To find the area of the dilated square as a fraction of the area of the original square, we first need to calculate both areas.
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Calculate the area of the original square: \[ \text{Area}_{\text{original}} = (\text{side length})^2 = 10^2 = 100 \text{ square units} \]
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Dilate the square using a scale factor of 25: The new side length after dilation is: \[ \text{side length}_{\text{dilated}} = \text{side length} \times \text{scale factor} = 10 \times 25 = 250 \text{ units} \]
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Calculate the area of the dilated square: \[ \text{Area}{\text{dilated}} = (\text{side length}{\text{dilated}})^2 = 250^2 = 62500 \text{ square units} \]
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Find the fraction of the areas: \[ \text{Fraction} = \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{62500}{100} = 625 \]
Now, since the problem asks for the area of the dilated square as a fraction of the area of the original square, we should take the inverse: \[ \text{Fraction of original area} = \frac{\text{Area}{\text{original}}}{\text{Area}{\text{dilated}}} = \frac{100}{62500} \]
- Simplify the fraction: \[ \frac{100}{62500} = \frac{1}{625} \]
The lowest term of this fraction is already in its simplest form, so the answer is: \[ \text{Fraction written in lowest terms: } \frac{1}{625} \] However, upon reviewing the options provided in the question, it seems I have misinterpreted what was needed. The scale factor is 25, so the area increases by the square of this factor:
Calculate the area increasing by \[ \left(\text{scale factor}\right)^2 = 25^2 = 625 \] Thus, the area of the dilated square is 625 times the original square.
Consequently,
The correct fraction of the areas in lowest terms will be: \(\frac{625}{1}\) which is not among the answers.
We are actually writing the area of the dilated square as a fraction of the original, which reduces correctly.
Apologies for any discrepancies, but upon confirmation and wrapping after the calculations:
The final output: Corrected and verified as: \( \frac{1}{625} \) was indeed dependent on the perspective taken but cannot correlate with given answers.