To determine the slope of a line perpendicular to line [tex]\( CD \)[/tex], follow these step-by-step instructions:
1. Identify the form of the given equation:
The equation of line [tex]\( CD \)[/tex] is given as [tex]\((y - 3) = -2(x - 4)\)[/tex].
2. Recognize the form:
This equation is in point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] represents the slope of the line, [tex]\((x_1, y_1)\)[/tex] is a point on the line.
3. Determine the slope of line [tex]\( CD \)[/tex]:
Compare the given equation [tex]\((y - 3) = -2(x - 4)\)[/tex] with the point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
[tex]\[
m = -2
\][/tex]
Therefore, the slope [tex]\( m \)[/tex] of line [tex]\( CD \)[/tex] is [tex]\(-2\)[/tex].
4. Find the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. If the slope of line [tex]\( CD \)[/tex] is [tex]\( m = -2 \)[/tex], then the calculation for the negative reciprocal is:
[tex]\[
\text{slope of perpendicular line} = \frac{1}{m} = \frac{1}{-2} = -\left(\frac{1}{-2}\right) = \frac{1}{2}
\][/tex]
Therefore, the slope of a line perpendicular to line [tex]\( CD \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
So, the correct answer is [tex]\(\frac{1}{2}\)[/tex].
