Given that $G$ is the midpoint of $FH$, it means that the distance from $F$ to $G$ is equal to the distance from $G$ to $H$. Therefore, $FH$ is twice the length of $GH$.

We are given: $GH = x + 9$ $FH = 11x + 9$

Since $G$ is the midpoint: $FH = 2 \times GH$

Substituting the given expressions: $11x + 9 = 2 \times (x + 9)$

Now, solve for $x$: $11x + 9 = 2x + 18$

Subtract $2x$ from both sides: $9x + 9 = 18$

Subtract 9 from both sides: $9x = 9$

Divide both sides by 9: $x = 1$

Now, substitute $x = 1$ back into the expression for $GH$: $GH = 1 + 9 = 10$

Thus, the length of $GH$ is $\boxed{10}$.

Would you like more details or have any questions?

Here are 8 related questions for further practice:

1. If $G$ is the midpoint and $FH = 30$, what is the value of $GH$?
2. If $G$ is the midpoint and $GH = 3x + 7$ and $FH = 6x + 14$, what is $GH$?
3. If $G$ is the midpoint and $GH = 2x - 5$ and $FH = 4x - 10$, find the value of $x$.
4. What would $GH$ be if $G$ is the midpoint, $GH = 5x$, and $FH = 10x + 4$?
5. How would the problem change if $GH = x + 7$ and $FH = 3x + 11$?
6. If $FH = 20$ and $G$ is the midpoint, how does the length of $GH$ compare to the length of $GF$?
7. Solve for $x$ if $GH = 2x + 4$ and $FH = 8x + 16$.
8. What happens if $G$ is not the midpoint of $FH$? How does this affect the equation?

Tip: When dealing with midpoint problems, remember that the midpoint divides a line segment into two equal parts. Use this property to set up your equations correctly.