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The problem asks to find \( FG \), given that \( G \) is the midpoint of \( FH \).

The diagram provides the following information:

- \( FG = 11x - 7 \)
- \( GH = 3x - 9 \)

Since \( G \) is the midpoint of \( FH \), the distances \( FG \) and \( GH \) must be equal.

Thus, we can set up the equation:
\[
11x - 7 = 3x - 9
\]
Let's solve for \( x \):

\[
11x - 7 = 3x - 9
\]

1. Subtract \( 3x \) from both sides:
\[
11x - 3x - 7 = -9
\]
\[
8x - 7 = -9
\]

2. Add 7 to both sides:
\[
8x = -9 + 7
\]
\[
8x = -2
\]

3. Divide both sides by 8:
\[
x = \frac{-2}{8}
\]
\[
x = -\frac{1}{4}
\]

Now that we have \( x \), we can find \( FG \) by substituting \( x = -\frac{1}{4} \) into the expression for \( FG \):

\[
FG = 11x - 7 = 11\left(-\frac{1}{4}\right) - 7
\]
\[
FG = -\frac{11}{4} - 7 = -\frac{11}{4} - \frac{28}{4} = -\frac{39}{4}
\]
\[
FG = -9.75
\]

Thus, the length of \( FG \) is \( -9.75 \).

Would you like more details or have any questions?

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Here are five related questions:
1. How would you find the length of \( GH \) given the value of \( x \)?
2. What does it mean for \( G \) to be the midpoint of \( FH \)?
3. Can the distance between two points ever be negative, and how does this affect interpretation?
4. How can you verify your solution by checking the length of the entire segment \( FH \)?
5. If the expressions for \( FG \) and \( GH \) were different, how would you solve for \( x \)?

**Tip:** Always check the symmetry in problems involving midpoints, as the two segments must be equal.

Learn how to solve for FG in a problem where G is the midpoint of FH. This problem uses the midpoint theorem and linear equations, providing step-by-step algebraic solutions. Suitable for students in Grades 8-10.

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