A line segment is bisected by a line with the equation #  2 y  5 x = 2 #. If one end of the line segment is at #( 8 , 7 )#, where is the other end?
1 Answer
Explanation:

Put the line in
#y# intercept form#y = mx + b# :
#2y = 5x + 2#
#y = 5/2x 1#
#m = 5/2#
The perpendicular bisector slope#= 1/m = 2/5# 
Find the equation for perpendicular bisector the line with
#(8,7)# :
#y = 2/5x + b#
#7 = 2/5*(8/1) + b#
#7 = 16/5 + b#
#35/5  16/5 = 19/5#
#y = 2/5x + 19/5# 
Find the midpoint (intersection point) of the two lines
#5/2x 1 = 2/5x + 19/5#
#5/2x  2/5x = 19/5 + 1#
#25/10x  4/10 x = 19/5 + 5/5#
#29/10x = 24/5#
#x = 24/5 * 10/29 = 24/1 * 2/29 = 48/29#
#y = 5/2 * 48/29
midpoint point

Length from midpoint to
#(8,7)# , 1/2 length of line segment:
#sqrt((7  91/29)^2 + (8 48/29)^2) = sqrt ((203/29  91/29)^2 + (232/29 + 48/29)^2) = sqrt((112/29)^2 + (280/29)^2) = sqrt((112^2+280^2)/(29^2) ) = sqrt(3136/29) = 56/sqrt(29) = (56sqrt(29))/29~~ 10.4# 
Length of the line segment:
#2 * 56/sqrt(29) = 112/sqrt(29) = (112sqrt(29))/29 ~~ 20.8# 
Use proportions to find the endpoint:
#x/(280/29) = (112/sqrt(29))/(56/sqrt(29)); " "x = 560/29#
Endpoint
CHECK using the line equation & finding length:
Length of 1/2 line from