To solve the problem, we need to determine which ratio is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex] given that [tex]\(\triangle RST \sim \triangle RYX\)[/tex] by the SSS similarity theorem.
When two triangles are similar, their corresponding sides are proportional. This means that the ratios of the lengths of corresponding sides of similar triangles are equal. For [tex]\(\triangle RST \sim \triangle RYX\)[/tex]:
[tex]\[
\frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{YX}
\][/tex]
We need to find which of the given ratios [tex]\(\frac{XY}{TS}\)[/tex], [tex]\(\frac{SY}{RY}\)[/tex], [tex]\(\frac{RX}{XT}\)[/tex], or [tex]\(\frac{ST}{YX}\)[/tex] is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
Let’s analyze each option:
1. [tex]\(\frac{XY}{TS}\)[/tex]:
- [tex]\(XY\)[/tex] and [tex]\(TS\)[/tex] are not corresponding sides in the two similar triangles. So, this ratio is not correct.
2. [tex]\(\frac{SY}{RY}\)[/tex]:
- [tex]\(SY\)[/tex] is not directly comparable to any side in [tex]\(\triangle RST\)[/tex], so this ratio doesn't match our required ratios.
3. [tex]\(\frac{RX}{XT}\)[/tex]:
- [tex]\(RX\)[/tex] and [tex]\(XT\)[/tex] are different types of segments, and [tex]\(XT\)[/tex] is not a side of the original triangle [tex]\(\triangle RST\)[/tex].
4. [tex]\(\frac{ST}{YX}\)[/tex]:
- [tex]\(ST\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to [tex]\(YX\)[/tex] in [tex]\(\triangle RYX\)[/tex]. Therefore, the correct ratio [tex]\(\frac{ST}{YX}\)[/tex] maintains the correspondence and ensures the proportional relationship required by triangle similarity.
Thus, the ratio [tex]\(\frac{ST}{YX}\)[/tex] correctly corresponds to the sides of the similar triangles, making it also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
