A line segment is bisected by a line with the equation # 4 y - 2 x = 5 #. If one end of the line segment is at #( 7 , 3 )#, where is the other end?

Question

A line segment is bisected by a line with the equation # 4  y  -  2  x   =   5  #.  If one end of the line segment is at #(    7  ,  3  )#, where is the other end?

Allison Allen · Accepted Answer

The other end is #(28/5,29/5)#

Given: #4y-2x=5" [1]"# is the equation of the bisector

Determine the bisector's slope:

#y = 2/4x+5/4#

#y = 1/2x+5/4#

The slope is #m_1= 1/2#

The line that is bisected is:

#m_2 = -1/m_1#

#m_2 = -1/(1/2)#

#m_2=-2#

To determine the equation of the bisected line, use the point-slope form of the equation of a line:

#y = m(x-x_0)+y_0#

#y = -2(x-7)+3#

#y = -2x+14+3#

#y = -2x+17#

#y + 2x= 17" [2]"#

Combine equations [2] and [1] as follows:

#4y+y-2x+2x=5+17#

#5y = 22#

#y = 22/5#

To determine the corresponding value of x, use equation [2]:

#22/5+2x=17#

#2x = 63/5#

#x = 63/10#

Apply the midpoint calculations:

#x_"midpoint" = (x_"end"+x_"start")/2#

#y_"midpoint" = (y_"end"+x_"start")/2#

Substitute #63/10# for #x_"midpoint"# and 7 for #x_"start"#:

#63/10 = (x_"end"+7)/2#

Substitute #22/5# for #y_"midpoint"# and 3 for #y_"start"#:

#22/5 = (y_"end"+3)/2#

Solve for #x_"end" and y_"end"#

#63/10 = (x_"end"+7)/2# #22/5 = (y_"end"+3)/2#

#x_"end" = 63/5-7# #y_"end"= 44/5-3#

#x_"end" = 28/5# #y_"end"= 29/5#

The other end is #(28/5,29/5)#

Claire Akin · Answer

undefined To find the other end of the line segment bisected by the line $4y - 2x = 5$, given one end at (7, 3), we can use the midpoint formula.

The midpoint formula states that for two endpoints $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint is given by $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

First, we need to find the coordinates of the midpoint. Then, knowing one end point and the midpoint, we can find the other end point by doubling the coordinates of the midpoint and subtracting the coordinates of the known end point.

Given the line equation $4y - 2x = 5$, we need to find the intersection point of this line with the line segment. We already have one point on the segment, (7, 3), so let's find the point of intersection by solving the system of equations formed by the line $4y - 2x = 5$ and the line containing the segment:

$$
\begin{cases}
4y - 2x = 5 \
y = \frac{1}{2}x - \frac{5}{2}
\end{cases}
$$

Solve this system to find the coordinates of the intersection point.

Once you have the coordinates of the intersection point, use them and the given point (7, 3) to find the midpoint. Then, double the coordinates of the midpoint and subtract the coordinates of the known point to find the other end of the line segment.