A line segment is bisected by line with the equation # 3 y - 3 x = 4 #. If one end of the line segment is at #(2 ,4 )#, where is the other end?

1 Answer
Douglas K.
Oct 31, 2016

The other end is at #(8/3, -2/3)#

Explanation:

Write the equation of the perpendicular bisector in slope-intercept form, #y = mx + b#:

#y = x + 4/3# [1]

The slope of the bisector is m = 1.

To find the slope of the bisected line, use the fact that the slopes of perpendicular are negative reciprocals of each other:

n = -1/m

Therefore, the slope of the bisected line is n = -1.

Use the slope-intercept form of the line, #y = nx + b#, the slope, #n = -1#, the given point, #(2,4)#, to write an equation that allows use to find the value of b:

#4 = -2 + b#

b = 6

The equation of the bisected line is:

#y = -x + 6# [2]

Because y = y, we can set the right side of equation [1] equal to the right side of equation [2]:

#x + 4/3 = -x + 6#

#2x = 14/3#

#x = 7/3#

This means that the line segment starts at 2, ( which is #6/3#), and goes to #7/3# to intersect with its bisector, therefore the other end of the segment must be twice that far , #8/3#

To find the corresponding y coordinate, substitute #8/3# for x in equation [2]

#y = -8/3 + 6#

#y = -2/3#

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