A line segment is bisected by a line with the equation # 4 y - 6 x = 8 #. If one end of the line segment is at #( 8 , 3 )#, where is the other end?

1 Answer
Aug 29, 2017

Exact value: #(-28/13,127/13)#

Decimal value (-2.16 9.77)

Explanation:

Bisectors are perpendicular - they bisect the line at a right angle. This means that the gradient of the line segment is the reciprocal of the other line:
#4y=6x+8#
#y=3/2x+2# has a gradient of #(3/2)#
#:.# unknown segment has a gradient of #color(blue)(-2/3)#

So our line segment has a point at #color(red)(8),color(green)(3)# and gradient #color(blue)(-2/3)#
#y-y_1=m(x-x_1#)
#:. y-color(green)(3)=color(blue)(-2/3)(x-color(red)8)#
#y-color(green)(3)=color(blue)(-2/3)x+color(red)16/3#
#y=color(blue)(-2/3)x+25/3# is the equation of our line segment.

Now to find the other endpoint:

The lines intersect where one equation = the other
#:.# where #3/2x+2=-2/3x+25/3#
#13/6x=19/3#
#x=38/13 # or #~~ 2.92#

Distance between point given #(x=8)# and midpoint found #(x=38/13)# is #66/13# or 5.08 units. So other endpoint of line is at #8-2*(66)/(13)=(-28)/(13)#, #=> y=(127/13)#