The equation of the perpendicular bisector between k(3,-6) and l(10,17) is
y = (23/13)x - 81/2.
To find the equation of perpendicular bisector we need the following:
- Midpoint of the line
[tex]x_{m}[/tex] = ( [tex]x_{1}+x_{2}[/tex] ) / 2 and [tex]y_{m} =( y_{1} +y_{2})[/tex] / 2
- A line with endpoints
[tex](x_{1},~x_{2})~and~(y_{1},~ y_{2} )[/tex]
- Slope of the line.
[tex]m_{1}[/tex] = ([tex]x_{2}-x_{1}[/tex] ) / ( [tex]y_{2} -y_{1}[/tex])
- Slope of perpendicular bisector.
[tex]m_{2}[/tex] = -1 / [tex]m_{1}[/tex]
The equation of perpendicular bisector is given by:
( y - [tex]y_{m}[/tex] ) = [tex]m_{2}[/tex] ( x - [tex]x_{m}[/tex] )
Let's calculate the following:
1.
[tex]x_{m}[/tex] = ( [tex]x_{1}+x_{2}[/tex] ) / 2 and [tex]y_{m} =( y_{1} +y_{2})[/tex] / 2
We have,
(3, -6) = ( [tex]x_{1},~y_{1}[/tex] )
(10, 17) = ( [tex]x_{2},~y_{2}[/tex] )
So,
[tex]x_{m}[/tex] = (3 + 10) / 2 = 13/2
[tex]y_{m}[/tex] = (-6+17) / 2 = 11/2
2.
[tex]m_{1}[/tex] = ([tex]x_{2}-x_{1}[/tex] ) / ( [tex]y_{2} -y_{1}[/tex])
= ( -10 - 3 ) / ( 17 - (-6))
= -13 / 23
3.
[tex]m_{2}[/tex] = -1 / [tex]m_{1}[/tex]
= -1 / (-13/23)
= 23 / 13
4.
( y - [tex]y_{m}[/tex] ) = [tex]m_{2}[/tex] ( x - [tex]x_{m}[/tex] )
( y - 11/2 ) = 23/13 ( x - 13/2 )
y - 11/2 = 23/13x - (23/13)(13/2)
y - 11/2 = 23/13x - 46
y = 23/13 x - 46 + 11/2
y = (23/13)x - 81/2
Thus the equation of the perpendicular bisector between k(3,-6) and l(10,17) is y = (23/13)x - 81/2.
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