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Let's break down the given problem step-by-step:

### Given Information:
- \( R \) is the midpoint of \( QS \).
- \( RS = 2x - 4 \)
- \( ST = 4x - 1 \)
- \( RT = 8x - 43 \)

### Objective:
Find the length of \( QS \).

### Step-by-Step Solution:
1. **Since \( R \) is the midpoint of \( QS \):**
   - \( R \) divides \( QS \) into two equal parts. Therefore, \( QR = RS \).

2. **Express \( QR \) in terms of \( RS \):**
   - Since \( R \) is the midpoint of \( QS \) and \( RS = 2x - 4 \), we have:
     \[
     QR = RS = 2x - 4
     \]

3. **Find \( QS \):**
   - Since \( QS = QR + RS \) and \( QR = RS \):
     \[
     QS = 2 \times RS = 2 \times (2x - 4) = 4x - 8
     \]

4. **Use \( RT = RS + ST \) to find \( x \):**
   - We know:
     \[
     RT = RS + ST
     \]
   - Substitute the given values:
     \[
     8x - 43 = (2x - 4) + (4x - 1)
     \]
   - Simplify the right-hand side:
     \[
     8x - 43 = 6x - 5
     \]
   - Solve for \( x \):
     \[
     8x - 6x = -5 + 43
     \]
     \[
     2x = 38
     \]
     \[
     x = 19
     \]

5. **Substitute \( x \) back to find \( QS \):**
   - Now that \( x = 19 \), substitute into the expression for \( QS \):
     \[
     QS = 4x - 8 = 4(19) - 8 = 76 - 8 = 68
     \]

### Final Answer:
The length of \( QS \) is \( 68 \).

Would you like further details or have any questions?

### Related Questions:
1. How do you find the midpoint of a line segment?
2. What is the formula for finding the midpoint in coordinate geometry?
3. How do you solve equations involving midpoints in a geometric line?
4. What other properties of midpoints can be used in coordinate geometry?
5. Can this problem be solved using coordinate geometry?

### Tip:
Always double-check the setup of your equations when solving problems involving midpoints and line segments.

If R is the midpoint of QS, RS = 2x - 4, ST = 4x - 1, and RT = 8x - 43, find QS.

Learn how to find the length of QS when R is the midpoint of QS, given RS = 2x - 4, ST = 4x - 1, and RT = 8x - 43. This problem involves applying the midpoint theorem and solving linear equations. Suitable for students in Grades 9-10.

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