To determine the scale factor for the dilation, we start with the relationship between the perimeter of a square and its side length.
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Dilation Scale Factor: If the perimeter of the square becomes 7 times as large, we can recall that the perimeter \( P \) of a square with side length \( s \) is given by \( P = 4s \). After dilation, if the perimeter is \( 7P = 7(4s) = 28s \), we set up the equation: \[ 4 \times \text{new side length} = 28s \] This indicates that: \[ \text{new side length} = 7s \] Therefore, the scale factor for the dilation is \( 7 \).
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Similarity of the Image: A dilation transformation results in a similar figure, meaning the original shape and the dilated shape have the same shape but different sizes. However, they are not congruent because the side lengths have changed. Thus, the image is similar but not congruent to the original square.
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Area of the Image: The area of a square is given by the formula \( A = s^2 \). When the side length is multiplied by a scale factor of \( k \), the area is multiplied by \( k^2 \). Since the scale factor is \( 7 \), the area of the image will be: \[ \text{New area} = 7^2 = 49 \text{ times the area of the original square}. \]
Putting it all together, we have:
Andy dilates the square by a scale factor of 7. The image is similar but not congruent to the original square. So, the area of the image is 49 times as large as the area of the original square.