If FG = 4x, GH = 3, and FH = x + 9, what is FG?


Given the triangle \( FGH \), we have the following information:

- \( FG = 4x \)
- \( GH = 3 \)
- \( FH = x + 9 \)

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. However, in this case, since we are dealing with the lengths, it seems like we need to apply the segment addition postulate instead. This postulate states that the sum of the lengths of two segments that form a straight line is equal to the length of the entire segment.

So, if point G lies between points F and H, then:

\[
FG + GH = FH
\]

Substituting the given values:

\[
4x + 3 = x + 9
\]

Now, solve for \( x \):

\[
4x - x = 9 - 3
\]

\[
3x = 6
\]

\[
x = 2
\]

Now that we have \( x \), we can find the length of \( FG \):

\[
FG = 4x = 4(2) = 8
\]

Thus, \( FG = 8 \).

Would you like any further details or have any other questions?

Here are some related questions to consider:
1. How can the segment addition postulate be used in other problems?
2. What happens if \( G \) is not between \( F \) and \( H \)?
3. Can you apply the triangle inequality theorem in this problem?
4. How do you solve linear equations with variables on both sides?
5. What is the significance of solving for \( x \) in geometry problems?
6. How would the solution change if \( GH \) were a variable instead of a constant?
7. What if the expression for \( FH \) was different, such as \( 2x + 5 \)?
8. How can you check your solution to ensure it's correct?

**Tip:** When solving for variables in geometry problems, always double-check if the context is dealing with straight lines or triangles to apply the correct postulate or theorem.

Explore how to apply the Segment Addition Postulate in geometry to find the length of FG in triangle FGH. Step-by-step solution provided for FG = 4x, GH = 3, and FH = x + 9. Suitable for Grades 7-9.

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