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)ResponsesThe 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, 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 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.The area increases more rapidly than the lengths of the sides increase.

Question

GPT-4o mini · Answer

When the sides of a pentagon (or any shape) are dilated by a scale factor less than 1, the lengths of the sides decrease. The area of a shape, on the other hand, is affected by the square of the scale factor.

If the scale factor is \( k \) (where \( 0 < k < 1 \)), the area of the pentagon will be multiplied by \( k^2 \). Since \( k^2 \) is also less than 1, the area will decrease.

Moreover, since the area scales with the square of the scale factor, it decreases more rapidly than the lengths of the sides, which scale linearly by \( k \).

Therefore, the correct response is:

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