To determine the equation of the line perpendicular to [tex]\(\overleftrightarrow{DB}\)[/tex], we proceed step by step as follows:
1. Convert the equation [tex]\( \frac{1}{2} x + 2 y = 12 \)[/tex] into slope-intercept form [tex]\( y = mx + c \)[/tex].
[tex]\[\frac{1}{2} x + 2 y = 12\][/tex]
Isolate [tex]\( y \)[/tex] on one side:
[tex]\[
2 y = -\frac{1}{2} x + 12
\][/tex]
[tex]\[
y = - \frac{1}{4} x + 6
\][/tex]
Thus, the equation is in the form [tex]\( y = mx + c \)[/tex] where the slope [tex]\( m \)[/tex] is [tex]\( -\frac{1}{4} \)[/tex].
2. Find the slope of the perpendicular line.
For two lines to be perpendicular, the product of their slopes must be [tex]\(-1\)[/tex]. If the slope of [tex]\(\overleftrightarrow{DB}\)[/tex] is [tex]\( -\frac{1}{4} \)[/tex], then let the slope of [tex]\(\overleftrightarrow{AC}\)[/tex] be [tex]\( m' \)[/tex].
[tex]\[
m \cdot m' = -1
\][/tex]
[tex]\[
\left( -\frac{1}{4} \right) \cdot m' = -1
\][/tex]
Solving for [tex]\( m' \)[/tex]:
[tex]\[
m' = 4
\][/tex]
Therefore, the slope of [tex]\(\overleftrightarrow{AC}\)[/tex] is [tex]\( 4 \)[/tex].
3. Construct the equation of the perpendicular line [tex]\( \overleftrightarrow{AC} \)[/tex] using the slope [tex]\( 4 \)[/tex].
The general form of the line's equation with slope [tex]\( 4 \)[/tex] is:
[tex]\[
y = 4x + c
\][/tex]
Convert this into standard form [tex]\(Ax + By = C\)[/tex]:
[tex]\[
4x - y = -c
\][/tex]
4. Compare the given options to determine which matches the form [tex]\( 4x - y = C \)[/tex].
- [tex]\( 2x + 8y = 12 \)[/tex]: This can be rewritten but doesn't fit the [tex]\( 4x - y = C \)[/tex] form directly.
- [tex]\( -4x + y = -28 \)[/tex]: When rearranged, this looks like [tex]\( y = 4x - 28 \)[/tex], but the signs are incorrect.
- [tex]\( 4x - y = -28 \)[/tex]: This is already in the correct form.
- [tex]\( 2x + y = 14 \)[/tex]: Doesn't fit the [tex]\( 4x - y = C \)[/tex] form directly.
Thus, the correct equation for the line [tex]\(\overleftrightarrow{AC}\)[/tex] is:
[tex]\[
4x - y = -28
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{4 x - y = -28}
\][/tex]
