A line segment is bisected by a line with the equation # 5 y + 2 x = 1 #. If one end of the line segment is at #(6 ,4 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #     5 y  +  2  x   =   1  #.  If one end of the line segment is at #(6  ,4  )#, where is the other end?

Savannah Aguilar · Accepted Answer

2nd endpoint: #(50/29, -194/29)#

Given: #5y + 2x = 1 " and line segment endpoint "(6,4)#.

A line segment that has been bisected is said to have been cut in half. The two lines will be perpendicular at the intersection point, which is the midpoint of the segment, meaning that their slopes will be negative reciprocals of one another.

#5y = -2x + 1; " "##y = -2/5x + 1/5#

Find the slope of the line segment #m_(segment) = -1/m = 5/2#

Find the equation of the line segment that goes through the point #(6, 4)#:

#y = 5/2 x + b#

#4 = 5/2 *6/1 + b#

#4 = 15 + b#

#b = 4 - 15 = -11#

#y_(segment) = 5/2 x - 11#

By making the two equations equal, find the midpoint of the line segment, or the intersection of the two lines:

#-2/5x + 1/5 = 5/2 x - 11 " Move like terms to the same side"#

#1/5 + 11 = 2/5x + 5/2x " Find common denominators"#

#1/5 + 11/1 * 5/5 = 2/5*2/2 x + 5/2 *5/5 x#

#56/5 = 29/10 x " Multiply by the reciprocal"#

#x_m = x_("midpoint") = 56/5 * 10/29 = 56/1 * 2/29 = 112/29#

#y_m = y_("midpoint") = 5/2 * 112/29 - 11/1 = 5/1 * 56/29 - 11/1 *29/29#

#y_m = -39/29#

To determine the second endpoint, apply the midpoint formula:

#(x_m, y_m) = ((x_1 + x_2)/2, (y_1 + y_2)/2)#

#(112/29, -39/29) = ((6 + x_2)/2, (4 + y_2)/2)#

#112/29 = (6+ x_2)/2; " " -39/29 = (4 + y_2)/2#

2nd endpoint: #(50/29, -194/29)#

CHECK:

The 2nd endpoint should lie on the line segment: #-194/29 = 5/2 * 50/29 - 11 #

# -194/29 = 5/1 * 25/29 - 11/1 * 29/29 #

#-194/29 = -194/29#

Each segment up to the halfway point ought to have the same length:

#d = sqrt((6 - 112/29)^2 + (4 - -39/29)^2) ~~ 5.756555483#

#d = sqrt((50/29 - 112/29)^2 + (-194/29- -39/29)^2) ~~ 5.756555483#

Parker Allen · Answer

undefined The other end of the line segment bisected by the line $5y + 2x = 1$ can be found by finding the intersection point of this line with the line segment. Given one end of the line segment at (6,4), we need to find the coordinates of the other end. First, we find the equation of the line segment using the midpoint formula, and then we find the intersection point of this line with $5y + 2x = 1$.

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

Using the midpoint formula with (6,4) and the unknown endpoint (let's denote it as $ (x, y) $), we get:
$$ \left( \frac{6 + x}{2}, \frac{4 + y}{2} \right) $$

Now, we substitute these values into the equation $5y + 2x = 1$ to find the coordinates of the other end of the line segment. 
$$ 5 \left( \frac{4 + y}{2} \right) + 2 \left( \frac{6 + x}{2} \right) = 1 $$

Solving this equation will give us the values of $x$ and $y$, which represent the coordinates of the other end of the line segment.