Question

.A straight line passes through point A(2,4) and B(6,12). Find the equation of the perpendicular bisector of line AB (write your answer in the form ax+by=c)​

Answer 1

x + 2y = 20

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 =
`(y_(2)-y_(1) )/(x_(2)-x_(1) )`

with (x₁, y₁ ) = A(2, 4) and (x₂, y₂ ) = B(6, 12)

m =
`(12-4)/(6-2)` =
`(8)/(4)` = 2

Given a line with slope m then the slope of a line perpendicular to it is

`m_(perpendicular)` = -
`(1)/(m)` = -
`(1)/(2)`

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Given endpoints (x₁, y₁ ) and (x₂, y₂ ) then the midpoint is

[
`(x_(1)+x_(2) )/(2)` ,
`(y_(1)+y_(2) )/(2)` ]

Thus midpoint of AB = (
`(2+6)/(2)`,
`(4+12)/(2)` ) = (4, 8 )

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y = -
`(1)/(2)` 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 = -
`(1)/(2)` 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