We are given three segments on the line: GI, HI, and GH. Let's express each segment in terms of $x$ based on the information provided.

• $GI = 5x$
• $HI = 3x - 8$
• $GH = 3x$

Since point $H$ lies on line segment $GI$, we know that:

$GH + HI = GI$

Substitute the expressions for $GH$, $HI$, and $GI$:

$3x + (3x - 8) = 5x$

Simplify the left-hand side:

$6x - 8 = 5x$

Now, solve for $x$:

$6x - 5x = 8$ $x = 8$

Now that we know $x = 8$, substitute this value back into the expression for $GH$:

$GH = 3x = 3(8) = 24$

Thus, the numerical length of $GH$ is 24 units.

Would you like further explanation or details?

Here are 5 related questions for practice:

1. What is the numerical length of $GI$?
2. What is the numerical length of $HI$?
3. If $H$ divides $GI$ in the ratio of $GH:HI$, what is that ratio?
4. If $GH = 24$ and $HI = 16$, what is the total length of $GI$?
5. Can you find $x$ if $GI = 10x$, $GH = 6x$, and $HI = 4x - 5$?

Tip: Always check if a point lies on a line segment by verifying that the sum of the smaller segments equals the total length.