To write the equation of a line perpendicular to line CD and passing through the point [tex]\((-1, 6)\)[/tex], we will follow these steps:
1. Determine the slope of the given line:
The equation of line CD is given as [tex]\( y = -0.5x - 5.5 \)[/tex]. Here, the slope ([tex]\(m\)[/tex]) is [tex]\(-0.5\)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
So, if the slope of line CD is [tex]\(-0.5\)[/tex], the slope ([tex]\(m_{\perpendicular}\)[/tex]) of the line perpendicular to it is:
[tex]\[
m_{\perpendicular} = -\frac{1}{-0.5} = 2
\][/tex]
3. Use the point-slope form of the line equation:
We know the slope of the perpendicular line is [tex]\(2\)[/tex] and it passes through the point [tex]\((-1, 6)\)[/tex]. The point-slope form of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting [tex]\(m = 2\)[/tex], [tex]\(x_1 = -1\)[/tex], and [tex]\(y_1 = 6\)[/tex], we get:
[tex]\[
y - 6 = 2(x - (-1)) \quad \Rightarrow \quad y - 6 = 2(x + 1)
\][/tex]
4. Simplify the equation to slope-intercept form:
Solve for [tex]\(y\)[/tex] to put the equation into the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y - 6 = 2(x + 1) \quad \Rightarrow \quad y - 6 = 2x + 2 \quad \Rightarrow \quad y = 2x + 2 + 6 \quad \Rightarrow \quad y = 2x + 8
\][/tex]
Hence, the equation of the line in slope-intercept form is:
[tex]\[
y = 2x + 8
\][/tex]
Thus, among the given choices, the equation that matches this is not provided directly. Correcting the choice, the final correct answer is:
[tex]\( y = 2x + 8 \)[/tex]
