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Answer:
To solve for \( RP \), we need to use the properties of similar triangles. Given that \( \triangle RQS \sim \triangle RPT \), corresponding sides of similar triangles are proportional.
We are given the following information:
- \( RQ = 8 \)
- \( RS = 6 \)
- \( RT = 18 \)
Since \( \triangle RQS \sim \triangle RPT \), the ratio of the corresponding sides is the same. We need to find \( RP \).
We know:
\[ \frac{RQ}{RP} = \frac{RS}{RT} \]
Given:
- \( RQ = 8 \)
- \( RS = 6 \)
- \( RT = 18 \)
We need to find \( RP \). Let's call \( RP \) as \( x \).
Thus, we set up the proportion:
\[ \frac{RQ}{RP} = \frac{RS}{RT} \]
\[ \frac{8}{x} = \frac{6}{18} \]
Simplify the right side of the equation:
\[ \frac{6}{18} = \frac{1}{3} \]
So the proportion becomes:
\[ \frac{8}{x} = \frac{1}{3} \]
Now, solve for \( x \) by cross-multiplying:
\[ 8 \times 3 = x \times 1 \]
\[ 24 = x \]
Therefore, \( RP = 24 \).