First, let's understand what it means for two triangles to be similar. Triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion.
We know that [tex]\( \triangle U V W \)[/tex] is a dilation of [tex]\( \triangle R S T \)[/tex] by a scale factor of 1. A dilation by a scale factor of 1 means that the size of the triangle does not change, which implies that [tex]\( \triangle U V W \)[/tex] and [tex]\( \triangle R S T \)[/tex] are the exact same size and hence congruent.
To verify similarity (or congruence in this case), we need to compare the corresponding sides of the triangles and show that the ratios of these sides are equal.
Let's examine the given options:
A. [tex]\( \frac{R S}{U V} = \frac{U W}{S R} \)[/tex]
This proportion does not compare corresponding sides correctly. It mixes different sides of the triangles in the ratios, which is not helpful for verifying similarity.
B. [tex]\( \frac{R S}{S T} = \frac{W V}{V U} \)[/tex]
In this option, [tex]\( \frac{R S}{S T} \)[/tex] is compared with [tex]\( \frac{W V}{V U} \)[/tex]. This correctly compares the corresponding sides of [tex]\( \triangle R S T \)[/tex] and [tex]\( \triangle U V W \)[/tex]. If these ratios are equal, it would indeed confirm that the triangles are similar.
C. [tex]\( \frac{R T}{S T} = \frac{U W}{U V} \)[/tex]
This option does not correctly compare the sides of the triangles. [tex]\( \frac{R T}{S T} \)[/tex] compares one ratio from the original triangle and [tex]\( \frac{U W}{U V} \)[/tex] shows another ratio from the dilated triangle. This does not establish a direct comparison between the corresponding sides.
D. [tex]\( \frac{P S}{U V} = \frac{Q T}{U W} \)[/tex]
Here, we see [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] appear which are not part of the triangles mentioned [tex]\( \triangle R S T \)[/tex] and [tex]\( \triangle U V W \)[/tex]. This option is invalid because the points [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] are irrelevant to the given triangles.
Given these analyses, we can conclude that:
To confirm that [tex]\( \triangle R S T \)[/tex] and [tex]\( \triangle U V W \)[/tex] are similar through their corresponding sides, the correct proportion is:
B. [tex]\( \frac{R S}{S T} = \frac{W V}{V U} \)[/tex]
Thus, the correct answer is option [tex]\( B \)[/tex].
