A line segment is bisected by a line with the equation # - 3 y + 2 x = 2 #. If one end of the line segment is at #( 7 , 9 )#, where is the other end?

1 Answer
Jacob T.
Jun 18, 2017

(151/13,27/13)

Explanation:

Rewrite the equation of the bisector in the form #y=mx+b#
#y=2/3x - 2/3#
Therefore its gradient equals to #2/3#.
Since the gradient product of two perpendicular lines is #-1#, the slope of the unknown line segment should be #(-1)/(2/3)=-3/2#

Solve for the equation of the line segment (which includes the point #(7,9)#. #y=-3/2(x-7)+9#, simplify (optional) and you get #y=-3/2x+39/2#

Solve for the #x#-coordinate for the intersection of the two perpendicular lines: Let #-3/2(x-7)+9=2/3x - 2/3 ", " x=121/13#

Now you can find the #x#-coordinate of the other end of the segment. Using the formula #x_1-x=-(x_2-x)#, where #x# is the #x#-coordinate of the point in the middle. #x_1=7# and #x_2=2*x-x_1#, #x_2=151/13#

Solve for the #y#-coordinate using the equation for the line segment. #y_2=-3/2(x-7)+9=27/13#
Therefore coordinate for the other end point is #(151/13,27/13)#
Generated using Mathematica 11. The blue line indicates the bisector.
https://www.mathsisfun.com/equation_of_line.html

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