Question

△RST ~ △RYX by the SSS similarity theorem.

Triangle R S T is shown. Line segment Y X is drawn from side R T to side R S to form triangle R Y X.

Which ratio is also equal to StartFraction R T Over R X EndFraction and StartFraction R S Over R Y EndFraction?

StartFraction X Y Over T S EndFraction
StartFraction S Y Over R Y EndFraction
StartFraction R X Over X T EndFraction
StartFraction S T Over Y X EndFraction

Answer 1

Given that triangles \( RST \) and \( RYX \) are similar by the SSS similarity theorem, the corresponding sides of the triangles maintain a constant ratio.

From the similarity of the triangles, we have:

\[
\frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{YX}
\]

To find which ratio is equal to \(\frac{RT}{RX}\) and \(\frac{RS}{RY}\), we can analyze the options given:

1. \(\frac{XY}{TS}\)
2. \(\frac{SY}{RY}\)
3. \(\frac{RX}{XT}\)
4. \(\frac{ST}{YX}\)

The key correspondences are \(RT\) corresponds to \(RX\), \(RS\) corresponds to \(RY\), and \(ST\) corresponds to \(YX\).

Thus, the correct ratio that is equal to both \(\frac{RT}{RX}\) and \(\frac{RS}{RY}\) will be \(\frac{ST}{YX}\).

Hence, the correct answer is:

\(\frac{ST}{YX}\)