The problem asks to find the values of $x$ and $EF$, given that line $EF$ bisects line $CD$, and the following algebraic expressions:

• $CG = 11x + 5$
• $GD = 13x - 15$
• $EF = 8x + 2$

Geometry Setup:

Since $EF$ bisects $CD$ at point $G$, this implies that $CG = GD$. Thus, we can set up the equation: $CG = GD$ $11x + 5 = 13x - 15$

Algebra Setup:

To solve for $x$, simplify the equation above: $11x + 5 = 13x - 15$ Subtract $11x$ from both sides: $5 = 2x - 15$ Add 15 to both sides: $20 = 2x$ Divide by 2: $x = 10$

Now that we know $x = 10$, substitute this into the expression for $EF$: $EF = 8x + 2 = 8(10) + 2 = 80 + 2 = 82$

Final Answers:

• $x = 10$
• $EF = 82$

Let me know if you need further explanation, or if you'd like to explore additional details on this problem!

Here are 5 related questions you might find helpful:

1. What is the relationship between bisectors and midpoint properties in geometry?
2. Can a line bisect a line segment in more than one point? Why or why not?
3. How can algebraic methods help in solving geometry problems involving line segments?
4. What is the difference between a bisector and a perpendicular bisector?
5. If the equation for $EF$ changed, how would it affect the overall solution?

Tip: When you see the term "bisects," always think about equal parts—this is key in setting up equations for geometric problems.