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)

The area decreases more rapidly
• than the lengths of the sides decrease.
The area increases, but less
• rapidly than the lengths of the sides increase.
The area increases more rapidly
• than the lengths of the sides increase.
The area decreases, but less
• rapidly than the lengths of the sides decrease.

1 answer

When a pentagon (or any geometric shape) is dilated by a scale factor less than 1, the lengths of its sides decrease. The area, however, is affected by the square of the scale factor.

If the scale factor is \( k \) (where \( 0 < k < 1 \)), then the area of the new pentagon will be \( k^2 \) times the original area. This means that the area decreases, but it decreases by the square of the scale factor. Since \( k^2 < k \) for \( 0 < k < 1 \), the area decreases more slowly than the lengths of the sides.

Based on this reasoning, the correct answer is:

The area decreases, but less rapidly than the lengths of the sides decrease.