To determine the area of the dilated figure, we first need to understand how dilation affects area.
When a figure is dilated with a scale factor \( k \), the area of the dilated figure changes by a factor of \( k^2 \).
Given a scale factor of 3, we calculate:
\[ k = 3 \quad \text{(scale factor)} \] \[ k^2 = 3^2 = 9 \]
This means that the area of the original figure will be multiplied by 9 when dilated.
Next, we need the original area of the composite figure. Based on the response options provided, we will assume the original area is one of the options you provided.
Assuming that the original area is equal to one of the options, we will denote the original area as \( A \).
To find the new area \( A' \) after dilation:
\[ A' = A \times k^2 \] \[ A' = A \times 9 \]
Now let's examine the given options:
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If the original area is 45 square units, \[ A' = 45 \times 9 = 405 \text{ square units} \]
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If the original area is 180 square units, \[ A' = 180 \times 9 = 1620 \text{ square units} \quad (\text{not an option}) \]
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If the original area is 270 square units, \[ A' = 270 \times 9 = 2430 \text{ square units} \quad (\text{not an option}) \]
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If the original area is 405 square units, \[ A' = 405 \times 9 = 3645 \text{ square units} \quad (\text{not an option}) \]
The only option that makes sense with a scale factor of 3 and fits the calculation is if the original area is 45 square units, resulting in a dilated area of \( 405 \) square units.
Therefore, the correct answer is 405 square units, and this is because the area of the original figure is multiplied by 9 during the dilation.