Equation of the Right Bisector of a Triangle

To find the equation of the right bisector of a triangle we examine the following example: Consider the triangle having vertices A\left( { - 3,2} \right), B\left( {5,4} \right) and C\left( {3, - 8} \right).

The equation of a perpendicular bisector is given as

y - \frac{{{y_1} + {y_2}}}{2} = - \frac{{{x_2} - {x_1}}}{{{y_2} - {y_1}}}\left( {x - \frac{{{x_1} + {x_2}}}{2}} \right)\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)

For the perpendicular bisector of A\left( { - 3,2} \right) and B\left( {5,4} \right), and putting these values in the above equation (i), we have

\begin{gathered} y - \frac{{2 + 4}}{2} = - \frac{{5 - \left( { - 3} \right)}}{{4 - 2}}\left( {x - \frac{{ - 3 + 5}}{2}} \right) \\ \Rightarrow y - \frac{6}{2} = - \frac{{5 + 3}}{2}\left( {x - \frac{2}{2}} \right) \\ \Rightarrow y - 3 = - 4\left( {x - 1} \right) \\ \Rightarrow 4x + y - 7 = 0 \\ \end{gathered}

This is the equation of the perpendicular bisector of A\left( { - 3,2} \right) and B\left( {5,4} \right)

For the perpendicular bisector of B\left( {5,4} \right) and C\left( {3, - 8} \right), and putting these values in the above equation (i), we have

\begin{gathered} y - \frac{{4 - 8}}{2} = - \frac{{3 - 5}}{{ - 8 - 4}}\left( {x - \frac{{5 + 3}}{2}} \right) \\ \Rightarrow y - \frac{{ - 4}}{2} = - \frac{{ - 2}}{{ - 12}}\left( {x - \frac{8}{2}} \right) \\ \Rightarrow y + 2 = - \frac{1}{6}\left( {x - 4} \right) \\ \Rightarrow x + 6y + 8 = 0 \\ \end{gathered}

This is the equation of the perpendicular bisector of B\left( {5,4} \right) and C\left( {3, - 8} \right)

For the perpendicular bisector of C\left( {3, - 8} \right) and A\left( { - 3,2} \right), and putting these values in the above equation (i), we have

\begin{gathered} y - \frac{{ - 8 + 2}}{2} = - \frac{{ - 3 - 3}}{{2 - \left( { - 8} \right)}}\left( {x - \frac{{3 - 3}}{2}} \right) \\ \Rightarrow y - \frac{{ - 6}}{2} = - \frac{{ - 6}}{{2 + 8}}\left( {x - 0} \right) \\ \Rightarrow y + 3 = \frac{3}{5}x \\ \Rightarrow 3x - 5y - 15 = 0 \\ \end{gathered}

This is the equation of the perpendicular bisector of C\left( {3, - 8} \right) and A\left( { - 3,2} \right).