Answer: [tex]\bold{y=-\dfrac{1}{3}x-1}[/tex]
Step-by-step explanation:
(4, 1) & (2, -5)
First, find the slope (m) and then the perpendicular (opposite reciprocal) slope:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}\\\\\\m=\dfrac{-5-1}{2-4} = \dfrac{-6}{-2}=3\quad \rightarrow \quad m_{\perp}=-\dfrac{1}{3}\\[/tex]
Next, find the midpoint of (4, 1) and (2, -5):
[tex]Midpoint=\bigg(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\bigg)\\\\\\.\qquad \qquad=\bigg(\dfrac{4+2}{2},\dfrac{1-5}{2}\bigg)\\\\\\.\qquad \qquad=\bigg(\dfrac{6}{2},\dfrac{-4}{2}\bigg)\\\\\\.\qquad \qquad=(3, -2)[/tex]
Lastly, input the perpendicular slope and the midpoint into the Point-Slope formula to find the equation of the line:
[tex]y - y_1 = m_{\perp}(x - x_1)\\\\y - (-2) = -\dfrac{1}{3}(x - 3)\\\\y + 2=-\dfrac{1}{3}x +1\\\\y =-\dfrac{1}{3}x - 1\\[/tex]
