To solve this problem, we use the concept of similarity between triangles. When two triangles are similar by the SSS (Side-Side-Side) similarity theorem, their corresponding sides are proportional.
In other words, corresponding sides of similar triangles have the same ratio.
Given triangles [tex]\( \triangle RST \)[/tex] and [tex]\( \triangle RYX \)[/tex] are similar by SSS similarity theorem, we know the corresponding sides are proportional:
[tex]\[
\frac{RT}{RX} = \frac{RS}{RY} = \frac{XY}{TS}
\][/tex]
We need to find which ratio is also equal to [tex]\( \frac{RT}{RX} \)[/tex] and [tex]\( \frac{RS}{RY} \)[/tex] from the provided 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]
Looking at the ratios given:
- [tex]\(\frac{XY}{TS}\)[/tex]: Since the corresponding sides for triangles [tex]\( \triangle RST \)[/tex] and [tex]\( \triangle RYX \)[/tex] are in proportion, this matches our requirement. Hence, [tex]\(\frac{XY}{TS}\)[/tex] is a correct ratio.
- [tex]\(\frac{SY}{RY}\)[/tex] and [tex]\(\frac{RX}{XT}\)[/tex] and [tex]\(\frac{ST}{rx}\)[/tex] do not represent the proportional sides of the similar triangles [tex]\( \triangle RST \)[/tex] and [tex]\( \triangle RYX \)[/tex] correctly.
Therefore, the correct corresponding ratio equal to [tex]\( \frac{RT}{RX} \)[/tex] and [tex]\( \frac{RS}{RY} \)[/tex] is:
[tex]\[
\frac{XY}{TS}
\][/tex]
To sum up, the ratio [tex]\( \frac{XY}{TS} \)[/tex] is equal to [tex]\( \frac{RT}{RX} \)[/tex] and [tex]\( \frac{RS}{RY} \)[/tex].
Thus, the answer is:
[tex]\[
\boxed{1}
\][/tex]
