A line segment is bisected by a line with the equation # 4 y + x = 8 #. If one end of the line segment is at #(5 ,6 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #     4 y  +    x   =   8  #.  If one end of the line segment is at #(5  ,6  )#, where is the other end?

Avery Allen · Accepted Answer

Any point on the line #color(red)(4y+x=-17)# together with #(5,6)# will provide end points for a line segment bisected by #4y+x=8#

A vertical line through #color(green)(""(5,6))# will intersect #color(purple)(4y+x=8)#
at #color(purple)(""(5,3/4))#

#color(purple)(""(5,3/4))# is #5 1/4# units below #color(green)(""(5,6)#

If #color(purple)(""(5,3/4))# is the bisect point of the vertical line,
the second end point must be #5 1/4# units lower than #color(purple)(""(5,3/4))#
That is the second end point must be at #color(red)(""(5,-5 1/2))#

The given line #color(purple)(4y+x=8)# bisects the vertical line segment between #color(green)(""(5,6))# and #color(red)(""(5,-5 1/2))#

Furthermore (based on similar triangles) any point on a line parallel to the bisector line #color(purple)(4y+x=8)# will also be bisected by #color(purple)(4y+x=8)#

#color(purple)(4y+x=8)# has a slope of #(-1/4)#
So all lines parallel to #color(purple)(4y+x=8)# will have a slope of #(-1/4)#

The equation for a line passing through #color(red)(""(5,-5 1/2))# with a slope of #(-1/4)# can be expressed in slope-point form as
#color(white)("XXX")color(red)(y- (-5 1/2) = -1/4(x-5))#
or after simplifying, as
#color(white)("XXX")color(red)(4y+x=-17)#

Sophia Alfred · Answer

undefined To find the other end of the line segment bisected by the line $4y + x = 8$ given that one end is at (5, 6), we can first find the equation of the line segment using the midpoint formula. Then we can find the point of intersection between this line and the given line $4y + x = 8$, which would give us the other end of the line segment.

First, let's find the midpoint of the line segment using the given point (5, 6) and the unknown point (let's call it $P$):

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

Substituting the given point (5, 6) and the unknown point $P$ into the midpoint formula, we have:
$$ \left(\frac{5 + x}{2}, \frac{6 + y}{2}\right) $$

Since the line segment is bisected by the line $4y + x = 8$, the midpoint must also lie on this line. So we can substitute the midpoint coordinates into the equation of the given line to find $P$.

$$ 4\left(\frac{6 + y}{2}\right) + \left(\frac{5 + x}{2}\right) = 8 $$

Solving this equation will give us the coordinates of point $P$, which represents the other end of the line segment.

$$ 4(6 + y) + (5 + x) = 16 $$

$$ 24 + 4y + 5 + x = 16 $$

$$ 4y + x = 8 $$

This equation is the same as the given equation of the line. Therefore, point $P$ will have the coordinates (5, 6).

So, the other end of the line segment is also at (5, 6).