A line segment is bisected by a line with the equation # - 3 y + 2 x = 5 #. If one end of the line segment is at #( 7 , 9 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #  - 3  y  + 2  x   =   5  #.  If one end of the line segment is at #(    7  ,  9  )#, where is the other end?

Sophia Alfred · Accepted Answer

First, find the midpoint of the segment by determining where the 2 lines intersect. From that, you can determine that the other end of the segment is at #P_2(163/13, 9/13)#

Given that this section's title refers to a perpendicular bisector, I'm assuming that the line is one.

As we are aware of one of the segment's endpoints, we can ascertain the other end by locating the midpoint, which is the point at which the line intersects the segment.

The equation of the bisecting line is: #-3y+2x=5# #3y=2x-5# #y=2/3x-5/3# From this we can see that the slope of this line is: #2/3#

If this line is perpendicular to the segment, then the slope of the segment is: #m=-3/2# (the negative reciprocal of the other one)

Now we have a point #(7,9)#, and the slope of the segment, so the line that extends this segment is given by: #y-y_1=m(x-x_1)# #y-9=-3/2(x-7)=-3/2x+21/2# #y=-3/2x+21/2+9# #y=-3/2x+39/2#

At the point where they intercect, the 2 lines have the same x and y values. Let's work with the y values being equal: #y=2/3x-5/3=-3/2x+39/2# #2/3x+3/2x=39/2+5/3# #13/6x=127/6# #13x=127# So the lines intercect at #x=127/13#

And that is the #x# coordinate of the midpoint of the segment.

The #x# value of the midpoint is the average of the 2 endpoints. So now we can determine the other endpoint: #x_m=(x_1+x_2)/2#

#127/13=(7+x_2)/2# #7+x_2=127/13*2# #x_2=254/13-7=254/13-91/13=163/13#

We use this #x# value to find the #y# value in the equation of the segment: #y_2=-3/2x_2+39/2# #y=-3/2(163/13)+39/2 = -489/26+39/2=-489/26+507/26# #y=18/26=9/13#

So, the other endpoint is: #P_2(163/13, 9/13)#

Claire Akin · Answer

undefined To find the other end of the line segment bisected by the line with the equation $ -3y + 2x = 5 $ when one end is at $(7, 9)$, follow these steps:

1. Find the slope of the given line by rearranging the equation into slope-intercept form ($y = mx + b$).
2. Since the line bisects the segment, the midpoint of the segment lies on the line.
3. Use the midpoint formula to find the coordinates of the midpoint.
4. Use the given endpoint and the midpoint to find the coordinates of the other end of the line segment.

By substituting the given endpoint $(7, 9)$ into the equation $ -3y + 2x = 5 $, you can find the slope of the line.

$$ -3(9) + 2(7) = 5 $$
$$ -27 + 14 = 5 $$
$$ -13 = 5 $$

The slope is $-13$.

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

Substitute the coordinates of the given endpoint $(7, 9)$ into the midpoint formula along with the slope-intercept equation to find the coordinates of the other end of the line segment.