Answer:
x + 2y = 20
Step-by-step explanation:
We require the slope and the midpoint of AB
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
Calculate m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = A(2, 4) and (x₂, y₂ ) = B(6, 12)
m = [tex]\frac{12-4}{6-2}[/tex] = [tex]\frac{8}{4}[/tex] = 2
Given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{2}[/tex]
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Given endpoints (x₁, y₁ ) and (x₂, y₂ ) then the midpoint is
[ [tex]\frac{x_{1}+x_{2} }{2}[/tex] , [tex]\frac{y_{1}+y_{2} }{2}[/tex] ]
Thus midpoint of AB = ( [tex]\frac{2+6}{2}[/tex], [tex]\frac{4+12}{2}[/tex] ) = (4, 8 )
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y = - [tex]\frac{1}{2}[/tex] x + c ← partial equation of perpendicular bisector
To find c substitute (4, 8) into the partial equation
8 = - 2 + c ⇒ c = 8 + 2 = 10
y = - [tex]\frac{1}{2}[/tex] x + 10 ← in slope- intercept form
Multiply through by 2
2y = - x + 20 ( add x to both sides )
x + 2y = 20 ← in the form ax + by = c
