Let's analyze the given triangles and the problem:
Given that [tex]\(\triangle R S T\)[/tex] is similar to [tex]\(\triangle R Y X\)[/tex] by the SSS (Side-Side-Side) similarity theorem, we can deduce that the ratios of the corresponding sides of these triangles are equal.
The Side-Side-Side (SSS) similarity theorem states that if the corresponding side lengths of two triangles are proportional, then the triangles are similar.
For [tex]\(\triangle R S T\)[/tex] and [tex]\(\triangle R Y X\)[/tex], this implies:
[tex]\[ \frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{XY}. \][/tex]
Now, let's check the given options to identify which ratio is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex]:
1. [tex]\(\frac{XY}{TS}\)[/tex]:
- This is the inverse of [tex]\(\frac{ST}{XY}\)[/tex], which means [tex]\(\frac{XY}{TS} = \frac{1}{\frac{ST}{XY}}\)[/tex]. This ratio is not equal to [tex]\(\frac{RT}{RX}\)[/tex] or [tex]\(\frac{RS}{RY}\)[/tex].
2. [tex]\(\frac{SY}{RY}\)[/tex]:
- This ratio involves [tex]\(\triangle R Y X\)[/tex] but does not align correctly with the corresponding sides of the two triangles. Thus, [tex]\(\frac{SY}{RY}\)[/tex] is not equal to [tex]\(\frac{RT}{RX}\)[/tex] or [tex]\(\frac{RS}{RY}\)[/tex].
3. [tex]\(\frac{RX}{XT}\)[/tex]:
- [tex]\(\frac{RX}{XT}\)[/tex] does not involve the correct pairs of corresponding sides. It mixes up the sides without proper alignment with [tex]\(\triangle R S T\)[/tex]. Therefore, this ratio is not equal to [tex]\(\frac{RT}{RX}\)[/tex] or [tex]\(\frac{RS}{RY}\)[/tex].
4. [tex]\(\frac{ST}{XY}\)[/tex]:
- [tex]\(\frac{ST}{XY}\)[/tex] is directly one of the corresponding sides' ratios according to the SSS similarity theorem. Thus, it is equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
Therefore, the correct ratio that is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex] is:
[tex]\[ \boxed{\frac{ST}{XY}} \][/tex]
