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.