# 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 $A\left( { – 3,2} \right)$.