Given that [tex]\(\triangle R S T \sim \triangle R Y X\)[/tex] by the SSS (Side-Side-Side) similarity theorem, we know that the ratios of the corresponding sides of these two triangles are equal.
The corresponding sides of the two triangles ∆RST and ∆RYX are:
- [tex]\(RS\)[/tex] corresponds to [tex]\(RY\)[/tex]
- [tex]\(RT\)[/tex] corresponds to [tex]\(RX\)[/tex]
- [tex]\(ST\)[/tex] corresponds to [tex]\(XY\)[/tex]
In the problem, we need to determine which given ratio is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
According to the similarity of triangles, the sides of the triangles have the following proportions:
[tex]\[ \frac{RS}{RY} = \frac{RT}{RX} = \frac{ST}{XY} \][/tex]
Now, let's review the choices given in the question:
1. [tex]\(\frac{XY}{TS}\)[/tex]
2. [tex]\(\frac{SY}{RY}\)[/tex]
3. [tex]\(\frac{RX}{XT}\)[/tex]
4. [tex]\(\frac{ST}{4x}\)[/tex]
Given that [tex]\(\frac{ST}{XY}\)[/tex] must be equal to the other ratios because they correspond to similar sides, we identify the correct ratio among the choices.
The ratio [tex]\(\frac{XY}{TS}\)[/tex] represents the inverse of [tex]\(\frac{ST}{XY}\)[/tex] but since these ratios are equal in magnitude, it is fundamentally the same ratio, just the other side of it, and still represents the corresponding sides in triangles ∆RST and ∆RYX.
Therefore, the ratio [tex]\(\frac{XY}{TS}\)[/tex] is indeed the ratio equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
Thus, the answer is the first choice:
[tex]\(\boxed{1}\)[/tex]
