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The problem asks us to determine the numerical length of the segment \( RT \). Let's analyze the given information:

1. \( ST = 3x - 8 \)
2. \( RT = 4x \)
3. \( RS = 4x - 7 \)

### Approach:

- Point \( S \) is on the line segment \( RT \), so we can use the fact that:
  \[
  RT = ST + RS
  \]
  Substituting the given expressions:
  \[
  4x = (3x - 8) + (4x - 7)
  \]
  Simplifying the right-hand side:
  \[
  4x = 3x - 8 + 4x - 7
  \]
  Combine like terms:
  \[
  4x = 7x - 15
  \]
  Now, solve for \( x \):
  \[
  4x - 7x = -15
  \]
  \[
  -3x = -15
  \]
  \[
  x = 5
  \]

### Step 2: Find the length of \( RT \)
Now that we have \( x = 5 \), we can substitute it into the expression for \( RT \):
\[
RT = 4x = 4(5) = 20
\]

Thus, the numerical length of \( RT \) is \( \boxed{20} \).

Would you like more details or have any questions?

Here are five related questions:
1. How would you solve for \( RS \) if \( x = 5 \)?
2. What is the length of \( ST \) when \( x = 5 \)?
3. Can you explain how to check if \( S \) lies on \( RT \)?
4. What is the difference between line segments and rays in geometry?
5. If \( x \) were different, how would the length of \( RT \) change?

**Tip**: Always double-check your calculations by plugging the values back into the original equations to ensure correctness.

Point S is on line segment RT. Given ST = 3x - 8, RT = 4x, and RS = 4x - 7, determine the numerical length of RT.

Learn how to find the length of segment RT when given the expressions for ST, RT, and RS. This step-by-step solution involves algebraic manipulation and application of the Segment Addition Postulate. Ideal for students in grades 8-10.

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