Answer:
\(y+3x-\frac{1}{2}=0\)
Step-by-step explanation:
The perpendicular bisector is basically the line that intercept the side HI in its midpoint with a right angle. To find the equation of such line, we first have to find the slope of side HI, which coordinates are H(-4,2) and I(2,4)
\(m_{HI}=\frac{4-2}{2-(-4)}=\frac{2}{6}=\frac{1}{3}\)
Now, the condition of perpendicularity is
\(m_{HI}m_{\perp}=-1\)
We replace the slope of HI to find the slope of the perpendicular bisector
\(m_{HI}m_{\perp}=-1\\\frac{1}{3}m_{\perp} =-1\\m_{\perp} =-3\)
We already have the slope of the perpendicular bisector, now we just a points that this line crosses. We know that a perpendicular bisector intercept the midpoint, so we have to find the midpoint of HI
\(M(\frac{4+(-4)}{2} ,\frac{-3+2}{2} )\\M(\frac{0}{2},\frac{-1}{2})\\M (0,-\frac{1}{2})\)
Now we know the point we use the point-slope from to find the equation
\(y-y_{1} =m(x-x_{1})\\y-\frac{1}{2}=-3(x-0) \\y=-3x+\frac{1}{2}\)
Therefore, the standard form of the perpendicular bisector of line HI is
\(y+3x-\frac{1}{2}=0\)
