To solve this problem, we need to identify the ratio that is equivalent to both [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex], given that [tex]\(\triangle RST \sim \triangle RYX\)[/tex].
1. Similarity of Triangles:
Since [tex]\(\triangle RST \sim \triangle RYX\)[/tex], corresponding sides of these triangles are proportional. This means that the ratios of corresponding sides are equal.
2. Corresponding Side Ratios:
- For [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex], the side [tex]\(RT\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to [tex]\(RX\)[/tex] in [tex]\(\triangle RYX\)[/tex].
- Similarly, the side [tex]\(RS\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to [tex]\(RY\)[/tex] in [tex]\(\triangle RYX\)[/tex].
3. Identify Equal Ratios:
According to the properties of similar triangles:
[tex]\[
\frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{XY}
\][/tex]
4. Match the given ratios:
- [tex]\(\frac{XY}{TS}\)[/tex]
- [tex]\(\frac{SY}{RY}\)[/tex]
- [tex]\(\frac{RX}{XT}\)[/tex]
- [tex]\(\frac{sT}{YX}\)[/tex]
From the above corresponding ratios, we need to check which one matches the provided equal ratios. The ratio [tex]\(\frac{XY}{TS}\)[/tex] corresponds directly to the inverse of our established corresponding sides [tex]\(\frac{ST}{XY}\)[/tex], hence it is equivalent to both [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
Thus, the answer is:
[tex]\[
\frac{XY}{TS}
\][/tex]
