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:
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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
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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