A line segment is bisected by a line with the equation # - 6 y + 5 x = 4 #. If one end of the line segment is at #( 2 , 5 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #  - 6  y  + 5  x   =   4  #.  If one end of the line segment is at #(    2  ,  5  )#, where is the other end?

Sophia Alfred · Accepted Answer

coordinates of the other end point #color (purple)(362/61, 17/61)#

The bisecting line is assumed to be a perpendicular bisector.

Standard form of equation #y=mx +c# Slope of perpendicular bisector m is given by #-6y + 5x= 4# #y= (5/6)x - (2/3)# #m = (5/6)#

Slope of line segment is #y - 5 = -(1/m)(x-2)# #y - 5= (-6/5)(x - 2)# #5y - 25 = -6x + 12#

#5y + 6x = 37 color (white)((aaaa)# Eqn (1) #-6y + 5x = 4 color (white)((aaaa)# Eqn (2)

Equations (1) and (2) solved,

#x= color (green)(242/61)#

#y= color (green)(161/61)#

Mid point #color(green)(242/61, 161/61)#

Let (x1,y1) the other end point. #(2+x1)/2= 242/37# #x1= (484/61) - 2 = color(red)( 362/61)# #(5+y1)/2= 161/61# #y1=(322/61) - 5 color(red)(17/61)#

Coordinates of other end point #color(red)(362/61, 17/61)#

Carson Arnold · Answer

undefined To find the other end of the line segment bisected by the line $ -6y + 5x = 4 $, you first need to find the midpoint of the line segment, which will also be on the given line. Then, you can use the midpoint formula to find the coordinates of the other end of the line segment.

1. Find the midpoint of the line segment using the given point and the equation of the bisecting line.
2. Once you have the midpoint, use it to determine the direction and distance to the other end of the line segment.

Here's how to do it:

1. Substitute the given coordinates $ (2, 5) $ into the equation of the bisecting line $ -6y + 5x = 4 $ to find the midpoint.
2. Solve the equation $ -6y + 5x = 4 $ for $ y $ to express $ y $ in terms of $ x $.
3. Substitute $ x = 2 $ into the expression you found for $ y $ to get the $ y $-coordinate of the midpoint.
4. You now have the coordinates of the midpoint. Let's denote them as $ (x_m, y_m) $.
5. Use the midpoint formula to find the coordinates of the other end of the line segment.
   - The midpoint formula is: $ (x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) $, where $ (x_1, y_1) $ and $ (x_2, y_2) $ are the coordinates of the endpoints of the line segment.
6. Substitute the coordinates of the given endpoint $ (2, 5) $ and the midpoint $ (x_m, y_m) $ into the midpoint formula to find the coordinates of the other endpoint.

Once you have followed these steps, you'll have the coordinates of the other end of the line segment.