To solve the question, let's carefully analyze the problem and the properties of similar triangles.
Given that \( \triangle RST \sim \triangle RYX \) by the SSS (Side-Side-Side) similarity theorem, we understand that the corresponding sides of these triangles are proportional. This means that the ratios of the lengths of corresponding sides of these triangles are equal.
We are provided with the following ratios:
[tex]\[ \frac{RT}{RX} \quad \text{and} \quad \frac{RS}{RY} \][/tex]
We need to determine which of the given ratios is also equal to these ratios.
Let's look closely at the provided options:
1. \(\frac{XY}{TS}\)
2. \(\frac{SY}{RY}\)
3. \(\frac{RX}{XT}\)
4. \(\frac{ST}{RX}\)
To check which ratio is correct, we use the concept of corresponding sides in similar triangles. Since \( \triangle RST \sim \triangle RYX \), the ratios of the corresponding sides should be equal.
Notice that:
- \( RT \) in \( \triangle RST \) corresponds to \( RX \) in \( \triangle RYX \).
- \( RS \) in \( \triangle RST \) corresponds to \( RY \) in \( \triangle RYX \).
- \( TS \) in \( \triangle RST \) corresponds to \( XY \) in \( \triangle RYX \).
So, we can write the following relationships:
[tex]\[ \frac{RT}{RX} = \frac{RS}{RY} = \frac{TS}{XY} \][/tex]
Rewriting these equivalences based on the ratios, we can conclude that:
[tex]\[ \frac{TS}{XY} \][/tex]
This means that \( \frac{XY}{TS} \) is also equal to \( \frac{RT}{RX} \) and \( \frac{RS}{RY} \), since ratios can be written in both ways, either as \(\frac{a}{b}\) or \(\frac{b}{a}\).
Therefore, the ratio that is also equal to \( \frac{RT}{RX} \) and \( \frac{RS}{RY} \) is:
[tex]\[ \frac{XY}{TS} \][/tex]
So, the correct option from the given choices is:
[tex]\[ \boxed{1} \][/tex]
