To determine which ratio is also equal to \(\frac{RT}{RX}\) and \(\frac{RS}{RY}\), let's first analyze the given triangles \(\triangle RST\) and \(\triangle RYX\). Because these triangles are similar by the SSS (Side-Side-Side) similarity theorem, the corresponding sides of these triangles are proportional.
The ratio \(\frac{RT}{RX}\) is given, and we’re also given that \(\frac{RS}{RY}\) is equal to this ratio. To find out which among the choices corresponds to this ratio, we should identify which sides of the triangles correspond.
Ratios of Corresponding Sides:
1. \(\frac{RT}{RX}\): This is the ratio of a side of \(\triangle RST\) to a corresponding side of \(\triangle RYX\).
2. \(\frac{RS}{RY}\): This is the ratio of another side of \(\triangle RST\) to a corresponding side of \(\triangle RYX\).
Given the choices:
1. \(\frac{XY}{TS}\)
2. \(\frac{SY}{RY}\)
3. \(\frac{RX}{XT}\)
4. \(\frac{ST}{VX}\)
Let's analyze each choice:
Choice 1: \(\frac{XY}{TS}\)
- In \(\triangle RST\) and \(\triangle RYX\), if we consider the side \(XY\) from \(\triangle RYX\) and \(TS\) from \(\triangle RST\), these are matching corresponding sides.
- Since \(\triangle RYX\) is similar to \(\triangle RST\), the ratio of their corresponding sides should be consistent with the given ratios \(\frac{RT}{RX}\) and \(\frac{RS}{RY}\).
- So, \(\frac{XY}{TS}\) should also equal the same proportional ratio.
Therefore, the ratio \(\frac{XY}{TS}\) is the correct choice that is equal to \(\frac{RT}{RX}\).
So, the ratio that is also equal to \(\frac{RT}{RX}\) and \(\frac{RS}{RY}\) is:
[tex]\(\boxed{\frac{XY}{TS}}\)[/tex].
