Let's break down the problem step by step:

We are given:

1. $\overline{HG} = 13x - 993$
2. $\overline{HF} = 3x - 129$
3. $\overline{GE} = 3x - 185$
4. $\overline{FE} = 4$

We are asked to find the length of $\overline{GE}$.

Step 1: Understanding the relationships

From the diagram and given information:

• $\overline{HE} = \overline{HG} + \overline{GE}$
• $\overline{HE}$ can also be written as $\overline{HF} + \overline{FE}$

So, $\overline{HE} = \overline{HG} + \overline{GE}$ and $\overline{HE} = \overline{HF} + \overline{FE}$.

Step 2: Set up the equation

From the two ways to express $\overline{HE}$:

$\overline{HG} + \overline{GE} = \overline{HF} + \overline{FE}$

Substitute the given values:

$(13x - 993) + (3x - 185) = (3x - 129) + 4$

Step 3: Simplify the equation

Combine like terms:

$(13x + 3x) - (993 + 185) = (3x - 129) + 4$ $16x - 1178 = 3x - 125$

Step 4: Solve for $x$

Move the $3x$ to the left side and constants to the right:

$16x - 3x = -125 + 1178$ $13x = 1053$ $x = \frac{1053}{13}$ $x = 81$

Step 5: Find the length of $\overline{GE}$

Now that we know $x = 81$, substitute it into the expression for $\overline{GE}$:

$\overline{GE} = 3x - 185 = 3(81) - 185 = 243 - 185 = 58$

Thus, the length of $\overline{GE}$ is 58.

Would you like more details on any part of this? Here are some related questions to extend your understanding:

1. What would be the length of $\overline{HG}$ if $x = 81$?
2. How would the equation change if $\overline{FE}$ were a different value?
3. Can you find the total length of $\overline{HE}$ using both expressions?
4. What does the equality of the two expressions for $\overline{HE}$ represent geometrically?
5. If $x$ were doubled, what would happen to the length of $\overline{GE}$?

Tip: When dealing with geometry problems involving segments, always look for multiple ways to write an equation for the same line or segment—it often reveals useful relationships.