To determine which line is perpendicular to a line with a given slope, we need to understand the relationship between the slopes of two perpendicular lines.
1. Slope Relationship: When two lines are perpendicular, the product of their slopes is \(-1\). If we denote the slope of the first line as \( m_1 \), and the slope of the perpendicular line as \( m_2 \), then:
[tex]\[
m_1 \times m_2 = -1
\][/tex]
2. Given Slope: In this problem, the slope of the first line (\( m_1 \)) is \(\frac{1}{2}\).
3. Finding the Perpendicular Slope:
[tex]\[
\frac{1}{2} \times m_2 = -1
\][/tex]
To find \( m_2 \), we solve for \( m_2 \) by isolating it on one side of the equation.
[tex]\[
m_2 = \frac{-1}{\frac{1}{2}}
\][/tex]
Dividing by a fraction is equivalent to multiplying by its reciprocal. Thus,
[tex]\[
m_2 = -1 \times \frac{2}{1}
\][/tex]
[tex]\[
m_2 = -2
\][/tex]
So, the slope of the line that is perpendicular to a line with slope \(\frac{1}{2}\) is \(-2\).
Final Answer: The line that is perpendicular to a line with slope \(\frac{1}{2}\) will have a slope of \(-2\). Therefore, to determine which of the lines \(AB\), \(CD\), \(FG\), and \(HJ\) is perpendicular:
- We must identify the line that has a slope of \(-2\).
Without additional details about the slopes of lines \(AB\), \(CD\), \(FG\), and \(HJ\) given in the problem, we need to match their slopes accordingly to determine which one possesses a slope of \(-2\).
Based on the slope criteria described, the correct line is the one aligned with having the slope of [tex]\(-2\)[/tex].
