A line segment is bisected by a line with the equation # 5 y -4 x = 1 #. If one end of the line segment is at #(3 ,8 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #     5 y  -4  x   =   1  #.  If one end of the line segment is at #(3  ,8  )#, where is the other end?

Evelyn Arboleda · Accepted Answer

The other end is at the point #(7,3)#

Since it is assumed that the bisector and the bisected line are perpendicular, we can determine the slope of the bisected line using the bisector's slope.

To enable us to observe the slope, write the equation for the bisector line in slope-intercept form:

#y = 4/5x + 1# [1]

The slope #m = 4/5#

The slope, n, of the bisected line is the negative reciprocal, #n = -5/4#

Substitute the slope and the point, #(3,8)#, and the slope, #-5/4# into the slope-intercept form of the equation of a line, #y = nx + b# and then solve for b:

#8 = -5/4(3) + b#

#b = 8 + 15/4#

#b = 47/4#

The bisected line's equation is as follows:

#y = -5/4x + 47/4# [2]

Equation [1] is subtracted from equation [2]:

#y - y = -5/4x - 4/5x + 47/4 - 1#

#0 = -5/4x - 4/5x + 47/4 - 1#

#0 = -41/20x + 43/4#

#41/20x = 43/4#

#x = 215/43 #

The intersection's x coordinate is:

#x = 5#

The starting point is two units to the right of the point of intersection, and an extra two units must separate the ending point:

#x = 7#

Enter 7 in place of x in equation [2]:

#y = -5/4(7) + 47/4#

#y = 3#

Sophia Alfred · Answer

undefined The other end of the line segment bisected by the line with the equation $5y - 4x = 1$ can be found by using the midpoint formula. Given that one end of the line segment is at the point (3, 8), and since the line bisects the segment, the midpoint of the segment is the same as the point (3, 8). Therefore, the other end of the line segment is also at the point (3, 8).