Respuesta :
[tex]y\text{ = x + 7}[/tex]
Here, we want to find the equation of the perpendicular bisector of th line segment with the given endpoints
We start by calculating the slope of the line segment
Mathematically, we can have that as;
[tex]\begin{gathered} m\text{ = }\frac{y_2-y_1}{x_2-x_1} \\ \\ m\text{ = }\frac{5-1}{-6-(-2)}=\text{ }\frac{4}{-4}\text{ = -1} \end{gathered}[/tex]So, we have the slope of the line as -1
Mathematically, the slopes of two lines which are perpendicular to each other have a product of -1
Thus;
[tex]\begin{gathered} m_2\text{ }\times\text{ (-1) = -1} \\ \\ m_2\text{ = 1} \end{gathered}[/tex]Now, we need the midpoint segment coordinates as it is the point through which the perpendicular bisector will pass through
We can get these coordinates using the mid-point formula
That will be;
[tex]\begin{gathered} (x,y)\text{ = (}\frac{x_2+x_1}{2},\frac{y_2+y_1}{2}) \\ \\ (x,y)\text{ = (}\frac{-2-6}{2},\frac{1+5}{2}) \\ \\ (x,y)\text{ = (-4,3)} \end{gathered}[/tex]So we use the point-slope formula to get the equation
That will be;
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ y-3\text{ = 1(x+4)} \\ y-3\text{ = x + 4} \\ y\text{ = x + 4 + 3} \\ y\text{ = x + 7} \end{gathered}[/tex]