To determine which ratio is equal to \(\frac{RT}{RX}\) and \(\frac{RS}{RY}\), let us use the properties of similar triangles. Since \(\triangle RST \sim \triangle RYX\) by the SSS similarity theorem, the corresponding sides of the triangles will be proportional. This means that:
[tex]\[
\frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{YX}
\][/tex]
Now, let's examine the given ratios to see which one corresponds to \(\frac{RT}{RX}\) and \(\frac{RS}{RY}\):
1. \(\frac{XY}{TS}\): This ratio compares non-corresponding sides of the triangles, so it is not correct.
2. \(\frac{5Y}{RY}\): This ratio is not in terms of corresponding sides of the triangles directly and involves \(5Y\) which has not been defined in the context of the problem.
3. \(\frac{RX}{XT}\): This ratio does not involve the right corresponding sides; it mixes corresponding sides from both triangles in a way that does not reflect the side ratios of similar triangles.
4. \(\frac{ST}{YX}\): This ratio compares the corresponding sides \(ST\) of \(\triangle RST\) and \(YX\) of \(\triangle RYX\).
Since \(\triangle RST \sim \triangle RYX\), the corresponding sides are proportional. Specifically:
[tex]\[
\frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{YX}
\][/tex]
Therefore, the ratio that is also equal to \(\frac{RT}{RX}\) and \(\frac{RS}{RY}\) is:
[tex]\[
\boxed{\frac{ST}{YX}}
\][/tex]
