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"20. Segment AB bisects CD at E. If AE=2x+7, BE=x+6, CE=3x-1, and DE=2x-3, find the length of AB."

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### Problem Breakdown:

Given:
- \( AE = 2x + 7 \)
- \( BE = x + 6 \)
- \( CE = 3x - 1 \)
- \( DE = 2x - 3 \)

Since AB bisects CD at E, this implies that \( CE = DE \).

### Step-by-Step Solution:

1. **Set the lengths CE and DE equal:**
   \[
   3x - 1 = 2x - 3
   \]

2. **Solve for \( x \):**
   Subtract \( 2x \) from both sides:
   \[
   x - 1 = -3
   \]
   Add 1 to both sides:
   \[
   x = -2
   \]

3. **Substitute \( x = -2 \) into the expressions for AE and BE to find the length of AB:**

   \[
   AE = 2(-2) + 7 = -4 + 7 = 3
   \]
   \[
   BE = -2 + 6 = 4
   \]

4. **Calculate the total length of AB:**
   \[
   AB = AE + BE = 3 + 4 = 7
   \]

### Final Answer:
The length of AB is **7 units**.

Do you need any more details or have any questions?

Here are some related questions:

1. How would the problem change if AB did not bisect CD?
2. What happens to the length of AB if AE and BE are equal?
3. What if CD was not bisected at E, but at a different point F? How would you find AB then?
4. How do you determine the length of CD?
5. Can you generalize this problem to three-dimensional geometry?

**Tip:** Always double-check your algebraic manipulations to ensure that you’ve isolated the variable correctly.

Segment AB bisects CD at E. If AE = 2x + 7, BE = x + 6, CE = 3x - 1, and DE = 2x - 3, find the length of AB.

Solve a geometry problem involving segment bisectors and algebraic expressions to find the length of AB. This solution includes key concepts in algebra and geometry suitable for students in Grades 6-8.

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