GisthemidpointofFH. If GH = x + 9 and FH = 11x + 9, what is GH?

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.

Explore how to apply the midpoint theorem to solve for GH given the equations GH = x + 9 and FH = 11x + 9. Learn step-by-step solutions and related practice problems. Suitable for high school students and beyond.

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