Respuesta :
Given:
AB is formed by A(-10, 3) and B(2,7). If line l is the perpendicular bisector of AB.
To find:
The equation of line l in slope intercept form.
Solution:
Slope formula:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
Slope of line AB is
[tex]m_1=\dfrac{7-3}{2-(-10)}[/tex]
[tex]m_1=\dfrac{4}{2+10}[/tex]
[tex]m_1=\dfrac{4}{12}[/tex]
[tex]m_1=\dfrac{1}{3}[/tex]
Product of slopes of two perpendicular lines is -1.
[tex]m_1\times m_2=-1[/tex]
[tex]\dfrac{1}{3}\times m_2=-1[/tex]
[tex]m_2=-3[/tex]
Midpoint of AB is
[tex]Midpoint=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)[/tex]
[tex]Midpoint=\left(\dfrac{-10+2}{2},\dfrac{3+7}{2}\right)[/tex]
[tex]Midpoint=\left(\dfrac{-8}{2},\dfrac{10}{2}\right)[/tex]
[tex]Midpoint=\left(-4,5\right)[/tex]
The perpendicular bisector of AB (i.e., line l)passes through he midpoint of AB, i.e., (-4,5) and having slope -3.
So, the equation of line l is
[tex]y-y_1=m(x-x_1)[/tex]
[tex]y-5=-3(x-(-4))[/tex]
[tex]y-5=-3(x+4)[/tex]
[tex]y=-3x-12+5[/tex]
[tex]y=-3x-7[/tex]
Therefore, the equation of line l in slope intercept form is [tex]y=-3x-7[/tex].
