To tackle this problem, let's systematically determine the length of [tex]\(\overline{EF}\)[/tex] after dilation, given the information.
Firstly, we need to understand the process of dilation and how it affects the lengths within the triangle. When a geometric figure undergoes dilation with a scale factor, all corresponding lengths are multiplied by that scale factor.
Here, we're given:
- The scale factor of the dilation is 4.
- The length of [tex]\(\overline{FD}\)[/tex] after dilation is 12 units.
Step-by-Step Solution:
1. Determine the original length of [tex]\(\overline{FD}\)[/tex]:
To find the original length of [tex]\(\overline{FD}\)[/tex] before the dilation, we divide its dilated length by the scale factor.
[tex]\[
\text{Original length of } \overline{FD} = \frac{\text{Dilated length of } \overline{FD}}{\text{Scale factor}} = \frac{12 \text{ units}}{4} = 3 \text{ units}
\][/tex]
2. Establish the relationship between [tex]\(\overline{FD}\)[/tex] and [tex]\(\overline{EF}\)[/tex]:
The problem suggests determining the length of [tex]\(\overline{EF}\)[/tex] corresponding to the dilation. As both [tex]\(\overline{FD}\)[/tex] and [tex]\(\overline{EF}\)[/tex] are affected by the same dilation process, [tex]\(\overline{EF}\)[/tex] would be dilated from its original length by the same scale factor.
3. Identify the original length of [tex]\(\overline{EF}\)[/tex]:
Given that the original lengths of [tex]\(\overline{EF}\)[/tex] and [tex]\(\overline{FD}\)[/tex] are the same (since both become proportional under dilation), we have:
[tex]\[
\text{Original length of } \overline{EF} = \text{Original length of } \overline{FD} = 3 \text{ units}
\][/tex]
4. Calculate the dilated length of [tex]\(\overline{EF}\)[/tex]:
Now, apply the scale factor to [tex]\(\overline{EF}\)[/tex] to find its dilated length:
[tex]\[
\text{Dilated length of } \overline{EF} = \text{Original length of } \overline{EF} \times \text{Scale factor} = 3 \text{ units} \times 4 = 12 \text{ units}
\][/tex]
5. Select the correct answer from the given options:
The dilated length of [tex]\(\overline{EF}\)[/tex] is 12 units. Therefore, we match this with the correct option among:
- [tex]\(\overline{EF} = 6\)[/tex] units
- [tex]\(\overline{EF} = 9\)[/tex] units
- [tex]\(\overline{EF} = 12\)[/tex] units
- [tex]\(\overline{EF} = 16\)[/tex] units
Clearly, the correct answer is:
[tex]\(\boxed{12 \text{ units}}\)[/tex]
