# 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 ( 7 , 3 ), where is the other end?

Mar 12, 2017

Any point on the line $\textcolor{p u r p \le}{2 y - 3 x = 23}$
or
if the given line $\textcolor{red}{4 y - 6 x = 8}$ is to be a perpendicular bisector then color(brown)(""(-23/15,46/5))

#### Explanation:

Apology:
Even omitting some details, this explanation is quite long.

Consider the vertical line $x = 7$ which passes through the given point $\left(7 , 3\right)$
This vertical line will intersect $\textcolor{red}{4 y - 6 x = 8}$ at $\left(7 , \frac{25}{2}\right)$, a point which is $\frac{25}{2} - 3 = \frac{19}{2}$ above the given point color(green)(""(7,3))
A point twice as far above color(green)(""(7,3)) would be at $\left(7 , 3 + 2 \times \frac{19}{2}\right) = \left(7 , 22\right)$
That is the line segment from color(green)(""(7,3)) to $\left(7 , 22\right)$ is bisected by color(red)("4y-6x=8) Furthermore, as we can see from similar triangles any line parallel to $\textcolor{red}{4 y - 6 x = 8}$ and through $\left(7 , 22\right)$ gives an infinite collection of points, any one of which would serve as an endpoint with color(green)(""(7,3)) for a line segment bisected by $\textcolor{red}{4 y - 6 x = 8}$ $\textcolor{red}{4 y - 6 x = 8}$ has a slope of $\frac{3}{2}$
so any line parallel to it must also have a slope of $\frac{3}{2}$
and
if such a line passes through $\left(7 , 22\right)$ then we can write its equation using the slope-point form:
$\textcolor{w h i t e}{\text{XXX}} y - 22 = \frac{3}{2} \left(x - 7\right)$
or simplified as
$\textcolor{w h i t e}{\text{XXX}} \textcolor{p u r p \le}{2 y - 3 x = 23}$

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It is possible, since this question was asked under the heading "Perpendicular Bisectors" that it was intended that $\textcolor{red}{4 y - 6 x = 8}$ should be the perpendicular bisector of a derived line segment.

In this case the perpendicular line to $\textcolor{red}{4 y - 6 x = 8}$ passing through color(green)(""(7,3))
would have a slope of $- \frac{2}{3}$ (the negative inverse of $\textcolor{red}{4 y - 6 x = 8}$ and (again, working through the slope-point form)
an equation of $\textcolor{b r o w n}{3 y + 2 x = 23}$ The system of equations:
$\textcolor{p u r p \le}{2 y - 3 x = 23}$
$\textcolor{b r o w n}{3 y + 2 x = 23}$
can be solved for the point of intersection: color(brown)(""(-23/15,46/5))