Given that [tex]\(\triangle RST \sim \triangle RYX\)[/tex] by the SSS similarity theorem, the corresponding sides of these triangles are proportional. This means the ratios of their corresponding sides are equal. Specifically, for the triangles [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex]:
[tex]\[
\frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{YX}
\][/tex]
Let's see which of the given ratios is equal to these.
1. [tex]\(\frac{A_T^n}{\text{TS}}\)[/tex]
- This ratio does not involve corresponding sides of the triangles [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex].
2. [tex]\(\frac{SY}{RY}\)[/tex]
- This ratio is not relevant as [tex]\(SY\)[/tex] and [tex]\(RY\)[/tex] do not correspond to complete sides of the given triangles.
3. [tex]\(\frac{RX}{XT}\)[/tex]
- This ratio involves sides within only [tex]\(\triangle RYX\)[/tex], not the corresponding sides between [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex].
4. [tex]\(\frac{ST}{YX}\)[/tex]
- This ratio directly involves the corresponding sides of [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex], and matches our derived ratio from the similarity condition.
We can see that the ratio [tex]\(\frac{ST}{YX}\)[/tex] is indeed equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
Therefore, the ratio that is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex] is:
[tex]\[
\boxed{\frac{ST}{YX}}
\][/tex]
