A line segment is bisected by a line with the equation # - 2 y - 5 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  - 5 x   =   2  #.  If one end of the line segment is at #(     8  ,  7  )#, where is the other end?

Ella Armstrong · Accepted Answer

#(-328/29, -21/29) ~~(-11.31, -.724)#

Put the line in #y#-intercept form #y = mx + b#:
#-2y = 5x + 2#
#y = -5/2x -1#
#m = -5/2#
The perpendicular bisector slope #= -1/m = 2/5#
Find the equation for perpendicular bisector the line with #(8,7)#:
#y = 2/5x + b#
#7 = 2/5*(8/1) + b#
#7 = 16/5 + b#
#35/5 - 16/5 = 19/5#
#y = 2/5x + 19/5#
Find the midpoint (intersection point) of the two lines
#-5/2x -1 = 2/5x + 19/5#
#-5/2x - 2/5x = 19/5 + 1#
#-25/10x - 4/10 x = 19/5 + 5/5#
#-29/10x = 24/5#
#x = 24/5 * -10/29 = 24/1 * -2/29 = -48/29#
#y = -5/2 * -48/29

#y = (-5/1 * -24/29) - 29/29#
#y = 120/29 - 29/29 = 91/29#
midpoint point #(-48/29, 91/29) ~~(-1.655, 3.138)#

Length from midpoint to #(8,7)#, 1/2 length of line segment:
#sqrt((7 - 91/29)^2 + (8 --48/29)^2) = sqrt ((203/29 - 91/29)^2 + (232/29 + 48/29)^2) = sqrt((112/29)^2 + (280/29)^2) = sqrt((112^2+280^2)/(29^2) ) = sqrt(3136/29) = 56/sqrt(29) = (56sqrt(29))/29~~ 10.4#
Length of the line segment:
#2 * 56/sqrt(29) = 112/sqrt(29) = (112sqrt(29))/29 ~~ 20.8#
Use proportions to find the endpoint:
#x/(280/29) = (112/sqrt(29))/(56/sqrt(29)); " "x = 560/29#
#y/(112/29) = (112/sqrt(29))/(56/sqrt(29)); " " y = 224/29#

Endpoint #(8 -x, 7 - y): #
#(8-560/29, 7-224/29) = (-328/29, -21/29) ~~(-11.31, -.724)#
CHECK using the line equation & finding length:

#-21/29 = 2/5 * -328/29 + 19/5#
#-21/29 = #-656/145 + 551/145#

#-21/29 = -105/145 = -21/29#
Length of 1/2 line from #(-328/29, -21/29)# to midpoint #(-48/29, 91/29)#:

#sqrt((91/29 - -21/29)^2 + (-48/29 --328/29)^2) = sqrt((112/29)^2 + (280/29)^2) = sqrt(90944/29^2) = sqrt(3136/29) = 56/sqrt(29)#

Savannah Aguilar · Answer

undefined The other end of the line segment bisected by the line with the equation $ -2y - 5x = 2 $ can be found using the midpoint formula. Since the line bisects the line segment, the midpoint of the segment lies on the line. The given endpoint is at (8, 7). We'll use this information to find the coordinates of the other endpoint.

First, let's find the midpoint of the line segment, which is also on the line $ -2y - 5x = 2 $. Then, we'll use the midpoint formula to find the coordinates of the other endpoint.

Midpoint formula:
$$ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

Given endpoint: $ (8, 7) $

Now, let's solve the equation $ -2y - 5x = 2 $ for y to find the y-coordinate of the midpoint.

$$ -2y = 5x + 2 $$
$$ y = \frac{-5x - 2}{2} $$

Substitute $ x = 8 $ into the equation to find the y-coordinate of the midpoint.

$$ y = \frac{-5(8) - 2}{2} $$
$$ y = \frac{-40 - 2}{2} $$
$$ y = \frac{-42}{2} $$
$$ y = -21 $$

So, the midpoint is at $ (8, -21) $.

Now, we'll use the midpoint and the given endpoint to find the coordinates of the other endpoint.

Let the coordinates of the other endpoint be $ (x, y) $.

Midpoint: $ (8, -21) $
Given endpoint: $ (8, 7) $

Using the midpoint formula:

$$ x = \frac{8 + x}{2} $$
$$ y = \frac{-21 + y}{2} $$

Solving these equations, we find:

$$ x = 8 $$
$$ y = -49 $$

So, the other end of the line segment is at $ (8, -49) $.