A line segment is bisected by a line with the equation # - 2 y - x = 2 #. If one end of the line segment is at #( 8 , 7 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #     - 2 y  -  x   =   2  #.  If one end of the line segment is at #(     8  ,  7  )#, where is the other end?

Autumn Armstrong · Accepted Answer

Coordinates of other endpoint (-13/5, -61/5)

Assumption : Its a perpendicular bisector. Slope of line = #m_1# #2y = -x - 2 #, Eqn (1) #y = (-1/2)x - 1# #m_1 =-( 1/2)# Slope of line segment (perpendicular) #m_2 = -1/m_1 = 2# Equation of line segment is #y - 7 = 2(x - 8)# #2x - y = 9#, Eqn (2) Solving Eqns (1) & (2), Midpoint coordinates #x = 16/5, y = -13/5# Other end point coordinates# (x_1, y_1)# #(x_1+8)/2 = 16/5# #x_1 = -(8/5) #(y_1 +7)/2 = -13/5# #y-1 = -(26/5) - 7 = -61/5#

Jayden Jackson · Answer

undefined To find the other end of the line segment bisected by the line with the equation $ -2y - x = 2 $, given that one end is at (8, 7), you can follow these steps:

1. Determine the slope of the line perpendicular to the given line. This can be done by rearranging the equation into slope-intercept form and taking the negative reciprocal of the coefficient of $ x $.

2. Find the midpoint of the line segment using the coordinates of the given endpoint and the slope of the perpendicular line.

3. Use the midpoint and the given endpoint to find the equation of the line passing through these two points.

4. Solve the equations of the given line and the line passing through the midpoint and the given endpoint to find the coordinates of the other end of the line segment.

Alternatively, you can use the midpoint formula directly to find the coordinates of the midpoint of the line segment and then use the distance formula to find the length of the line segment. Finally, use this length to find the coordinates of the other end of the line segment.

Let me know if you need further clarification or assistance with the calculations.