Let's consider the given information and infer the solution step-by-step.
1. Understanding the Dilation:
- A triangle is dilated by a scale factor of 6. This means every side of the original triangle is multiplied by 6 to obtain the dilated triangle.
2. Given Measurements:
- After the dilation, the length of [tex]\(\overline{DE}\)[/tex] is measured to be 30 units.
3. Relationship Between Original and Dilated Lengths:
- If the given length [tex]\(\overline{DE}\)[/tex] is already 30 units after dilation, and the dilation factor is 6, then the original length of [tex]\(\overline{DE}\)[/tex] before dilation must have been:
[tex]\[
\text{Original Length of } \overline{DE} = \frac{30 \text{ units}}{6} = 5 \text{ units}
\][/tex]
4. Calculating [tex]\(\overline{EF}\)[/tex] after Dilation:
- However, the length given as 30 units is after the dilation. Since the same dilation factor applies equally to all parts of the triangle proportionally, the corresponding original length for [tex]\(\overline{EF}\)[/tex] before dilation should have been:
[tex]\[
\text{Original Length of } \overline{EF} = \frac{\text{Length of } \overline{DE}}{\text{Scale Factor}}
\][/tex]
But since [tex]\(\overline{DE}\)[/tex] and [tex]\(\overline{EF}\)[/tex] are part of the corresponding sides in similar triangles, their dilated lengths remain proportional.
5. Final Confirmation:
- Given the choices, the length of [tex]\(\overline{EF}\)[/tex] after applying the same scale factor would end up being:
[tex]\[
\text{Length of } \overline{EF} = 30 \text{ units}
\][/tex]
Thus, the length of [tex]\(\overline{EF}\)[/tex] should be:
[tex]\[
\boxed{30 \text{ units}}
\][/tex]
