SOLUTION: I need your help about this: Find the equation of the line with points equidistant from (5,-2) and (4,3).

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Question 203940: I need your help about this:
Find the equation of the line with points equidistant from (5,-2) and (4,3).
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
you want to have a perpendicular bisector of the line that is formed by those 2 points.
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the points are:
(5,-2)
(4,3)
slope of the line through those points is (y2-y1)/(x2-x1) = (3-(-2))/(4-5) = 5/-1 = -5
equation of the line through those points is:
y =
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the slope of the line perpendicular to it would be -(1/-5) = 1/5
this line would have to bisect the line it is perpendicular to.
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midpoint of the original line is at ((x1+x2)/2,(y1+y2)/2)
this midpoint turns out to be ((5+4)/2,(3-2)/2) which equals (9/2,1/2)
the equation of the perpendicular line would be:
y =
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since this line is a perpendicular bisector of the original line, then all points on this line will be equidistant from the two given points because any of those points will form an isosceles triangle with the given points on original line.
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a graph of the original line formed by the 2 given points and the line that is the perpendicular of that original line is shown below.
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as an exercise, show that any point on that line taken at random will be equidistant from the 2 given points.
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the equation of the perpendicvular line is
y = 1/5x - .4
take x = 20
that makes y = 3.6
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the coordinates of this point are:
(20,3.6)
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we need to show that this point is equidistant from:
(5,-2)
and
(4,3)
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the length of either of these lines is given by the formula:
L =
for (20,3.6) and (5,-2) this comes out to be: 16.01124605
for (20,3.6) and (4,3) this comes out to be: 16.01124605
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they are both equidistant.
any other point on the perpenduclar bisector of the original line will be the same.
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in general, given 2 points, you can find the slope by using the equation:
slope =
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in general, given the slope and one of the points on the line, you can find the y intercept by solving for b in the general equation of:
y = m*x + b
where m is the slope
and b is the y intercept
all you do is replace the x with the known value for x and replace the y with the known value for y and replace the m with the known value of the slope and solve for b.
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in general, you can find the midpoint of a line by using the equation:
x = (x1+x2)/2
y = (y1+y1)/2
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in general, you can find the length of a line by using the formula:
length of the line =
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Answer by ikleyn(53369)   (Show Source): You can put this solution on YOUR website!
.
Find the equation of the line with points equidistant from (5,-2) and (4,3).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


        As an alternative to the geometric solution by @Theo,  below is simple algebraic solution. 


Let (x,y) be the point equidistant from these given points (5,-2) and (4,3).


It means that the distances from (x,y) to these points are equal.

Hence, the squares of distances are equal.


So, we write

     +  =  + .

or

     +  =  + .


Now we simplify it

    x^2 - 10x + 25 + y^2 + 4y + 4 = x^2 - 8x + 16 + y^2 - 6y + 9,

    -10x + 4y + 29 = -8x - 6y + 25,

    (-10x + 8x) + (4y + 6y) = 25 - 29,

    -2x + 10y = -4,

     -x +  5y = -2,

      x -  5y = 2.

Solved.



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