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Answered

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Given that [tex]\(\triangle XYZ\)[/tex] is a dilation of [tex]\(\triangle ABC\)[/tex] by a scale factor of 5, which of the following proportions verifies that [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle XYZ\)[/tex] are similar?

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]\(BC = AC\)[/tex]

Sagot :

GinnyAnsweridn
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]
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