find the equation of the perpendicular bisector of (2,-5) and (6,9), expressing your answer in the form: ax + by + c = 0

thanks for any help i can quite seem to get this one...

and its in for tomorrow!! gah!

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- Oct 6th 2008, 09:42 AMpop_91perpenicular bisector...?
find the equation of the perpendicular bisector of (2,-5) and (6,9), expressing your answer in the form: ax + by + c = 0

thanks for any help i can quite seem to get this one...

and its in for tomorrow!! gah! - Oct 6th 2008, 10:01 AMmasters
Find the midpoint of the line segment through (2, -5) and (6, 9):

$\displaystyle Midpoint=\left(\frac{2+6}{2} \ \ , \ \ \frac{9-5}{2}\right)=(4, 2)$

Find the slope of the line through (2, -5) and (6, 9).

slope = $\displaystyle \frac{9-^-5}{6-2}=\frac{14}{4}=\frac{7}{2}$

The slope of the perpendicular bisector is the negative reciprocal of that slope, so it would be $\displaystyle -\frac{2}{7}$ and pass through the midpoint (4, 2).

Using y=mx+b and m = $\displaystyle -\frac{2}{7}$ through (4, 2).

$\displaystyle 2=-\frac{2}{7}(4)+b$

$\displaystyle \frac{22}{7}=b$

$\displaystyle y=-\frac{2}{7}x+\frac{22}{7}$

$\displaystyle 7y=-2x+22$

$\displaystyle 2x+7y-22=0$ - Oct 6th 2008, 10:03 AMpop_91
thank you soo much!!!! (Rofl)

- Oct 6th 2008, 10:03 AMSoroban
Hello, pop_91!

All the necessary information is there ... just dig it up.

Quote:

Find the equation of the perpendicular bisector of $\displaystyle A(2,-5)$ and $\displaystyle B(6,9),$

expressing your answer in the form: $\displaystyle ax + by + c \:=\: 0$

The perpendicular bisector of $\displaystyle AB$ goes through the midpoint of $\displaystyle AB$

. . and is perpendicular to $\displaystyle AB.$

The midpoint of $\displaystyle AB$ is: .$\displaystyle \left(\frac{2+6}{2},\;\frac{\text{-}5+9}{2}\right) \;=\;(4,\:2)$

The slope of $\displaystyle AB$ is: .$\displaystyle m_1 \;=\;\frac{9-(\text{-}5)}{6-2} \:=\:\frac{14}{4} \:=\:\frac{7}{2}$

The perpendicular slope is: .$\displaystyle m_2 \:=\:\text{-}\frac{2}{7}$

The line through $\displaystyle (4,\:2)$ with slope $\displaystyle \text{-}\frac{2}{7}$ is:

. . $\displaystyle y - 2 \;=\;\text{-}\frac{2}{7}(x - 4) \quad\Rightarrow\quad y - 2 \;=\;\text{-}\frac{2}{7}x + \frac{8}{7}$

Multiply by 7: .$\displaystyle 7y - 14 \;=\;\text{-}2x + 8$

. . Therefore: . $\displaystyle {\color{blue}2x + 7y - 22 \;=\;0}$