To determine the proportion that verifies the similarity between [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle DEF\)[/tex] after dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex], we need to check if the corresponding sides have consistent ratios, in accordance with the scale factor.
First, let's understand what dilation implies in this context. Dilation takes each length of [tex]\(\triangle ABC\)[/tex] and multiplies it by the scale factor [tex]\(\frac{1}{3}\)[/tex] to obtain the corresponding lengths in [tex]\(\triangle DEF\)[/tex].
Given the initial side lengths of [tex]\(\triangle ABC\)[/tex] and their corresponding sides after dilation:
- If [tex]\(AB\)[/tex] and [tex]\(AC\)[/tex] are original sides of [tex]\(\triangle ABC\)[/tex], then the corresponding sides [tex]\(DE\)[/tex] and [tex]\(DF\)[/tex] of [tex]\(\triangle DEF\)[/tex] would be [tex]\( \frac{1}{3}AB\)[/tex] and [tex]\( \frac{1}{3}AC\)[/tex], respectively.
To confirm the similarity, we need to ensure the ratio of the corresponding sides between the two triangles remains consistent:
- Consider the ratio [tex]\(\frac{AB}{DE}\)[/tex]:
Since [tex]\(DE = \frac{1}{3}AB\)[/tex], we get:
[tex]\[
\frac{AB}{DE} = \frac{AB}{\frac{1}{3}AB} = 3
\][/tex]
- Now consider the ratio [tex]\(\frac{AC}{DF}\)[/tex]:
Since [tex]\(DF = \frac{1}{3}AC\)[/tex], we get:
[tex]\[
\frac{AC}{DF} = \frac{AC}{\frac{1}{3}AC} = 3
\][/tex]
Since both ratios [tex]\(\frac{AB}{DE}\)[/tex] and [tex]\(\frac{AC}{DF}\)[/tex] are equal (both equate to 3), the proportion that verifies the similarity of the triangles is:
[tex]\[
\frac{AB}{DE} = \frac{AC}{DF}
\][/tex]
Thus, the correct choice is:
D. [tex]\(\frac{AB}{DE} = \frac{AC}{DF}\)[/tex]
