To solve this problem, we first need to understand the given condition. The problem involves two triangles, [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex], which are similar by the Side-Side-Side (SSS) similarity theorem. This means that the corresponding sides of these triangles are proportional.
The ratios given are:
[tex]\[
\frac{RT}{RX} \quad \text{and} \quad \frac{RS}{RY}
\][/tex]
We need to identify which of the provided options also represents a ratio equal to these.
Recall that by the definition of similar triangles, the corresponding sides have equal ratios. In this case:
[tex]\[
\frac{RT}{RX} = \frac{RS}{RY} = \frac{XY}{TS}
\][/tex]
Let's check each option:
1. [tex]\(\frac{XY}{TS}\)[/tex]:
As stated, [tex]\(\frac{XY}{TS}\)[/tex] represents the ratio of the measure of one side of [tex]\(\triangle RYX\)[/tex] to its corresponding side in [tex]\(\triangle RST\)[/tex]. Given the similarity condition, this is correct. Therefore,
[tex]\[
\frac{RT}{RX} = \frac{RS}{RY} = \frac{XY}{TS}
\][/tex]
2. [tex]\(\frac{SY}{RY}\)[/tex]:
This ratio involves sides SY and RY. However, these sides do not form corresponding sides in the similar triangles [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex], so this ratio cannot be guaranteed to match [tex]\(\frac{RT}{RX}\)[/tex] or [tex]\(\frac{RS}{RY}\)[/tex].
3. [tex]\(\frac{RX}{XT}\)[/tex]:
This ratio involves side RX and a side XT that isn't mentioned in the context of either of the given triangles directly. So, it does not represent a corresponding side ratio of the similar triangles [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex].
4. [tex]\(\frac{ST}{VX}\)[/tex]:
This ratio includes side ST from [tex]\(\triangle RST\)[/tex] and side VX whose triangle context is not clearly defined. Again, this does not necessarily correspond with the sides defined in the given problem.
After evaluating each choice, the ratio that correctly represents the equality as per the similarity theorem is:
[tex]\[
\frac{XY}{TS}
\][/tex]
Therefore, the correct answer to which ratio is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex] is:
[tex]\[
\boxed{\frac{XY}{TS}}
\][/tex]
