When a shape, such as a pentagon, is dilated by a scale factor less than 1, all linear dimensions of the shape, including its sides, are reduced. The key relationship to consider is how the area of the shape changes in relation to the linear dimensions when the shape is dilated.
For two-dimensional shapes, when the scale factor for dilation is \(k\) (where \(0 < k < 1\)), the new area \(A'\) of the dilated shape is related to the original area \(A\) by the formula:
\[ A' = k^2 \cdot A \]
Because \(k\) is a number less than 1, \(k^2\) will also be less than 1. This means that the area \(A'\) will decrease when compared to the original area \(A\).
In this context, the lengths of the sides decrease proportionally by the scale factor \(k\), while the area decreases by the factor \(k^2\). Since \(k^2\) decreases more rapidly than \(k\) does, we can say 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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