To find the slope of line [tex]\( t \)[/tex] given that it is perpendicular to line [tex]\( s \)[/tex], we can use a key property of perpendicular lines. Specifically, the slopes of two perpendicular lines multiply to [tex]\(-1\)[/tex]. This means if one line has a slope [tex]\( m_1 \)[/tex], the other line must have a slope [tex]\( m_2 \)[/tex] such that:
[tex]\[ m_1 \cdot m_2 = -1 \][/tex]
Here, we are given that the slope of line [tex]\( s \)[/tex] (denoted as [tex]\( m_1 \)[/tex]) is 5. We need to determine the slope of line [tex]\( t \)[/tex] (denoted as [tex]\( m_2 \)[/tex]).
Following the property of perpendicular slopes, we can set up the equation:
[tex]\[ 5 \cdot m_2 = -1 \][/tex]
Next, we solve for [tex]\( m_2 \)[/tex] by dividing both sides of the equation by 5:
[tex]\[ m_2 = \frac{-1}{5} \][/tex]
So, the slope of line [tex]\( t \)[/tex] is [tex]\( -\frac{1}{5} \)[/tex].
Out of the given options, this corresponds to:
A. [tex]\( -\frac{1}{5} \)[/tex]
Thus, the correct answer is [tex]\( m_2 = -\frac{1}{5} \)[/tex], which matches option A.
