Answer:
[tex]y = -\frac{3}{2} x + 4[/tex]
Step-by-step explanation:
First, let us find the gradient of AB:
Gradient of AB = [tex]\frac{-1-3}{-1-5}[/tex]
= [tex]\frac{2}{3}[/tex]
We also need to know that The product of gradients which are perpendicular to each other is -1. Using this idea, we can find the gradient of the perpendicular bisector:
(Gradient of perpendicular bisector)( [tex]\frac{2}{3}[/tex] ) = -1
Gradient of perpendicular bisector = [tex]-\frac{3}{2}[/tex]
Now, we need to know at which coordinates the perpendicular bisector intersects AB. A perpendicular bisector bisects a line to two equal parts. Hence the coordinates of the intersection point is the midpoint of AB. Thus,
Coordinates of intersection = ( [tex]\frac{-1 + 5}{2}[/tex], [tex]\frac{-1 + 3}{2}[/tex] )
= ( 2, 1 )
Now, we can construct our equation. The equation of a line can be formed using the formula [tex](y - y_{1}) = m(x - x_{1})[/tex] where [tex]m[/tex] is the gradient and the line passes through [tex](x_{1} , y_{1} )[/tex]. Hence by substituting the values, we get:
[tex](y - 1) = -\frac{3}{2} (x - 2)[/tex]
[tex]y - 1 = -\frac{3}{2} x + 3[/tex]
[tex]y = -\frac{3}{2} x + 4[/tex]
