https://file.aiquickdraw.com/mathbot/file/1726450828526f0rqtg2g.jpg

The problem in the image states:

- \( \overline{EF} \) bisects \( \overline{CD} \).
- \( CG = 5x - 1 \)
- \( GD = 7x - 13 \)
- \( EF = 6x - 4 \)
- \( GF = 13 \)

The task is to find \( EG \).

### Step-by-step solution:

1. Since \( \overline{EF} \) bisects \( \overline{CD} \), we know that:
   \[
   CG = GD
   \]
   So, we can set the two expressions equal to each other:
   \[
   5x - 1 = 7x - 13
   \]
   
2. Solve for \( x \):
   \[
   5x - 1 = 7x - 13
   \]
   Subtract \( 5x \) from both sides:
   \[
   -1 = 2x - 13
   \]
   Add 13 to both sides:
   \[
   12 = 2x
   \]
   Divide both sides by 2:
   \[
   x = 6
   \]

3. Now substitute \( x = 6 \) back into the expressions for \( CG \), \( GD \), and \( EF \).

   - \( CG = 5(6) - 1 = 30 - 1 = 29 \)
   - \( GD = 7(6) - 13 = 42 - 13 = 29 \) (which confirms that \( \overline{EF} \) bisects \( \overline{CD} \)).
   - \( EF = 6(6) - 4 = 36 - 4 = 32 \)

4. Since \( GF = 13 \), and \( EF = 32 \), we can find \( EG \) using the relation \( EF = EG + GF \):
   \[
   EG + 13 = 32
   \]
   Subtract 13 from both sides:
   \[
   EG = 19
   \]

Thus, \( EG = 19 \).

Would you like any further details or clarifications?

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Here are five related questions for practice:
1. How would the solution change if \( EF \) did not bisect \( CD \)?
2. What is the significance of the midpoint when a line bisects another line?
3. If \( EF \) were longer, how would that affect the calculation for \( EG \)?
4. Can you verify that \( CG \) and \( GD \) are indeed equal by substituting the found value of \( x \)?
5. What would happen if \( GF \) were given as 15 instead of 13?

**Tip:** In geometry, when a line bisects another, both segments are equal, which simplifies many problems involving midpoints.

If EF bisects CD, CG = 5x - 1, GD = 7x - 13, EF = 6x - 4, and GF = 13, find EG.

This geometry problem involves finding EG when EF bisects CD. The equations provided are CG = 5x - 1, GD = 7x - 13, EF = 6x - 4, and GF = 13. Solve for x and use the bisector theorem to calculate EG with step-by-step algebraic reasoning. Suitable for Grades 8-10.

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