To determine which line is perpendicular to a line that has a slope of [tex]\(\frac{1}{2}\)[/tex], we need to use the mathematical relationship between the slopes of perpendicular lines. The slopes of two perpendicular lines are negative reciprocals of each other.
1. Original Slope:
The slope of the given line is [tex]\(\frac{1}{2}\)[/tex].
2. Negative Reciprocal:
To find the slope of the line that is perpendicular to this one, we take the negative reciprocal of [tex]\(\frac{1}{2}\)[/tex].
- The reciprocal of [tex]\(\frac{1}{2}\)[/tex] is [tex]\(2\)[/tex], because [tex]\(\frac{1}{\frac{1}{2}} = 2\)[/tex].
- The negative of this reciprocal is [tex]\(-2\)[/tex].
Thus, the slope of a line that is perpendicular to a line with a slope of [tex]\(\frac{1}{2}\)[/tex] is [tex]\(-2\)[/tex].
We conclude that any line with this slope, [tex]\(-2\)[/tex], will be perpendicular to the line with the slope of [tex]\(\frac{1}{2}\)[/tex].
Therefore, the line we are looking for has a slope of [tex]\(-2\)[/tex].
