When a pentagon (or any geometric shape) undergoes dilation by a scale factor, the change in area can be determined by the square of that scale factor.
If the sides of the pentagon are dilated by a scale factor \( k \) (where \( 0 < k < 1 \)), the following occurs:
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Linear Dimensions: Each linear dimension of the pentagon (such as the lengths of its sides) is multiplied by the scale factor \( k \).
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Area Calculation: The area of the original pentagon is affected by the square of the scale factor because area is a two-dimensional measurement. Therefore, if the original area of the pentagon is \( A \), the area \( A' \) of the dilated pentagon will be calculated as follows: \[ A' = k^2 \cdot A \]
Since \( k \) is a fraction (less than 1), \( k^2 \) will also be a fraction less than 1. This means:
- The area of the dilated pentagon will be smaller than the original area.
- Specifically, the new area will be \( k^2 \) times the original area, indicating a reduction in area.
In summary, when the sides of a pentagon are dilated by a scale factor less than 1, the area of the pentagon decreases, specifically by a factor of \( k^2 \).
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