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

Given the problem, we need to determine the length of \( QS \).

**Given:**
- \( R \) is the midpoint of \( QS \), so \( QR = RS \).
- \( RS = 2x - 4 \)
- \( ST = 4x - 1 \)
- \( RT = 8x - 43 \)

Since \( R \) is the midpoint of \( QS \), the entire length of \( QS \) is twice the length of \( RS \). Therefore, \( QS = 2 \times RS \).

To find \( QS \), we will first use the given relationship \( RT = RS + ST \):

\[
RT = RS + ST
\]

Substitute the given expressions:

\[
8x - 43 = (2x - 4) + (4x - 1)
\]

Now, combine like terms on the right side:

\[
8x - 43 = 6x - 5
\]

Next, solve for \( x \):

\[
8x - 6x = -5 + 43
\]

\[
2x = 38
\]

\[
x = 19
\]

Now that we have \( x \), substitute it back into the expression for \( RS \) to find \( QS \):

\[
RS = 2x - 4 = 2(19) - 4 = 38 - 4 = 34
\]

Since \( QS = 2 \times RS \):

\[
QS = 2 \times 34 = 68
\]

Thus, the length of \( QS \) is \( \boxed{68} \).

Would you like more details or have any questions? Here are some related problems to consider:

1. If \( QS = 68 \) and \( RT = 8x - 43 \), what is the length of \( RT \)?
2. If \( QS = 68 \), what would be the value of \( x \) if \( ST = 5x + 2 \)?
3. What is the length of \( QR \) if \( QS = 68 \) and \( R \) is the midpoint?
4. How would the length of \( QS \) change if \( R \) were not the midpoint but a different point along \( QS \)?
5. If the value of \( ST \) were \( 7x - 3 \) instead of \( 4x - 1 \), how would that change \( QS \)?

**Tip:** When solving problems with midpoints, remember that the segments on either side of the midpoint are equal in length. This can simplify calculations significantly.

Learn how to find the length of segment QS when R is the midpoint and RS, ST, and RT are given lengths. This problem involves applying the Midpoint Theorem in geometry and solving for the length of QS using algebraic equations. Suitable for Grades 9-12.

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