To solve the problem where a triangle was dilated by a scale factor of 4, and we are given that [tex]\(\overline{FD}\)[/tex] measures 12 units, we need to find the length of [tex]\(\overline{EF}\)[/tex] in the original triangle.
Here are the steps:
1. Understand that dilation by a scale factor changes all corresponding side lengths by that factor. So, if [tex]\(\overline{FD}\)[/tex] in the dilated triangle measures 12 units, we need to find the corresponding side length [tex]\(\overline{EF}\)[/tex] in the original triangle before dilation.
2. Given that the scale factor is 4, it means that every length in the original triangle is increased by a factor of 4 in the dilated triangle.
3. Therefore, to find the original length [tex]\(\overline{EF}\)[/tex], we need to divide the dilated side length [tex]\(\overline{FD}\)[/tex] by the scale factor. So, we do:
[tex]\[
\overline{EF} = \frac{\overline{FD}}{\text{scale factor}}
\][/tex]
4. Substitute the known values:
[tex]\[
\overline{EF} = \frac{12}{4} = 3
\][/tex]
Therefore, the length of [tex]\(\overline{EF}\)[/tex] in the original triangle is 3 units. Since none of the provided answers (6, 9, 12, 16) match, it's important to cross-check the problem setup. Given the provided numerical answer, the length should specifically be:
[tex]\[
\overline{EF} = 3 \text{ units}
\][/tex]
Thus, the choices listed do not include the correct value of [tex]\(\overline{EF}\)[/tex] based on the given information.
---
If this discrepancy arises from a misunderstanding of the problem or potential other details not specified, it's crucial to revisit exactly what is being asked or confirm all given values and relations in the problem.
