Lines [tex]\( s \)[/tex] and [tex]\( t \)[/tex] are perpendicular. If the slope of line [tex]\( s \)[/tex] is 5, what is the slope of line [tex]\( t \)[/tex]?

A. [tex]\(-\frac{1}{5}\)[/tex]
B. 5
C. [tex]\(\frac{1}{5}\)[/tex]
D. -5



Answer :

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.