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?

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.