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Find the equation of the perpendicular bisector of $AB$ for: $A(1, 3)$ and $B(-3, 5)$.

What I did: $m=\frac{3-5}{1+3}=-\frac12$ for the slope of $AB$

$(\frac{3+5}2, \frac{1-3}2)=(4, -1)$ for the midpoint.

Equation of perpendicular-bisector of $AB$ is: $y=\frac{x}2+-2$?

Paras Khosla
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Tonny Dong
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3 Answers3

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Hint: From the data, you know a vector $\vec n$ normal to the perpendicular bisector an a point $I $ on it. If$O$ is the origin and $M$ is any point on the perpendicular bisector, a vector equation is: $$\vec n\cdot \overrightarrow{OM}=\vec n\cdot \overrightarrow{OI}.$$

Bernard
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  • sorry I don't quite get it, its too complex, but the answer I should get at the end is Y=mx+c. I got everything except for the "C" part – Tonny Dong Jul 05 '15 at 07:48
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    In my method, if you translate it into coordinates, you obtain $ax+by=c$. Do you know what a dot product is? – Bernard Jul 05 '15 at 07:53
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If the OP doesn't have the tools used in @Bernard's answer: The slope of $AB$ is directed-y-difference divided by directed-x-difference, or $\frac{5-3}{-3-1}$ or $\frac2{-4}$ or $-\frac12$. The midpoint of $AB$ is $(\frac{1-3}2\mid\frac{3+5}2)$ or $(-1\mid 4)$. The slope perpendicular to the slope of $AB$ is the negative reciprocal of $-\frac12$ or $2$. Then apply the point-slope form of a line, $2=\frac{y-4}{x+1}$.

Senex Ægypti Parvi
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Hint

  1. Note the fact that the perpendicular bisector must pass through the mid-point of the two given points.
  2. The slope of the perpendicular bisector must be the negative of the reciprocal of the slope of the line joining the two points.

In your case the correct answer is $y=2x+6$. You've incorrectly determined the slope of the perpendicular bisector and the mid-point.

Bonus (Generalization)

If $(x_1, y_1)$ and $(x_2, y_2)$ are two points which are joined to form a line segment, the equation of whose perpendicular bisector is to be determined. Make note of the above provided hints. Consider the slope-intercept form of a line $[y=mx+b]$. Finding $m$ and plugging in the coordinates $\bigl(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\bigr)$. You get the equation for the perpendicular bisector as (which can be further simplified): $$y=-\biggl(\dfrac{x_2-x_1}{y_2-y_1}\biggr)x+\dfrac{y_2^2-y_1^2+x_2^2-x_1^2}{2(y_2-y_1)}$$

Paras Khosla
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