https://file.aiquickdraw.com/mathbot/file/1726794167548xcko74p3.jpg

We are given the following information from the image:

- \( FG = 7x + 11 \)
- \( GH = 9x \)
- \( FH = 12x + 19 \)

We know from the diagram that the entire length of \( FH \) is the sum of \( FG \) and \( GH \). Therefore, we can write the equation:

\[
FG + GH = FH
\]

Substituting the given expressions for \( FG \), \( GH \), and \( FH \):

\[
(7x + 11) + (9x) = 12x + 19
\]

Now let's solve for \( x \):

\[
7x + 11 + 9x = 12x + 19
\]
\[
16x + 11 = 12x + 19
\]

Subtract \( 12x \) from both sides:

\[
4x + 11 = 19
\]

Subtract 11 from both sides:

\[
4x = 8
\]

Now divide by 4:

\[
x = 2
\]

Now that we know \( x = 2 \), we can substitute this back into the expression for \( FG \):

\[
FG = 7x + 11 = 7(2) + 11 = 14 + 11 = 25
\]

Thus, the length of \( FG \) is \( \boxed{25} \).

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. What is the length of \( GH \) when \( x = 2 \)?
2. What is the length of \( FH \) when \( x = 2 \)?
3. How do we verify that the sum of \( FG \) and \( GH \) equals \( FH \)?
4. What happens if we change the expression for \( FH \)? How would it affect the solution?
5. What if a different part of the segment was unknown? How could we solve for it?

**Tip:** Always check that the sum of the parts equals the whole in these types of geometry problems to avoid mistakes.

If FG = 7x + 11, GH = 9x, and FH = 12x + 19, what is FG?

In this problem, learn how to solve for FG using the segment addition postulate and linear equations. Given that FG = 7x + 11, GH = 9x, and FH = 12x + 19, we derive FG step-by-step by solving a linear equation. This problem is ideal for students in Grades 7-10.

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