Given that [tex]\( \triangle DEF \)[/tex] is a dilation of [tex]\( \triangle ABC \)[/tex] by a scale factor of [tex]\( \frac{1}{3} \)[/tex], it means that the lengths of the sides of [tex]\( \triangle DEF \)[/tex] are [tex]\( \frac{1}{3} \)[/tex] of the corresponding sides of [tex]\( \triangle ABC \)[/tex]. That is, if [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( CA \)[/tex] are the sides of [tex]\( \triangle ABC \)[/tex], then:
- [tex]\( DE = \frac{1}{3} \cdot AB \)[/tex]
- [tex]\( DF = \frac{1}{3} \cdot AC \)[/tex]
- [tex]\( EF = \frac{1}{3} \cdot BC \)[/tex]
To determine which proportion verifies that [tex]\( \triangle ABC \)[/tex] and [tex]\( \triangle DEF \)[/tex] are similar, we need to check the corresponding sides and their ratios.
### Option A: [tex]\( \frac{DE}{AB} = \frac{BC}{DF} \)[/tex]
For Option A:
- [tex]\( DE = \frac{1}{3} \cdot AB \)[/tex]
- [tex]\( DF = \frac{1}{3} \cdot AC \)[/tex]
- [tex]\( \frac{DE}{AB} = \frac{\frac{1}{3} \cdot AB}{AB} = \frac{1}{3} \)[/tex]
- [tex]\( \frac{BC}{DF} = \frac{BC}{\frac{1}{3} \cdot AC} = 3 \cdot \frac{BC}{AC} \)[/tex]
There is no consistent ratio here.
### Option B: [tex]\( \frac{AB}{AC} = \frac{DE}{EF} \)[/tex]
For Option B:
- [tex]\( \frac{AB}{AC} \)[/tex] is a ratio of the sides of [tex]\( \triangle ABC \)[/tex]
- [tex]\( DE \neq EF \)[/tex], but [tex]\( EF = \frac{1}{3} \cdot BC \)[/tex]
Comparing non-corresponding parts of the triangles is incorrect.
### Option C: [tex]\( \frac{AB}{EF} = \frac{DE}{BC} \)[/tex]
For Option C:
- [tex]\( EF = \frac{1}{3} \cdot BC \)[/tex]
- [tex]\( \frac{AB}{EF} = \frac{AB}{\frac{1}{3} \cdot BC} = 3 \cdot \frac{AB}{BC} \)[/tex]
- [tex]\( \frac{DE}{BC} = \frac{\frac{1}{3} \cdot AB}{BC} \)[/tex]
This does not guarantee a consistent ratio either.
### Option D: [tex]\( \frac{AB}{DE} = \frac{AC}{DF} \)[/tex]
For Option D:
- [tex]\( DE = \frac{1}{3} \cdot AB \)[/tex]
- [tex]\( DF = \frac{1}{3} \cdot AC \)[/tex]
- [tex]\( \frac{AB}{DE} = \frac{AB}{\frac{1}{3} \cdot AB} = 3 \)[/tex]
- [tex]\( \frac{AC}{DF} = \frac{AC}{\frac{1}{3} \cdot AC} = 3 \)[/tex]
This comparison shows that [tex]\( \frac{AB}{DE} = \frac{AC}{DF} = 3 \)[/tex], and both sides of the triangles are in the same ratio.
Therefore, the correct proportion that verifies that [tex]\( \triangle ABC \)[/tex] and [tex]\( \triangle DEF \)[/tex] are similar is:
[tex]\[ \boxed{D: \frac{AB}{DE} = \frac{AC}{DF}} \][/tex]
