To solve this problem, we need to use the properties of similar triangles. When two triangles are similar, the ratios of the lengths of their corresponding sides are equal.
Given:
[tex]\(\triangle R S T \sim \triangle R Y X\)[/tex] by the SSS (Side-Side-Side) similarity theorem.
This implies that:
[tex]\[
\frac{R T}{R X} = \frac{R S}{R Y}
\][/tex]
We are asked to find which other ratio is also equal to [tex]\(\frac{R T}{R X}\)[/tex] and [tex]\(\frac{R S}{R Y}\)[/tex].
When two triangles are similar, not only are the ratios of the lengths of corresponding sides equal, but the order of the letters also indicates which side in one triangle corresponds to which side in the other triangle.
- [tex]\(R T\)[/tex] in [tex]\(\triangle R S T\)[/tex] corresponds to [tex]\(R X\)[/tex] in [tex]\(\triangle R Y X\)[/tex].
- [tex]\(R S\)[/tex] in [tex]\(\triangle R S T\)[/tex] corresponds to [tex]\(R Y\)[/tex] in [tex]\(\triangle R Y X\)[/tex].
- Consequently, [tex]\(S T\)[/tex] in [tex]\(\triangle R S T\)[/tex] corresponds to [tex]\(Y X\)[/tex] in [tex]\(\triangle R Y X\)[/tex].
Thus, we need to establish which ratio among the given choices matches this correspondence:
1. [tex]\(\frac{X Y}{T S}\)[/tex]
2. [tex]\(\frac{S Y}{R Y}\)[/tex]
3. [tex]\(\frac{R X}{X T}\)[/tex]
4. [tex]\(\frac{S T}{Y X}\)[/tex]
We determined that [tex]\(S T\)[/tex] corresponds to [tex]\(Y X\)[/tex], so:
[tex]\[
\frac{S T}{Y X}
\][/tex]
Hence, the ratio [tex]\(\frac{S T}{Y X}\)[/tex] is the one that also equals [tex]\(\frac{R T}{R X}\)[/tex] and [tex]\(\frac{R S}{R Y}\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
