Answer to Question #5854 in Abstract Algebra for nayeli correa

Question

Write an equation of the perpendicular bisector of the line segment whose endpoints are (-1,1) and (7,-5)

Answer 1

1) Let’s call the one endpoint (-1,1) as point A and another endpoint (7,-5) as point B
2) Let’s find the middle point of the line segment AB – the point, called C
<img style="width: 169px; height: 49px;" src="data:image/png;base64,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" alt="">
We can use it to make the perpendicular bisector.
3) Let’s find the equation of the line , on which there are points A and B. Lets called this line L
<img style="width: 89px; height: 104px;" src="data:image/png;base64,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" alt="">
The normal vector of this line L is (3,4). It also would be a directional vector of the perpendicular line K to the line L
4) Let’s find the equation of the line K using that C is lay on the line K
<img style="width: 171px; height: 101px;" src="data:image/png;base64,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" alt="">
But :
If you want the equation of the perpendicular line without using the point C(the general look of the equation), it would be
4x-3y+z=0 ,
where z is any integer.