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.