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If cd
 is the perpendicular bisector of 
ab, what is the value of x?

In this problem, we know that line segment \( CD \) is the perpendicular bisector of \( AB \). This implies two main things:

1. \( AB \) is bisected, so \( AC = AD \).
2. \( CD \) is perpendicular to \( AB \), which doesn't necessarily affect the length equality but confirms the orientation.

### Step 1: Set up the equation based on the bisector property
Since \( CD \) bisects \( AB \) at the midpoint, we have:
\[
AC = AD
\]
Given:
- \( AC = 5 - 2x \)
- \( AD = 4x + 3 \)

So, we can set up the equation:
\[
5 - 2x = 4x + 3
\]

### Step 2: Solve for \( x \)
1. Rearrange the equation to bring terms involving \( x \) to one side:
   \[
   5 - 3 = 4x + 2x
   \]
   Simplifying this:
   \[
   2 = 6x
   \]

2. Divide both sides by 6:
   \[
   x = \frac{1}{3}
   \]

### Conclusion
The value of \( x \) is \( \frac{1}{3} \).

Would you like a step-by-step breakdown of the solution or have further questions on this concept?

Here are some additional questions to explore:

1. If \( CD \) bisects \( AB \), what can we say about the relationship between segments \( AC \) and \( AD \)?
2. How do we verify if \( CD \) is the perpendicular bisector based on segment lengths?
3. What would change in our solution if \( CD \) were not perpendicular to \( AB \)?
4. Can we determine the lengths of \( AC \) and \( AD \) with the found value of \( x \)?
5. How does changing the orientation of \( CD \) affect the calculations?

**Tip:** When given that a line is a perpendicular bisector, focus on setting up equal segments on either side of the midpoint for an efficient solution.

If CD is the perpendicular bisector of AB, what is the value of x?

Learn how to solve for x when CD is the perpendicular bisector of AB. This problem covers algebraic concepts and properties of perpendicular bisectors, suitable for students in Grades 8-10.

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