Answer :
Given the triangles [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex] that are similar by the SSS (Side-Side-Side) similarity theorem, we need to find the ratio that is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
Step-by-step solution:
1. Identify corresponding sides in the similar triangles:
- Since [tex]\(\triangle RST \sim \triangle RYX\)[/tex], the sides of these triangles are proportional. Corresponding sides must be identified correctly.
- In these similar triangles:
- [tex]\(RT\)[/tex] corresponds to [tex]\(RX\)[/tex],
- [tex]\(RS\)[/tex] corresponds to [tex]\(RY\)[/tex],
- [tex]\(ST\)[/tex] corresponds to [tex]\(YX\)[/tex].
2. Form the ratios using corresponding sides:
- The ratio of sides [tex]\(RT/\)[/tex] and [tex]\(RX\)[/tex] pertains to corresponding sides in both triangles.
- Similarly, the ratio of sides [tex]\(RS/\)[/tex] and [tex]\(RY\)[/tex] also pertains to corresponding sides in both triangles.
3. Check each given option for correspondence and confirm proportional relationships:
- Check if [tex]\(\frac{XY}{TS}\)[/tex] is a valid ratio:
- [tex]\(XY\)[/tex] and [tex]\(TS\)[/tex] are not corresponding sides in [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex], so this ratio is not valid.
- Check if [tex]\(\frac{SY}{RY}\)[/tex] is a valid ratio:
- [tex]\(SY\)[/tex] is not a corresponding side in either triangle, making this option invalid.
- Check if [tex]\(\frac{RX}{XT}\)[/tex] is a valid ratio:
- This ratio is not a valid analogy for corresponding sides in the similar triangles.
- Check if [tex]\(\frac{ST}{YX}\)[/tex] is a valid ratio:
- Here, [tex]\(ST\)[/tex] and [tex]\(YX\)[/tex] are corresponding sides of the similar triangles.
- Because [tex]\(ST\)[/tex] corresponds to [tex]\(YX\)[/tex], the ratio [tex]\(\frac{ST}{YX}\)[/tex] is a valid ratio that must be equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
4. Conclusion:
- The correct ratio that corresponds to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex] is [tex]\(\frac{ST}{YX}\)[/tex].
Thus, the answer is [tex]\(\frac{ST}{YX}\)[/tex], which is option 4.
Step-by-step solution:
1. Identify corresponding sides in the similar triangles:
- Since [tex]\(\triangle RST \sim \triangle RYX\)[/tex], the sides of these triangles are proportional. Corresponding sides must be identified correctly.
- In these similar triangles:
- [tex]\(RT\)[/tex] corresponds to [tex]\(RX\)[/tex],
- [tex]\(RS\)[/tex] corresponds to [tex]\(RY\)[/tex],
- [tex]\(ST\)[/tex] corresponds to [tex]\(YX\)[/tex].
2. Form the ratios using corresponding sides:
- The ratio of sides [tex]\(RT/\)[/tex] and [tex]\(RX\)[/tex] pertains to corresponding sides in both triangles.
- Similarly, the ratio of sides [tex]\(RS/\)[/tex] and [tex]\(RY\)[/tex] also pertains to corresponding sides in both triangles.
3. Check each given option for correspondence and confirm proportional relationships:
- Check if [tex]\(\frac{XY}{TS}\)[/tex] is a valid ratio:
- [tex]\(XY\)[/tex] and [tex]\(TS\)[/tex] are not corresponding sides in [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle RYX\)[/tex], so this ratio is not valid.
- Check if [tex]\(\frac{SY}{RY}\)[/tex] is a valid ratio:
- [tex]\(SY\)[/tex] is not a corresponding side in either triangle, making this option invalid.
- Check if [tex]\(\frac{RX}{XT}\)[/tex] is a valid ratio:
- This ratio is not a valid analogy for corresponding sides in the similar triangles.
- Check if [tex]\(\frac{ST}{YX}\)[/tex] is a valid ratio:
- Here, [tex]\(ST\)[/tex] and [tex]\(YX\)[/tex] are corresponding sides of the similar triangles.
- Because [tex]\(ST\)[/tex] corresponds to [tex]\(YX\)[/tex], the ratio [tex]\(\frac{ST}{YX}\)[/tex] is a valid ratio that must be equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
4. Conclusion:
- The correct ratio that corresponds to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex] is [tex]\(\frac{ST}{YX}\)[/tex].
Thus, the answer is [tex]\(\frac{ST}{YX}\)[/tex], which is option 4.
