https://file.aiquickdraw.com/mathbot/file/1726098698425wylp88j3.jpg

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.

If EF bisects CD, CG = 11x + 5, GD = 13x - 15, and EF = 8x + 2, find x and EF.

Learn how to solve a geometry problem where EF bisects CD. This problem involves finding x and EF using algebraic methods and geometric principles. Suitable for students in Grades 8-10.

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