Answer :
Given that [tex]\(\triangle RST \sim \triangle RYX\)[/tex] by the SSS (Side-Side-Side) similarity theorem, the corresponding sides of similar triangles are proportional. Let's identify the corresponding sides in these similar triangles:
- [tex]\(RT\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to [tex]\(RX\)[/tex] in [tex]\(\triangle RYX\)[/tex]
- [tex]\(RS\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to [tex]\(RY\)[/tex] in [tex]\(\triangle RYX\)[/tex]
- [tex]\(ST\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to [tex]\(YX\)[/tex] in [tex]\(\triangle RYX\)[/tex]
Given this proportionality, we can write the following ratios between corresponding sides:
- [tex]\(\frac{RT}{RX} = \frac{RS}{RY}\)[/tex]
- [tex]\(\frac{RS}{RY} = \frac{ST}{YX}\)[/tex]
Thus, all three ratios [tex]\(\frac{RT}{RX}\)[/tex], [tex]\(\frac{RS}{RY}\)[/tex], and [tex]\(\frac{ST}{YX}\)[/tex] are equal. The question asks which ratio among the given options is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
From our analysis, we see that the ratio [tex]\(\frac{ST}{YX}\)[/tex] is equal to these ratios because:
[tex]\[ \frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{YX} \][/tex]
Therefore, the ratio [tex]\(\frac{ST}{YX}\)[/tex] is the one that is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
Hence, the correct answer is:
[tex]\(\frac{ST}{YX}\)[/tex]
- [tex]\(RT\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to [tex]\(RX\)[/tex] in [tex]\(\triangle RYX\)[/tex]
- [tex]\(RS\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to [tex]\(RY\)[/tex] in [tex]\(\triangle RYX\)[/tex]
- [tex]\(ST\)[/tex] in [tex]\(\triangle RST\)[/tex] corresponds to [tex]\(YX\)[/tex] in [tex]\(\triangle RYX\)[/tex]
Given this proportionality, we can write the following ratios between corresponding sides:
- [tex]\(\frac{RT}{RX} = \frac{RS}{RY}\)[/tex]
- [tex]\(\frac{RS}{RY} = \frac{ST}{YX}\)[/tex]
Thus, all three ratios [tex]\(\frac{RT}{RX}\)[/tex], [tex]\(\frac{RS}{RY}\)[/tex], and [tex]\(\frac{ST}{YX}\)[/tex] are equal. The question asks which ratio among the given options is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
From our analysis, we see that the ratio [tex]\(\frac{ST}{YX}\)[/tex] is equal to these ratios because:
[tex]\[ \frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{YX} \][/tex]
Therefore, the ratio [tex]\(\frac{ST}{YX}\)[/tex] is the one that is also equal to [tex]\(\frac{RT}{RX}\)[/tex] and [tex]\(\frac{RS}{RY}\)[/tex].
Hence, the correct answer is:
[tex]\(\frac{ST}{YX}\)[/tex]
