Determine how dilation affects the area of a pentagon when the sides of the pentagon are dilated by a scale factor less than 1.(1 point) Responses The area decreases, but less rapidly than the lengths of the sides decrease. The area decreases, but less rapidly than the lengths of the sides decrease. The area increases, but less rapidly than the lengths of the sides increase. The area increases, but less rapidly than the lengths of the sides increase. The area decreases more rapidly than the lengths of the sides decrease. The area decreases more rapidly than the lengths of the sides decrease. The area increases more rapidly than the lengths of the sides increase.

Question

GPT-4o mini · Answer

When a polygon, including a pentagon, is dilated by a scale factor \( k \), the area of the polygon is affected by the square of the scale factor. Specifically, if the original area is \( A \), then the new area \( A' \) after dilation is given by:

\[ A' = k^2 \cdot A \]

If the scale factor is less than 1 (i.e., \( 0 < k < 1 \)), this means that \( k^2 \) will also be less than 1. Thus, the new area \( A' \) will be smaller than the original area \( A \).

Since the effect of the dilation on the area is determined by squaring the scale factor, the area decreases more rapidly than the lengths of the sides. The lengths of the sides decrease by a factor of \( k \), while the area decreases by a factor of \( k^2 \).

Therefore, the correct response is:

The area decreases more rapidly than the lengths of the sides decrease.