https://file.aiquickdraw.com/mathbot/file/17267912203118xa3je3z.jpg

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.

If HG = 13x - 993, HF = 3x - 129, GE = 3x - 185, and FE = 4, find the length of GE.

Learn how to find the length of segment GE using algebraic expressions for HG, HF, and FE. This step-by-step solution covers key concepts in algebra and geometry, suitable for students in Grades 8-10.

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