To find the relationship between the area of the original square and the dilated square, we first need to calculate the areas of both squares.
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Area of the original square: The original square has a side length of 4 units. The area \(A_1\) is calculated as follows: \[ A_1 = \text{side}^2 = 4^2 = 16 \text{ square units} \]
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Area of the dilated square: The dilated square has a side length of 6 units. The area \(A_2\) is calculated as follows: \[ A_2 = \text{side}^2 = 6^2 = 36 \text{ square units} \]
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Fraction to express the relationship: We need to express the relationship by forming a fraction where the numerator is the area of the dilated square and the denominator is the area of the original square: \[ \text{Fraction} = \frac{A_2}{A_1} = \frac{36}{16} \]
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Simplifying the fraction: We can simplify \(\frac{36}{16}\): \[ \frac{36}{16} = \frac{9}{4} \]
The fraction expressing the relationship between the areas in lowest terms is: \[ \frac{9}{4} \]
Now, let’s express it in the form of the possible answers provided:
- \(16/36\) is equivalent to \(\frac{4}{9}\) (not in lowest terms).
- \(2\) is a whole number.
- \(\frac{1}{4}\) does not reflect the areas accurately.
- \(\frac{4}{9}\) is also not correct based on our calculations.
Therefore, since \(\frac{9}{4}\) isn't one of the exact options, the closest relationship involving both areas is given as: \[ \frac{36}{16} \] So the best choice based on the incorrect options but aligning with our simplification would be:
- \(16/36\) (this simplifies to \(\frac{4}{9}\), which doesn't match but reflects a part of the area change).
The proper fraction representing the area comparison is: \(\frac{9}{4}\) but none of the options match directly with that.