To determine which ratio is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex], we need to rely on the properties of similar triangles, specifically the Side-Side-Side (SSS) similarity theorem.
The SSS similarity theorem states that if two triangles have their corresponding sides in proportion, then the triangles are similar.
Given:
- [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex] are similar triangles based on the SSS similarity theorem.
So, the corresponding sides of these similar triangles should be proportional:
[tex]\[
\frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{XY}
\][/tex]
We are asked to identify the correct ratio that matches [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
By the properties of similar triangles:
1. The ratio [tex]\(\frac{RT}{RX}\)[/tex] involves sides [tex]\(RT\)[/tex] in [tex]\(\triangle RST\)[/tex] and [tex]\(RX\)[/tex] in [tex]\(\triangle RYX\)[/tex].
2. Similarly, the ratio [tex]\(\frac{RS}{RY}\)[/tex] involves sides [tex]\(RS\)[/tex] in [tex]\(\triangle RST\)[/tex] and [tex]\(RY\)[/tex] in [tex]\(\triangle RYX\)[/tex].
For these ratios to be equal, there must be a corresponding ratio involving the remaining pair of sides.
Let's evaluate the options:
1. [tex]\(\frac{XY}{TS}\)[/tex]: This suggests that the ratio of side [tex]\(XY\)[/tex] in [tex]\(\triangle RYX\)[/tex] to side [tex]\(TS\)[/tex] in [tex]\(\triangle RST\)[/tex] is consistent with the given ratios [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex]. This fits the pattern of corresponding sides in similar triangles.
2. [tex]\(\frac{SY}{RY}\)[/tex]: This ratio involves part of one side in [tex]\(\triangle RYX\)[/tex] and one side in [tex]\(\triangle RYX\)[/tex], not fitting the proportional relationships between [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex].
3. [tex]\(\frac{RX}{XT}\)[/tex]: This ratio involves parts of two different sides in [tex]\(\triangle RYX\)[/tex], which doesn't follow the pattern of similar triangle sides.
4. [tex]\(\frac{ST}{YX}\)[/tex]: This ratio involves the side [tex]\(ST\)[/tex] from [tex]\(\triangle RST\)[/tex] and side [tex]\(YX\)[/tex] from [tex]\(\triangle RYX\)[/tex]. Structurally it might look similar, but directly it looks reversed based on standard proportional relationships in similar triangles.
Given these options, the only ratio that correctly forms and matches the proportional relationships derived from similar triangles is:
[tex]\[
\boxed{\frac{XY}{TS}}
\][/tex]
