A line segment is bisected by a line with the equation $- 6 y + 5 x = 4$ . If one end of the line segment is at $( 2 , 5 )$ , where is the other end?

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Answer 1 · 2017-12-31T09:19:33

coordinates of the other end point $color (purple)(362/61, 17/61)$

Assumption : Bisecting line is a perpendicular bisector

Standard form of equation $y=mx +c$
Slope of perpendicular bisector m is given by
$-6y + 5x= 4$
$y= (5/6)x - (2/3)$
$m = (5/6)$

Slope of line segment is
$y - 5 = -(1/m)(x-2)$
$y - 5= (-6/5)(x - 2)$
$5y - 25 = -6x + 12$

$5y + 6x = 37 color (white)((aaaa)$ Eqn (1)
$-6y + 5x = 4 color (white)((aaaa)$ Eqn (2)

Solving Eqns (1) & (2),

$x= color (green)(242/61)$

$y= color (green)(161/61)$

Mid point $color(green)(242/61, 161/61)$

Let (x1,y1) the other end point.
$(2+x1)/2= 242/37$
$x1= (484/61) - 2 = color(red)( 362/61)$
$(5+y1)/2= 161/61$
$y1=(322/61) - 5 color(red)(17/61)$

Coordinates of other end point $color(red)(362/61, 17/61)$

A line segment is bisected by a line with the equation - 6 y + 5 x = 4 - 6 y + 5 x = 4 . If one end of the line segment is at ( 2 , 5 )( 2 , 5 ), where is the other end?