Point G is on line segment start overline, F, H, end overline*FH*. Given F, H, equals, 4, x, plus, 8, comma*FH*=4*x*+8, F, G, equals, x, comma*FG*=*x*, and G, H, equals, 5, x, comma*GH*=5*x*, determine the numerical length of start overline, G, H, end overline, .*GH*.

We are given the following information:

- The total length of segment \( FH = 4x + 8 \).
- The length of segment \( FG = x \).
- The length of segment \( GH = 5x \).

Since point \( G \) is on the line segment \( FH \), we know that:

\[
FG + GH = FH
\]

Substitute the given values into the equation:

\[
x + 5x = 4x + 8
\]

Simplify the left-hand side:

\[
6x = 4x + 8
\]

Now, subtract \( 4x \) from both sides:

\[
2x = 8
\]

Solve for \( x \):

\[
x = 4
\]

Now that we know \( x = 4 \), substitute this value back into the expression for \( GH \):

\[
GH = 5x = 5(4) = 20
\]

Thus, the numerical length of segment \( GH \) is \( \boxed{20} \).

Would you like further details or have any questions?

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Here are 5 related questions to expand on this:

1. How would you determine the length of \( FG \) using the same method?
2. What is the total length of \( FH \) given that \( x = 4 \)?
3. If another point \( J \) was added between \( G \) and \( H \), how would you calculate the lengths of the new segments?
4. Can the same method be applied if \( G \) is not between \( F \) and \( H \)?
5. What happens if the relationship between the segment lengths changes, such as making \( FG = 2x \) instead?

**Tip:** Always check if points lie on the same line or segment to avoid overcomplicating problems involving distances!

Point G is on line segment FH. Given FH = 4x + 8, FG = x, and GH = 5x, determine the numerical length of GH.

Learn how to calculate the length of segment GH when point G lies on line segment FH. The problem involves key algebraic and geometric concepts, such as the segment addition formula and solving linear equations. Suitable for students in Grades 8-10.

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