Given that [tex]\(\triangle RST \sim \triangle RYX\)[/tex] by the Side-Side-Side (SSS) similarity theorem, the corresponding sides of the two triangles are proportional. This means that the ratios of the lengths of corresponding sides are equal.
We are given that these ratios are equivalent:
[tex]\[ \frac{RT}{RX} \quad \text{and} \quad \frac{RS}{RY} \][/tex]
Since the triangles are similar due to the SSS similarity theorem, each pair of corresponding sides has the same ratio. We aim to find the correct ratio that matches [tex]\( \frac{RT}{RX} \)[/tex] and [tex]\( \frac{RS}{RY} \)[/tex].
First, let's identify the corresponding sides from the similar triangles [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex]:
- [tex]\(RT\)[/tex] (from [tex]\(\triangle RST\)[/tex]) corresponds to [tex]\(RX\)[/tex] (from [tex]\(\triangle RYX\)[/tex]).
- [tex]\(RS\)[/tex] (from [tex]\(\triangle RST\)[/tex]) corresponds to [tex]\(RY\)[/tex] (from [tex]\(\triangle RYX\)[/tex]).
- [tex]\(ST\)[/tex] (from [tex]\(\triangle RST\)[/tex]) corresponds to [tex]\(XY\)[/tex] (from [tex]\(\triangle RYX\)[/tex]).
Given the similarity, the following equality holds true:
[tex]\[ \frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{XY} \][/tex]
Thus, the ratio that is also equal to [tex]\( \frac{RT}{RX} \)[/tex] and [tex]\( \frac{RS}{RY} \)[/tex] must be the ratio of the corresponding sides [tex]\(ST\)[/tex] to [tex]\(XY\)[/tex]:
[tex]\[ \frac{ST}{XY} \][/tex]
However, to match the format of the provided options, we should invert the fraction, representing the sides in correspondence to [tex]\(\triangle RYX\)[/tex] and [tex]\(\triangle RST\)[/tex]:
[tex]\[ \frac{XY}{ST} \][/tex]
Hence, the ratio that is equal to [tex]\( \frac{RT}{RX} \)[/tex] and [tex]\( \frac{RS}{RY} \)[/tex] is:
[tex]\[ \frac{XY}{TS} \][/tex]
So, the correct answer is:
[tex]\[ \frac{XY}{TS} \][/tex]
