https://file.aiquickdraw.com/mathbot/file/173013880300943fi6mvb.jpg

To solve this problem, let's analyze the information given and apply some geometry principles.

### Problem Analysis
1. We are given that \(AB\) is a **perpendicular bisector** of \(CD\).
2. Being a perpendicular bisector means that:
   - \(AB\) intersects \(CD\) at a right angle (90°).
   - \(AB\) divides \(CD\) into two equal segments, meaning \(CA = AD\).

### Solution Steps
Since \(AB\) bisects \(CD\), we can set up the equation:
\[
CA = AD
\]
According to the diagram:
- \(CA = 2x + 3\)
- \(AD = 5x - 12\)

So, we set \(CA\) equal to \(AD\):
\[
2x + 3 = 5x - 12
\]

### Solving the Equation
1. Subtract \(2x\) from both sides:
   \[
   3 = 3x - 12
   \]

2. Add \(12\) to both sides:
   \[
   15 = 3x
   \]

3. Divide by \(3\) to solve for \(x\):
   \[
   x = 5
   \]

### Final Answer
The value of \(x\) is:
\[
x = 5
\]

Would you like further details on any of these steps, or do you have any questions?

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Here are five related questions to explore further:
1. What is the length of \(CA\) or \(AD\) when \(x = 5\)?
2. How does the property of a perpendicular bisector help solve similar geometry problems?
3. What is the significance of setting two expressions equal when they represent equal segments?
4. What other methods could be used to verify if \(AB\) is a perpendicular bisector?
5. How can we find the length of \(CD\) using the given values?

**Tip:** When dealing with bisectors in geometry, look for opportunities to set expressions equal, as bisectors imply equal segments.

In the figure, AB is a perpendicular bisector of CD. Find the value of x.

Learn how to find the value of x in a geometry problem involving a perpendicular bisector. This solution explains the concept of perpendicular bisectors and algebraic equation setup. Suitable for students in Grades 8-10.

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