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

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 = $\frac{y_{2}-y_{1} }{x_{2}-x_{1} }$

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

m = $\frac{12-4}{6-2}$ = $\frac{8}{4}$ = 2

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

$m_{perpendicular}$ = - $\frac{1}{m}$ = - $\frac{1}{2}$

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

[ $\frac{x_{1}+x_{2} }{2}$ , $\frac{y_{1}+y_{2} }{2}$ ]

Thus midpoint of AB = ( $\frac{2+6}{2}$, $\frac{4+12}{2}$ ) = (4, 8 )

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y = - $\frac{1}{2}$ 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 = - $\frac{1}{2}$ 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