Two lines are perpendicular if the product of their slopes is -1.
We'll write the equation of the line that passs through the points (4,2) and (4,6).
Let's note the points above as A and B.
We'll write the formula of the equation:
(xB-xA)/(x-xA) = (yB-yA)/(y-yA)
(4-4)/(x-4) = (6-2)/(y-2)
0/(x-4) = (4)/(y-2)
We'll cross multiply:
4(x-4) = 0
We'll divide by 4 both sides:
x-4 = 0
We'll add 4 both sides:
x = 4
We'll write the equation of the line in the standard form:
y = mx + n, where m is the slope and n is the y intercept.
From the equation x = 4 it results that:
y = 4
is the equation of the line that is perpendicular to the line whose equation is x = 4.
The equation of the line between two points (x1, y1) and (x2, y2) is:
y - y1 = [( y2 -y1)/ (x2 -x1)] * (x - x1).
Here we have the points ( 4,2) and (4,6)
The line passing through them has a slope. (6 - 2) / (4 - 4) = inf
Or it is a vertical line.
The line perpendicular to a vertical line is a horizontal line, with slope 0.
Also this line passes through the point lying between ( 4,2) and (4,6) which is (4 , 4 )
So the slope of the required line is 0 and the y- intercept is 4.
The line is y= 0*x + 4.
So the equation of the required line is y=4.
To find the equation of the perpendicular bisector for line segment joining A(4,2) and B(4,6).
The perpendicular bisector passes through the midpoint of the line segment joining A and B . and is perpendicular to AB.
The mid point of the line AB is (Mx , My) = ( (Ax+Bx)/2 , (Ay+By)/2) ) = ( (4+4)/2 , (2+6)/2 ) = (4,4).
The slope of the line through the mid point M(4,4) shoulf be perpendicular to AB.
The slope of AB = (By-Ay)/(Bx-Ax) = (6-2)/(4-4) = infinite or AB is parallel to y axis.
So the perpendicular to AB should be parallel to x axis ( or perpendicular to y axis). Ao the equation of this line is y = k.
Since y=k should pass thruogh mid point of AB , that is M(4,4).
So y = 4 is the line which is the perpendicular bisector of A(4,2) and B(4,6)