Solution
Given that
[tex]\begin{gathered} A(2,4) \\ B(5,-4) \end{gathered}[/tex]
To find the slope, m, of the line passing through the given points, the formula is
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Where
[tex]\begin{gathered} (x_1,y_1)\Rightarrow A(2,4) \\ (x_2,y_2)\Rightarrow B(5,-4) \end{gathered}[/tex]
Substitute the coordinates into the formula to find the slope, m, of a line
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{-4-4}{5-2}=\frac{-8}{3}=-\frac{8}{3} \\ m=-\frac{8}{3} \end{gathered}[/tex]
The slope of the line AB passing through the given points is m = -8/3
A) If two lines are parallel, their slopes are equal.
Hence, the slope, m₁ of the line that is parellel to line AB is
[tex]m_1=-\frac{8}{3}[/tex]
Thus, the slope of a line parallel to line AB is m₁ = -8/3
B) If two lines are perpendicular, the formula to find the slope m₂ of the line perpedicular to the slope of a given line
[tex]m_2=-\frac{1}{m_{}}[/tex]
Where m = -8/3, the slope, m₂, of a line perpendicular to line AB will be
[tex]\begin{gathered} m_2=-\frac{1}{m_{}} \\ m_2=-\frac{1}{\frac{-8}{3}_{}}=\frac{3}{8} \\ m_2=\frac{3}{8} \end{gathered}[/tex]
Thus, the slope of a line perpendicular to line AB is m₂ = 3/8
