Given that [tex]\(\triangle RST \sim \triangle RYX\)[/tex] by the SSS (Side-Side-Side) similarity theorem, we know that the ratios of the corresponding sides of these similar triangles are equal. The SSS similarity theorem states that if the corresponding side lengths of two triangles are proportional, then the triangles are similar.
### Let's identify the corresponding sides of the triangles:
1. Side [tex]\(RS\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to side [tex]\(RY\)[/tex] in [tex]\(\triangle RYX\)[/tex].
2. Side [tex]\(RT\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to side [tex]\(RX\)[/tex] in [tex]\(\triangle RYX\)[/tex].
3. Side [tex]\(ST\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to side [tex]\(YX\)[/tex] in [tex]\(\triangle RYX\)[/tex].
### Write the proportional relationships:
1. [tex]\(\frac{RS}{RY}\)[/tex]
2. [tex]\(\frac{RT}{RX}\)[/tex]
3. [tex]\(\frac{ST}{YX}\)[/tex]
According to similarity, all these ratios should be equal:
[tex]\[ \frac{RS}{RY} = \frac{RT}{RX} = \frac{ST}{YX} \][/tex]
### Now, analyze the given choices:
- [tex]\(\frac{XY}{TS}\)[/tex]
- [tex]\(\frac{SY}{RY}\)[/tex]
- [tex]\(\frac{RX}{XT}\)[/tex]
- [tex]\(\frac{ST}{YX}\)[/tex]
### Verify each choice:
- [tex]\(\frac{XY}{TS}\)[/tex]: This ratio does not correspond to the sides in the given triangles as [tex]\(\frac{XY}\)[/tex] and [tex]\(\frac{TS}\)[/tex] do not appear directly as a pair.
- [tex]\(\frac{SY}{RY}\)[/tex]: This is also not a corresponding side ratio since it mixes two different corresponding sides.
- [tex]\(\frac{RX}{XT}\)[/tex]: While [tex]\(\frac{RX}\)[/tex] is a side in [tex]\(\triangle RYX\)[/tex], [tex]\(\frac{XT}\)[/tex] is not a side length that directly relates in the proportional sides specified by similarity theorem.
- [tex]\(\frac{ST}{YX}\)[/tex]: This ratio directly corresponds to the sides [tex]\(ST\)[/tex] and [tex]\(YX\)[/tex], which are known to be proportional from [tex]\( \frac{RS}{RY} = \frac{RT}{RX} = \frac{ST}{YX} \)[/tex].
Therefore, the correct ratio that is equal to both [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex] is [tex]\(\frac{ST}{YX}\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
