okuism
Answered

Lines [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are perpendicular. If the slope of line [tex]\( a \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex], what is the slope of line [tex]\( b \)[/tex]?

A. [tex]\( \frac{3}{2} \)[/tex]

B. [tex]\( -\frac{3}{2} \)[/tex]

C. [tex]\( -\frac{2}{3} \)[/tex]

D. [tex]\( \frac{2}{3} \)[/tex]



Answer :

To determine the slope of line [tex]\( b \)[/tex], we need to use the relationship between the slopes of two perpendicular lines. If two lines are perpendicular, the product of their slopes is [tex]\(-1\)[/tex].

Given that the slope of line [tex]\( a \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex], let's denote the slope of line [tex]\( b \)[/tex] by [tex]\( m_b \)[/tex].

The relationship between the slopes of the two lines can be expressed by the equation:
[tex]\[ m_a \cdot m_b = -1 \][/tex]

Substitute the given slope of line [tex]\( a \)[/tex] into the equation:
[tex]\[ -\frac{2}{3} \cdot m_b = -1 \][/tex]

To solve for [tex]\( m_b \)[/tex], divide both sides of the equation by [tex]\(-\frac{2}{3}\)[/tex]:
[tex]\[ m_b = \frac{-1}{-\frac{2}{3}} \][/tex]

Simplify the right-hand side of the equation:
[tex]\[ m_b = \frac{-1 \div -1}{\frac{2}{3}} = \frac{1}{\frac{2}{3}} = 1 \cdot \frac{3}{2} = \frac{3}{2} \][/tex]

Therefore, the slope of line [tex]\( b \)[/tex] is:
[tex]\[ m_b = \frac{3}{2} \][/tex]

So, the slope of line [tex]\( b \)[/tex] corresponding to the correct answer is:
[tex]\[ \boxed{\frac{3}{2}} \][/tex]