To verify that [tex]$\triangle ABC$[/tex] and [tex]$\triangle XYZ$[/tex] are similar by a dilation with a scale factor of 5, we need to ensure that corresponding sides of these triangles are proportional to each other by the scale factor. Here are the given options:
A. [tex]\(\frac{AB}{XY} = \frac{AC}{XZ}\)[/tex]
B. [tex]\(BC = AB\)[/tex]
C. [tex]\(\frac{AB}{AC} + \frac{XZ}{XY}\)[/tex]
D. [tex]\(V2 = AC\)[/tex]
Let's analyze each option one by one:
### Option A: [tex]\(\frac{AB}{XY} = \frac{AC}{XZ}\)[/tex]
For the triangles to be similar under dilation, corresponding side lengths need to be proportional. Given the scale factor is 5, it means the sides of [tex]$\triangle ABC$[/tex] are 1/5th the length of the corresponding sides in [tex]$\triangle XYZ$[/tex]:
[tex]\[
\frac{AB}{XY} = \frac{AC}{XZ} = \frac{BC}{YZ} = \frac{1}{5}
\][/tex]
This option is correct because it expresses the proportional relationship required for similarity under dilation.
### Option B: [tex]\(BC = AB\)[/tex]
This option implies that two sides of [tex]$\triangle ABC$[/tex] are equal, which is generally not a requirement for similarity by dilation. It does not relate to the scale factor or proportionality between corresponding sides of the two triangles. So, this option is incorrect.
### Option C: [tex]\(\frac{AB}{AC} + \frac{XZ}{XY}\)[/tex]
This option combines ratios of the sides from each triangle in a way that does not represent the proportionality required for similarity. The sum of these ratios does not give any meaningful information regarding dilation or similarity. Thus, this option is incorrect.
### Option D: [tex]\(V2 = AC\)[/tex]
This option does not make any geometric sense in the context of verifying similarity through dilation. It mentions a variable "V2" which seems unrelated or incorrect in this context. Therefore, this option is incorrect.
### Conclusion
Given the analysis above, the correct option that verifies the similarity of [tex]$\triangle ABC$[/tex] and [tex]$\triangle XYZ$[/tex] under a dilation with a scale factor of 5 is:
A. [tex]\(\frac{AB}{XY} = \frac{AC}{XZ}\)[/tex]
