Point H is on line segment start overline, G, I, end overline*GI*. Given G, I, equals, 5, x, comma*GI*=5*x*, H, I, equals, 3, x, minus, 8, comma*HI*=3*x*−8, and G, H, equals, 3, x, comma*GH*=3*x*, determine the numerical length of start overline, G, H, end overline, .*GH*.

We are given the following information:

- \( GI = 5x \)
- \( HI = 3x - 8 \)
- \( GH = 3x \)

Since point \( H \) is on the line segment \( \overline{GI} \), the total length of \( GI \) is the sum of the lengths of \( GH \) and \( HI \), i.e.,

\[
GH + HI = GI
\]

Substitute the values for \( GH \), \( HI \), and \( GI \) into this equation:

\[
3x + (3x - 8) = 5x
\]

Now, simplify the left-hand side:

\[
3x + 3x - 8 = 5x
\]
\[
6x - 8 = 5x
\]

Subtract \( 5x \) from both sides:

\[
6x - 5x = 8
\]
\[
x = 8
\]

Now that we know \( x = 8 \), we can find the length of \( GH \):

\[
GH = 3x = 3(8) = 24
\]

Thus, the numerical length of \( \overline{GH} \) is 24 units.

Would you like further clarification or details?

Here are 5 related questions to explore:
1. What is the length of \( HI \) when \( x = 8 \)?
2. How do we verify that point \( H \) lies on \( \overline{GI} \)?
3. If \( GH = 24 \), what is the total length of \( GI \)?
4. How does the relationship between segment lengths change if \( H \) is the midpoint of \( \overline{GI} \)?
5. Can the equation setup change if \( H \) were outside \( \overline{GI} \)?

**Tip**: When dealing with line segments, always check if the sum of the smaller segments equals the total length.

Point H is on line segment GI. Given GI = 5x, HI = 3x - 8, and GH = 3x, determine the numerical length of GH.

Learn how to calculate the length of line segment GH, given GI = 5x, HI = 3x - 8, and GH = 3x. This step-by-step solution involves solving linear equations and applying the Segment Addition Postulate. Ideal for students in Grades 8-10.

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