- #1

mindauggas

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## Homework Statement

(b) Show that the equation of the perpendicular bisector of the line segment with endpoints (x[itex]_{1}[/itex], y[itex]_{1}[/itex]) and (x[itex]_{2}[/itex],y[itex]_{2}[/itex]) can be written as [itex]\frac{x-x^{-}}{y_{2}-y_{1}}[/itex]+[itex]\frac{y-y^{-}}{x_{2}-x_{1}}[/itex]= 0, where (x[itex]^{-}[/itex],y[itex]^{-}[/itex])are coordinates of the midpoint of the segment

## The Attempt at a Solution

Because:

(1) [itex]\frac{x-x^{-}}{y_{2}-y_{1}}[/itex]+[itex]\frac{y-y^{-}}{x_{2}-x_{1}}[/itex]= 0

We have:

(2)[itex]\frac{x-x^{-}}{y_{2}-y_{1}}[/itex]=-[itex]\frac{y-y^{-}}{x_{2}-x_{1}}[/itex]

Does this the minus sing in front of the term on the right side of the equation express the relative slope of the perpendicular bisector? (Relative to the line segment).

I also realized that the perpendicular bisector formula does not work, or it seems to me, when line segment is parallel to x axis, hence y=y[itex]^{-}[/itex] which makes the first term undefined. Hence the formula does not work when the line segment is perpendicular to the y-axis for similar reason.

Yeat still I don't know how to show the equation to be true in special cases where y[itex]\neq[/itex]y[itex]^{-}[/itex]

I reason than symmetry has to be somehow involved, but is it?