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Question Number 38880 by Rio Mike last updated on 30/Jun/18
Find the equation of the perpendicular  bisector of the line segment joining  the points (1,3) and (5,1)
$${Find}\:{the}\:{equation}\:{of}\:{the}\:{perpendicular} \\ $$$${bisector}\:{of}\:{the}\:{line}\:{segment}\:{joining} \\ $$$${the}\:{points}\:\left(\mathrm{1},\mathrm{3}\right)\:{and}\:\left(\mathrm{5},\mathrm{1}\right) \\ $$
Answered by MrW3 last updated on 30/Jun/18
eqn of line joining (1,3) and (5,1):  ((y−1)/(3−1))=((x−5)/(1−5))  y=−(1/2)(x−5)+1=−(1/2)x+(7/2)    midpoint of (1,3) and (5,1):  (3,2)    eqn. of perpendicular bisector:  ((y−2)/(x−3))=−(1/(−(1/2)))=2  ⇒y=2x−4
$${eqn}\:{of}\:{line}\:{joining}\:\left(\mathrm{1},\mathrm{3}\right)\:{and}\:\left(\mathrm{5},\mathrm{1}\right): \\ $$$$\frac{{y}−\mathrm{1}}{\mathrm{3}−\mathrm{1}}=\frac{{x}−\mathrm{5}}{\mathrm{1}−\mathrm{5}} \\ $$$${y}=−\frac{\mathrm{1}}{\mathrm{2}}\left({x}−\mathrm{5}\right)+\mathrm{1}=−\frac{\mathrm{1}}{\mathrm{2}}{x}+\frac{\mathrm{7}}{\mathrm{2}} \\ $$$$ \\ $$$${midpoint}\:{of}\:\left(\mathrm{1},\mathrm{3}\right)\:{and}\:\left(\mathrm{5},\mathrm{1}\right): \\ $$$$\left(\mathrm{3},\mathrm{2}\right) \\ $$$$ \\ $$$${eqn}.\:{of}\:{perpendicular}\:{bisector}: \\ $$$$\frac{{y}−\mathrm{2}}{{x}−\mathrm{3}}=−\frac{\mathrm{1}}{−\frac{\mathrm{1}}{\mathrm{2}}}=\mathrm{2} \\ $$$$\Rightarrow{y}=\mathrm{2}{x}−\mathrm{4} \\ $$

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