Let's go through the problem step-by-step to find the value of \( x \) and the length of segment \( BD \).
### Step-by-Step Solution:
1. Understanding the Problem:
- We are given that \( AD \) is the perpendicular bisector of segment \( BC \). A key property of a perpendicular bisector is that it divides the segment it bisects (in this case, \( BC \)) into two equal parts. Thus, \( BD \) equals \( DC \).
2. Setting Up the Equation:
- We are given that:
[tex]\[
BD = 12x - 2
\][/tex]
[tex]\[
DC = 4x + 6
\][/tex]
- Since \( BD \) and \( DC \) are equal, we can write the equation:
[tex]\[
12x - 2 = 4x + 6
\][/tex]
3. Solving for \( x \):
- To solve this equation, we first isolate the terms involving \( x \) on one side of the equation and the constant terms on the other side:
[tex]\[
12x - 4x = 6 + 2
\][/tex]
- Simplifying both sides:
[tex]\[
8x = 8
\][/tex]
- Dividing both sides by 8 to solve for \( x \):
[tex]\[
x = 1
\][/tex]
4. Finding the Length of Segment \( BD \):
- Now that we have found \( x = 1 \), we can substitute this value back into the expression for \( BD \):
[tex]\[
BD = 12x - 2
\][/tex]
- Substituting \( x = 1 \):
[tex]\[
BD = 12(1) - 2 = 12 - 2 = 10
\][/tex]
### Conclusion:
- The value of \( x \) is:
[tex]\[
x = 1
\][/tex]
- The length of segment \( BD \) is:
[tex]\[
BD = 10
\][/tex]
Thus, we have successfully determined that [tex]\( x = 1 \)[/tex] and the length of segment [tex]\( BD \)[/tex] is 10.
