Given that [tex]\(\triangle RST \sim \triangle RYX\)[/tex] by the Side-Side-Side (SSS) similarity theorem, we know that the corresponding sides of these triangles are proportional. This implies that the ratios of the corresponding sides are equal.
We are given the ratios:
[tex]\[
\frac{RT}{RX} \quad \text{and} \quad \frac{RS}{RY}
\][/tex]
We need to determine which ratio is also equal to these ratios from the given options:
1. [tex]\(\frac{XY}{TS}\)[/tex]
2. [tex]\(\frac{SY}{RY}\)[/tex]
3. [tex]\(\frac{RX}{XT}\)[/tex]
4. [tex]\(\frac{sT}{rx}\)[/tex]
To solve this, let's identify each pair of corresponding sides in the similar triangles [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex].
- [tex]\(RT\)[/tex] corresponds to [tex]\(RX\)[/tex]
- [tex]\(RS\)[/tex] corresponds to [tex]\(RY\)[/tex]
- [tex]\(ST\)[/tex] corresponds to [tex]\(YX\)[/tex]
We look for the ratio that mirrors these correspondences. Since the triangles are similar, corresponding side lengths have equivalent ratios. Hence, the ratio we are asked to find [tex]\(\frac{XY}{TS}\)[/tex]:
- Corresponds to the sides of [tex]\(\triangle RYX\)[/tex] and [tex]\(\triangle RST\)[/tex].
So, the ratio [tex]\(\frac{XY}{TS}\)[/tex] must be equivalent to the corresponding sides in the original equalities given:
[tex]\[
\frac{XY}{TS} = \frac{YX}{ST}
\][/tex]
Since the triangles are similar, the ratios of corresponding sides are maintained:
[tex]\[
\frac{XY}{TS} = \frac{RT}{RX} = \frac{RS}{RY}
\][/tex]
Thus, the ratio [tex]\(\frac{XY}{TS}\)[/tex] is also equal to the given ratios:
[tex]\[
\boxed{1}
\][/tex]
