https://file.aiquickdraw.com/mathbot/file/1726546054354p0bidq11.jpg

The problem from the image states:

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

### Solution:

Since \(EF\) bisects \(CD\) at point \(G\), we know that:

\[
CG = GD
\]

Given:

- \(CG = 5x - 1\)
- \(GD = 7x - 13\)

Since \(CG = GD\):

\[
5x - 1 = 7x - 13
\]

Now, solve for \(x\):

1. Subtract \(5x\) from both sides:

\[
-1 = 2x - 13
\]

2. Add 13 to both sides:

\[
12 = 2x
\]

3. Divide by 2:

\[
x = 6
\]

Now that we have \(x = 6\), substitute this value back into the expressions for \(CG\) and \(GD\):

- \(CG = 5(6) - 1 = 30 - 1 = 29\)
- \(GD = 7(6) - 13 = 42 - 13 = 29\)

Thus, \(CG = GD = 29\), confirming the bisection.

### Find \(EG\):

Given that \(EF = 6x - 4\) and \(GF = 13\), substitute \(x = 6\) into the expression for \(EF\):

\[
EF = 6(6) - 4 = 36 - 4 = 32
\]

Since \(EG = EF - GF\), we have:

\[
EG = 32 - 13 = 19
\]

### Final Answer:
\[
EG = 19
\]

Would you like more details or have any questions?

Here are some related questions to further your understanding:
1. How would the solution change if \(EF\) didn’t bisect \(CD\)?
2. Can you find the length of \(CF\) using the information from the problem?
3. What happens to \(EG\) if the value of \(x\) increases or decreases?
4. How is the concept of line bisection important in solving geometry problems?
5. If the coordinates of these points were given, could you use a coordinate geometry approach to solve the problem?

**Tip:** In geometry, identifying equal segments or angles can simplify complex problems by reducing them to solvable algebraic equations.

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

Learn how to solve a problem where EF bisects CD at G, and CG = 5x - 1, GD = 7x - 13, EF = 6x - 4, GF = 13. Using algebra and geometry, solve for the length of EG. This detailed solution is suitable for students in Grades 8-10.

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