Given that [tex]\(\triangle RST \sim \triangle RYX\)[/tex] by the SSS (Side-Side-Side) similarity theorem, this means that all corresponding sides of the triangles are proportional and corresponding angles are equal.
The ratios of the corresponding sides are equal since the triangles are similar. The sides corresponding to [tex]\(RT, RX\)[/tex] and [tex]\(RS\)[/tex] in [tex]\(\triangle RYX\)[/tex] are [tex]\(ST, YX\)[/tex] and [tex]\(RY\)[/tex] respectively.
To find the ratio that is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex], we need to look at the sides that will match the proportions.
Let's consider the ratios given:
1. [tex]\(\frac{XY}{TS}\)[/tex]
2. [tex]\(\frac{SY}{RY}\)[/tex]
3. [tex]\(\frac{RX}{XT}\)[/tex]
4. [tex]\(\frac{ST}{YX}\)[/tex]
Given the similar triangles, the corresponding side to [tex]\(RT\)[/tex] in [tex]\(\triangle RST\)[/tex] is [tex]\(ST\)[/tex] in [tex]\(\triangle RYX\)[/tex] and the corresponding side to [tex]\(RX\)[/tex] in [tex]\(\triangle RST\)[/tex] is [tex]\(YX\)[/tex] in [tex]\(\triangle RYX\)[/tex].
Thus, the ratio [tex]\(\frac{ST}{YX}\)[/tex] is the correct one because:
[tex]\[ \frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{YX} \][/tex]
Therefore, the answer is:
[tex]\[
\boxed{\frac{ST}{YX}}
\][/tex]
