Dilation affects the dimensions of a shape in a proportional manner based on the scale factor used. When the sides of a pentagon (or any polygon) are dilated by a scale factor less than 1, the lengths of the sides decrease.
The area of a polygon scales by the square of the scale factor. Thus, if the scale factor is \( k \) (where \( 0 < k < 1 \)), the new area \( A' \) can be expressed as:
\[ A' = k^2 \times A \]
where \( A \) is the original area of the pentagon. Since \( k^2 \) will also be less than 1, the new area \( A' \) will be less than the original area \( A \).
This means that while the lengths of the sides decrease, the area decreases more significantly (actually, it decreases quadratically relative to the scale factor) because the area is affected by the square of the scale factor.
Thus, the correct answer is:
D. The area decreases more rapidly than the lengths of the sides decrease.
1 answer