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Since, it is given that [tex]\Delta RST \sim \Delta RYX[/tex] by SSS similarity theorem.

We have to identify the ratio which is equal to [tex]\frac{RT}{RX}[/tex] and [tex]\frac{RS}{RY}[/tex].

When the triangles are similar, then the ratio of the corresponding sides are always equal and the congruent angles are also equal.

Since,  [tex]\Delta RST \sim \Delta RYX[/tex]

So, [tex]\frac{RT}{RX} = \frac{RS}{RY} = \frac{ST}{YX}[/tex]

So, the given ratios are equal to the ratio [tex]\frac{ST}{YX}[/tex].

So, Option D is the correct answer.

The triangle STX is similar to triangle YXR by the SSS similarity theorem. Option (A) is correct.

Further Explanation:

The similar triangles are those in which all the corresponding angles are equal and the sides are proportional.

There are many similarity rules and are as follows.

1. Angle Angle Angle (AAA)

2. Side Side Side (SSS)

3. Side Angle Side (SAS)

Given:

The options are as follows.

(A). [tex]\frac{{{\text{XY}}}}{{{\text{TS}}}}[/tex]

(B). [tex]\frac{{{\text{SY}}}}{{{\text{RY}}}}[/tex]

(C). [tex]\frac{{{\text{RX}}}}{{{\text{XT}}}}[/tex]

(D). [tex]\frac{{{\text{ST}}}}{{{\text{YX}}}}[/tex]

Explanation:

The STX is similar to triangle YXR then the sides are proportional.

The proportional of corresponding side can be obtained as follows,

[tex]\frac{{{\text{RT}}}}{{{\text{RX}}}}=\frac{{{\text{RS}}}}{{{\text{RY}}}}=\frac{{{\text{XY}}}}{{{\text{TS}}}}[/tex].

The corresponding sides are proportional. Therefore the triangles are similar.

The triangle STX is similar to triangle YXR by the SSS similarity theorem. Option (A) is correct.

Option (A) is correct as the ratio [tex]\frac{{{\text{XY}}}}{{{\text{TS}}}}[/tex] is equal to [tex]\frac{{{\text{RT}}}}{{{\text{RX}}}}[/tex] and [tex]\frac{{{\text{RS}}}}{{{\text{RY}}}}[/tex].

Option (b) is not correct as the ratio [tex]\frac{{{\text{SY}}}}{{{\text{RY}}}}[/tex] is not equal to [tex]\frac{{{\text{RT}}}}{{{\text{RX}}}}[/tex] and [tex]\frac{{{\text{RS}}}}{{{\text{RY}}}}[/tex].

Option (c) is not correct as the ratio [tex]\frac{{{\text{RX}}}}{{{\text{XT}}}}[/tex] is not equal to [tex]\frac{{{\text{RT}}}}{{{\text{RX}}}}[/tex] and [tex]\frac{{{\text{RS}}}}{{{\text{RY}}}}[/tex].

Option (d) is not correct as the ratio [tex]\frac{{{\text{ST}}}}{{{\text{YX}}}}[/tex] is not equal to [tex]\frac{{{\text{RT}}}}{{{\text{RX}}}}[/tex] and [tex]\frac{{{\text{RS}}}}{{{\text{RY}}}}[/tex].

Learn more:

1. Learn more about inverse of the function https://brainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function https://brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Triangle

Keywords: congruent, angles, triangle, ASA, angle side angle, congruent sides, acute angle, side, corresponding angles, congruent triangle, similarity theorem.

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