A line segment is bisected by a line with the equation # 4 y - 6 x = 8 #. If one end of the line segment is at #( 1 , 8 )#, where is the other end?

Question

A line segment is bisected by a line with the equation # 4  y  -  6  x   =   8  #.  If one end of the line segment is at #(    1  ,  8  )#, where is the other end?

Savannah Aguilar · Accepted Answer

#color(blue)((67/13,68/13)#

First rearrange #4y-6x=8# to the form #y=mx+b#

#y=3/2x+2 \ \ \ \[1]#

This will be perpendicular to the line through the point #(1,8)#

We need to find the equation of this line. We know that if two lines are perpendicular then the product of their gradients is #bb(-1)#

Gradient of #[1]# is: #3/2#

Let #bbm# be the gradient of the line through #(1,8)#

Then:

#m*3/2=-1=>m=-2/3#

Using point slope form of a line and point #(1,8)#:

#(y_2-y_1)=m(x_2-x_1)#

#y-8=-2/3(x-1)#

#y=-2/3x+26/3 \ \ \ [2]#

The intersection of #[1]# and #[2]# will be the midpoint of the line segment. Solving these simultaneously:

#3/2x+2=-2/3x+26/3=>x=40/13#

Substitute in #[1]#

#y=3/2(40/13)+2=86/13#

Co-ordinates of midpoint #(40/13,86/13)#

The midpoint's coordinates are provided by:

#((x_1+x_2)/2,(y_1+y_2)/2)#

So:

#((x_1+x_2)/2,(y_1+y_2)/2)=(40/13,86/13)#

If the end points are #(x_1,y_1)# and #(x_2,y_2)#

We have #(1,8)# and #(x_2,y_2)#

#((1+x_2)/2,(8+y_2)/2)=(40/13,86/13)#

Hence:

#(1+x_2)/2=40/13=>x_2=67/13#

and

#(8+y_2)/2=86/13=>y_2=68/13#

Thus, an additional endpoint is:

#color(blue)((67/13,68/13)#

Claire Akin · Answer

undefined To find the other end of the line segment bisected by the line $4y - 6x = 8$, we first need to determine the coordinates of the midpoint of the line segment.

The midpoint formula is:

$$ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

Given one end of the line segment as (1, 8), we'll use this point along with the midpoint formula to find the midpoint.

Substituting the given point (1, 8) into the midpoint formula:

$$ x_1 = 1, \ y_1 = 8 $$

We don't have the coordinates of the other end, so we denote them as (x, y). Now we have:

$$ \frac{1 + x}{2} = x $$
$$ \frac{8 + y}{2} = y $$

We can simplify these equations:

$$ 1 + x = 2x $$
$$ 8 + y = 2y $$

Rearranging terms:

$$ x = 1 $$
$$ y = 8 $$

Now we have found the coordinates of the midpoint of the line segment, which is (1, 8). Since the line segment is bisected, the other end of the line segment will also have the same coordinates as the midpoint. Therefore, the other end of the line segment is also at (1, 8).