Answer :
Given that \(\triangle RST \sim \triangle RYX\) by the SSS (Side-Side-Side) similarity theorem, we need to determine which ratio is also equal to \(\frac{RT}{RX}\) and \(\frac{RS}{RY}\).
### Step-by-Step Solution
1. Understanding Similar Triangles:
Since \(\triangle RST\) is similar to \(\triangle RYX\), the corresponding sides of these triangles are in proportion. This means:
[tex]\[ \frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{YX} \][/tex]
2. Ratios of Corresponding Sides:
- \( RT \) corresponds to \( RX \)
- \( RS \) corresponds to \( RY \)
- \( ST \) corresponds to \( YX \)
3. Figuring Out the Equal Ratios:
Next, we need to identify which option reflects the ratio of one set of corresponding sides.
4. Analyzing the Options:
- \(\frac{XY}{TS}\): This compares the side \( XY \) from \(\triangle RYX\) with side \( TS \) from \(\triangle RST\) which are not corresponding sides in the similar triangles.
- \(\frac{SY}{RY}\): Not relevant as it compares a mix of non-corresponding sides.
- \(\frac{RX}{XT}\): This is not a correct relationship of corresponding sides.
- \(\frac{ST}{YX}\): This correctly compares the corresponding sides \( ST \) from \(\triangle RST\) and \( YX \) from \(\triangle RYX\).
5. Conclusion:
From the properties of similar triangles mentioned and the above analysis, we see that \(\frac{ST}{YX}\) is indeed in the same ratio as \(\frac{RT}{RX}\) and \(\frac{RS}{RY}\).
Hence, the ratio that is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex] is [tex]\(\frac{ST}{YX}\)[/tex].
### Step-by-Step Solution
1. Understanding Similar Triangles:
Since \(\triangle RST\) is similar to \(\triangle RYX\), the corresponding sides of these triangles are in proportion. This means:
[tex]\[ \frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{YX} \][/tex]
2. Ratios of Corresponding Sides:
- \( RT \) corresponds to \( RX \)
- \( RS \) corresponds to \( RY \)
- \( ST \) corresponds to \( YX \)
3. Figuring Out the Equal Ratios:
Next, we need to identify which option reflects the ratio of one set of corresponding sides.
4. Analyzing the Options:
- \(\frac{XY}{TS}\): This compares the side \( XY \) from \(\triangle RYX\) with side \( TS \) from \(\triangle RST\) which are not corresponding sides in the similar triangles.
- \(\frac{SY}{RY}\): Not relevant as it compares a mix of non-corresponding sides.
- \(\frac{RX}{XT}\): This is not a correct relationship of corresponding sides.
- \(\frac{ST}{YX}\): This correctly compares the corresponding sides \( ST \) from \(\triangle RST\) and \( YX \) from \(\triangle RYX\).
5. Conclusion:
From the properties of similar triangles mentioned and the above analysis, we see that \(\frac{ST}{YX}\) is indeed in the same ratio as \(\frac{RT}{RX}\) and \(\frac{RS}{RY}\).
Hence, the ratio that is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex] is [tex]\(\frac{ST}{YX}\)[/tex].
