To solve this problem, we need to consider the properties of similar triangles and the Side-Side-Side (SSS) similarity theorem. According to the SSS similarity theorem, if the corresponding sides of two triangles are proportional, then the triangles are similar.
Given that [tex]\(\triangle RST \sim \triangle RYX\)[/tex], we know that the corresponding sides of these triangles are proportional. This implies the following relationships:
[tex]\[
\frac{RT}{RX} = \frac{RS}{RY}
\][/tex]
To find which ratio among the given options is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex], we need to determine the corresponding side in [tex]\(\triangle RYX\)[/tex] for the remaining side [tex]\(ST\)[/tex] in [tex]\(\triangle RST\)[/tex].
1. Identify Corresponding Sides:
- [tex]\(RT\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to [tex]\(RX\)[/tex] in [tex]\(\triangle RYX\)[/tex].
- [tex]\(RS\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to [tex]\(RY\)[/tex] in [tex]\(\triangle RYX\)[/tex].
- Remaining side [tex]\(ST\)[/tex] in [tex]\(\triangle RST\)[/tex] should correspond to the side between the non-included angles of [tex]\(\triangle RYX\)[/tex], which is [tex]\(XY\)[/tex].
Therefore, the ratio involving the remaining sides [tex]\(ST\)[/tex] and [tex]\(XY\)[/tex] is:
[tex]\[
\frac{XY}{TS}
\][/tex]
This ratio must also be equal to the previously mentioned proportional side ratios because it involves the corresponding third sides of the similar triangles.
2. Verify the Corresponding Side Ratio:
- Check the given options to see which one matches our identified ratio:
- [tex]\(\frac{XY}{TS}\)[/tex]
- [tex]\(\frac{SY}{RY}\)[/tex]
- [tex]\(\frac{PX}{XT}\)[/tex]
- [tex]\(\frac{ST}{VX}\)[/tex]
Only the first option [tex]\(\frac{XY}{TS}\)[/tex] fits the identified ratio for the corresponding sides:
[tex]\[
\frac{XY}{TS}
\][/tex]
Thus, the correct answer is [tex]\(\frac{XY}{TS}\)[/tex].
