Given:
Endpoints of a segment are J(−24,−4) and K(−4,6).
To find:
The equation in point-slope form for the perpendicular bisector of JK.
Solution:
Slope of JK is
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\dfrac{6-(-4)}{-4-(-24)}[/tex]
[tex]m=\dfrac{6+4}{-4+24}[/tex]
[tex]m=\dfrac{10}{20}[/tex]
[tex]m=\dfrac{1}{2}[/tex]
Product of slopes of two perpendicular lines is -1. So, product of perpendicular line of JK is -2.
Perpendicular bisector of JK passes through the midpoint of JK.
[tex]Midpoint=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)[/tex]
[tex]Midpoint=\left(\dfrac{-24-4}{2},\dfrac{-4+6}{2}\right)[/tex]
[tex]Midpoint=\left(\dfrac{-28}{2},\dfrac{2}{2}\right)[/tex]
[tex]Midpoint=\left(-14,1\right)[/tex]
Point slope form:
[tex]y-y_1=m(x-x_1)[/tex]
where, m is slope.
Perpendicular bisector of JK passes through (-14,1) with slope -2. So, the point slope form of JK is
[tex]y-1=-2(x-(-14))[/tex]
[tex]y-1=-2(x+14)[/tex]
Therefore, the correct option is D.
