Answer:
D
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
To calculate m of AB
m = [tex]\frac{rise}{run}[/tex] = [tex]\frac{3}{2}[/tex]
Given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{\frac{3}{2} }[/tex] = - [tex]\frac{2}{3}[/tex]
To find the midpoint of AB use the midpoint formula
M = [ [tex]\frac{1}{2}[/tex](x₁ + x₂ ), [tex]\frac{1}{2}[/tex] (y₁ + y₂ ) ]
with (x₁, y₁ ) = )0, 0) and (x₂, y₂ ) = (2, 3), thus
M = [ [tex]\frac{1}{2}[/tex](0 + 2), [tex]\frac{1}{2}[/tex](0 + 3 )] = (1, [tex]\frac{3}{2}[/tex] )
Partial equation of perpendicular bisector is
y = - [tex]\frac{2}{3}[/tex] x + c
To find c substitute (1, [tex]\frac{3}{2}[/tex] ) into the partial equation
[tex]\frac{3}{2}[/tex] = - [tex]\frac{2}{3}[/tex] + c ⇒ c = [tex]\frac{3}{2}[/tex] + [tex]\frac{2}{3}[/tex] = [tex]\frac{13}{6}[/tex]
y = - [tex]\frac{2}{3}[/tex] x + [tex]\frac{13}{6}[/tex] → D
