# A line segment is bisected by a line with the equation  4 y + 3 x = 2 . If one end of the line segment is at ( 2 , 1 ), where is the other end?

Jun 14, 2016

One possible other end would be at color(green)(""(2,-3))
The equation of all possible answers is $\textcolor{g r e e n}{2 x + 4 y = - 6}$
If the given equation is to be the perpendicular bisector: color(green)(""(2/25,-39/25))

#### Explanation:

$K : 4 y + 3 x = 2$ (the given bisecting line)

Let $H$ be a vertical line through the given end point $\left(2 , 1\right)$.
Since $H$ is a vertical line all values of $x \in H$ are equal to $2$
and $H$ intersects $K$ at $\left(2 , - 1\right)$

The distance from $\left(2 , 1\right)$ to the intersection point $\left(2 , - 1\right)$ is $2$ units.
Moving another $2$ units down $H$ would bring us to $\left(2 , - 3\right)$

$K$ bisects the line segment from $\left(2 , 1\right)$ to color(green)(""(2,-3))
i.e. color(green)(""(2,-3)) is one possible solution value to the given question.

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If we wanted the equation for all solution values
Consider a line $L$ drawn through the point $\left(2 , - 3\right)$ parallel to $K$

Note the labeling of Line Segment $H$ into $A C$ and $C D$
which we know from the previous work are in the ratio $1 : 1$

Consider any other arbitrary Line Segment $J$ from $A$ at $\left(2 , 1\right)$ to the Line $L$.
Note again the labeling that divides $J$ into segments $A B$ and $B C$

Since $\triangle A B C$ and $\triangle A D E$ are similar,
the ratio of $\left\mid A B \right\mid : \left\mid B D \right\mid = \left\mid A C \right\mid : \left\mid C E \right\mid = 1 : 1$

That is any arbitrary line segment connecting $\left(2 , 1\right)$ and line $L$ is bisected by $K$

Since $L$ has the same slope as $K$ and passes through $\left(2 , - 3\right)$
its equation is
$\textcolor{w h i t e}{\text{XXX}} y + 3 = \left(- \frac{3}{4}\right) \left(x - 2\right)$
or
$\textcolor{w h i t e}{\text{XXX}} 3 x + 4 y = - 6$

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If we wanted $K$ to be a $\underline{\textcolor{b l a c k}{\text{perpendicular}}}$ bisector of the line segment

Consider the line $M$ perpendicular to $K$ through $\left(2 , 1\right)$

Since the slope of $K$ is ${m}_{K} = - \frac{3}{4}$
the slope of $M$ must be ${m}_{M} = \frac{4}{3}$

and the equation of $M$ must be
$\textcolor{w h i t e}{\text{XXX}} \left(y + 3\right) = \frac{4}{3} \left(x - 2\right)$
or
$\textcolor{w h i t e}{\text{XXX}} 4 x - 3 y = 5$

The endpoint of this perpendicular line segment can be derived as the intersection of $M$ and $L$

$\left\{\begin{matrix}4 x - 3 y = 5 \\ 3 x + 4 y = - 6\end{matrix}\right.$

Which gives $\left(x , y\right) = \left(\frac{2}{25} , - \frac{39}{25}\right)$

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My apologies for the length of this solution.

If anyone can provide a complete solution more briefly, please post it.