A line segment is bisected by a line with the equation # - 4 y + x = 1 #. If one end of the line segment is at #( 7 , 2 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #  -  4 y  +   x   =   1  #.  If one end of the line segment is at #(     7  ,  2  )#, where is the other end?

Sadie Adams · Accepted Answer

The other end is at #(123/17, 18/17)#

Let's write the given line in the form #ax + by = c#

#x - 4y = 1" [1]"#

The general equation of lines perpendicular to this line is:

#4x + y = c#

To find the value of c, substitute 7 for x and 2 for y:

#4(7) + 2 = c#

#c = 30#

The equation of the bisected line segment is:

#4x + y = 30" [2]"#

The midpoint is at the intersection of of these two lines:

#x - 4y = 1" [1]"#
#4x + y = 30" [2]"#

Multiply equation [1] by 4 and subtract from equation [2]

#17y = 26#

#y = 26/17#

Let #Deltay = # the change in the y coordinate = #26/17 - 2#

The y coordinate of the other end of the line, #y_1# will have 2 twice the change from the starting y coordinate:

#y_1 = 2Deltay + y_0#

#y_1 = 2(26/17 - 2) + 2#

#y_1 = 52/17 - 2#

#y_1 = 18/17#

To find the corresponding x coordinate, substitute #18/17# for y into equation [2]:

#4x + 18/17 = 30#

#4x = 30 - 18/17#

#x_1 = 123/17#

The other end is at #(123/17, 18/17)#

Here is a graph of two lines and two points:

Evelyn Arboleda · Answer

undefined To find the other end of the line segment bisected by the line with the equation $ -4y + x = 1 $, given that one end is at (7, 2), follow these steps:

1. Find the midpoint of the line segment using the given point (7, 2) and the equation of the line.

2. Use the midpoint formula to find the coordinates of the midpoint.

3. Use the fact that the line bisects the segment to find the other end of the segment.

Here are the detailed steps:

1. Find the midpoint:
   - The midpoint formula for a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by:
     $$ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$
   - Given one end of the segment as (7, 2), and since the line bisects the segment, the midpoint lies on the line. Plug in these values into the equation $ -4y + x = 1 $ to find the $x$-coordinate of the midpoint.

2. Once you find the $x$-coordinate of the midpoint, substitute it back into the equation of the line to find the corresponding $y$-coordinate.

3. The coordinates of the other end of the line segment are now determined, consisting of the midpoint you found and the given endpoint (7, 2).

Perform these calculations to find the coordinates of the other end of the line segment bisected by the given line.