lengths of the sides decrease.
Therefore, the correct answer is:
The area decreases more rapidly than the lengths of the sides decrease.
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