To determine the slope of line [tex]\( q \)[/tex], given that lines [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are perpendicular and the slope of line [tex]\( p \)[/tex] is 2, we can use a key property of perpendicular lines.
Property of Perpendicular Lines:
The slopes of two perpendicular lines, when multiplied together, equal [tex]\(-1\)[/tex]. This can be expressed as:
[tex]\[ m_p \cdot m_q = -1 \][/tex]
where [tex]\( m_p \)[/tex] is the slope of line [tex]\( p \)[/tex] and [tex]\( m_q \)[/tex] is the slope of line [tex]\( q \)[/tex].
Given:
[tex]\[ m_p = 2 \][/tex]
We need to find [tex]\( m_q \)[/tex] such that:
[tex]\[ 2 \cdot m_q = -1 \][/tex]
To solve for [tex]\( m_q \)[/tex], we divide both sides of the equation by 2:
[tex]\[ m_q = \frac{-1}{2} \][/tex]
Therefore, the slope of line [tex]\( q \)[/tex] is:
[tex]\[ m_q = -\frac{1}{2} \][/tex]
From the given options, we see that the correct answer is:
C. [tex]\(-\frac{1}{2}\)[/tex]
