To solve the problem, we need to find the equation of the line that passes through the point [tex]\((-1, 6)\)[/tex] and is perpendicular to line [tex]\(CD\)[/tex].
Step 1: Determine the slope of line [tex]\(CD\)[/tex].
- The slope of the perpendicular line ([tex]\(m_\perp\)[/tex]) will be the negative reciprocal of the slope of line [tex]\(CD\)[/tex]. Assuming the slope of line [tex]\(CD\)[/tex] is [tex]\(m_{CD} = -\frac{1}{2}\)[/tex].
Step 2: Find the slope of the perpendicular line.
- The negative reciprocal of [tex]\(-\frac{1}{2}\)[/tex] is:
[tex]\[
m_\perp = -\left(- \frac{1}{2}\right)^{-1} = -(-2) = 2
\][/tex]
Step 3: Formulate the equation of the perpendicular line in point-slope form.
- The point-slope form of a line is given by [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is the point the line passes through.
- Plug in the point [tex]\((-1, 6)\)[/tex] and the slope [tex]\(2\)[/tex]:
[tex]\[
y - 6 = 2(x + 1)
\][/tex]
Step 4: Convert the equation to slope-intercept form ([tex]\(y = mx + b\)[/tex]).
- Distribute and simplify:
[tex]\[
y - 6 = 2(x + 1) \implies y - 6 = 2x + 2 \implies y = 2x + 8
\][/tex]
Step 5: Verify the available options.
- The equation we derived is [tex]\(y = 2x + 8\)[/tex].
None of the provided options match [tex]\(y = 2x + 8\)[/tex]. Hence, the result is none of the options match.
Therefore, the solution concludes that none of the given options match the equation [tex]\(y = 2x + 8\)[/tex].
