Given that [tex]\(\triangle RST \sim \triangle RYX\)[/tex] by the SSS (Side-Side-Side) similarity theorem, let's determine which ratio among the given options is also equal to the ratios [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
When two triangles are similar, their corresponding sides are proportional. This means that the ratios of the lengths of corresponding sides of the triangles are equal.
From the given information, we know:
[tex]\[
\frac{RT}{RX} = \frac{RS}{RY}
\][/tex]
We need to identify which of the given ratios is equivalent to these.
Let's examine the given options:
1. [tex]\(\frac{XY}{TS}\)[/tex]
2. [tex]\(\frac{SY}{RY}\)[/tex]
3. [tex]\(\frac{RX}{XT}\)[/tex]
4. [tex]\(\frac{ST}{rx}\)[/tex] (assuming [tex]\(rx\)[/tex] is meant to be [tex]\(RX\)[/tex])
Since [tex]\(\triangle RST \sim \triangle RYX\)[/tex], the sides of [tex]\(\triangle RST\)[/tex] (i.e., [tex]\(RT, RS, ST\)[/tex]) correspond proportionally to the sides of [tex]\(\triangle RYX\)[/tex] (i.e., [tex]\(RX, RY, XY\)[/tex]).
Among the given options:
1. [tex]\(\frac{XY}{TS}\)[/tex]: [tex]\(XY\)[/tex] in [tex]\(\triangle RYX\)[/tex] corresponds to [tex]\(TS\)[/tex] in [tex]\(\triangle RST\)[/tex]. However, this ratio does not fit the corresponding proportional sides directly as expressed by [tex]\(\frac{RT}{RX}\)[/tex] or [tex]\(\frac{RS}{RY}\)[/tex].
2. [tex]\(\frac{SY}{RY}\)[/tex]: This ratio is not expressing a correspondence of sides from [tex]\(\triangle RST\)[/tex] to [tex]\(\triangle RYX\)[/tex], and does not fit our requirement.
3. [tex]\(\frac{RX}{XT}\)[/tex]: This maintains the proportional relationship of corresponding sides since [tex]\(RX\)[/tex] is a side in [tex]\(\triangle RYX\)[/tex] and [tex]\(XT\)[/tex] can be viewed proportional to [tex]\(RT\)[/tex] but not directly corresponding here; thus, it fits the similarity criteria.
4. [tex]\(\frac{ST}{rx}\)[/tex]: Interpreting [tex]\(rx\)[/tex] as [tex]\(RX\)[/tex], this ratio is [tex]\(\frac{ST}{RX}\)[/tex], aligning with corresponding side relationships; however, it's more indirect than direct correspondence ratios.
From these options, [tex]\(\frac{RX}{XT}\)[/tex] stands out as aligned with maintaining triangle side proportionality. Hence, maintaining the proportional relationship:
[tex]\[
\boxed{\frac{RX}{XT}}
\][/tex]
