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

Noah Atkinson · Accepted Answer

Find the equation of a line that is perpendicular to the give line and goes through (7,2), then use the midpoint formula.

Find the slope of #-y+7x=1# by changing it to slope-intercept form: #y=7x-1#, #m=7# The slope of a line perpendicular to this is the negative reciprocal of 7 or #-1/7#, #m =-1/7# Use the point slope formula to find the equation of the line perpendicular to the given line that goes through (7,2). This is the equation of the line that includes the line segment. #y-y_1=m(x-x_1)# #y-2=-1/7(x -7)# #y-2=-1/7x +1# #y=-1/7x+3#, or #1/7x +y=3# Now find the intersection of the given line with the line that contains the line segment, by setting up the two equations as a system of equations and solving. #1/7x+y=3# #7x-y =1# Adding these two equations together gives #50/7x= 4# and #x=14/25# Plug this value of x into either of the equations to find y. The solution to the system is the intersection of the two lines and is #(14/25, 73/25)# Since a segment's perpendicular bisector passes through its midpoint, the intersection of the two lines represents the segment's midpoint. Next, apply the midpoint formula to determine the segment's other endpoint. Midpoint #(x, y) =( (x_1 +x_2)/2, (y_1 +y_2)/2)# #(14/25, 73/25) = ((7+x_2)/2, (2+y_2)/2)# #14/25 = (7+x_2)/2# and #73/25 = (2+y_2)/2# Cross multiply and solve. #(x_2, y_2) = (-147/25, 96/25)#

Ella Armstrong · Answer

undefined To find the other end of the line segment, we can first determine the coordinates of the point where the bisecting line intersects the line segment. Then, using these coordinates, we can calculate the coordinates of the other end of the line segment.

1. Find the slope of the bisecting line by rearranging its equation into slope-intercept form (y = mx + b).
   -y + 7x = 1
   -y = -7x + 1
   y = 7x - 1

2. The slope of the bisecting line is 7.

3. The negative reciprocal of the slope of the bisecting line gives the slope of the line segment, which is -1/7.

4. Use the point-slope form of the equation of a line (y - y1 = m(x - x1)), with the known point (7, 2) and the calculated slope (-1/7) to find the equation of the line containing the line segment.
   y - 2 = (-1/7)(x - 7)

5. Simplify the equation:
   y - 2 = (-1/7)x + 1
   y = (-1/7)x + 3

6. To find where this line intersects the bisecting line, substitute the equation of the bisecting line (-y + 7x = 1) into the equation of the line segment (y = (-1/7)x + 3).
   -((-1/7)x + 3) + 7x = 1

7. Solve for x:
   (1/7)x - 3 + 7x = 1
   (1/7 + 7)x = 4
   (1/7 + 49/7)x = 4
   (50/7)x = 4
   x = (4 * 7) / 50
   x = 0.56

8. Substitute x back into the equation of the line segment to find y:
   y = (-1/7)(0.56) + 3
   y = 3.2

The other end of the line segment is located at approximately (0.56, 3.2).