# 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 ( 8 , 3 ), where is the other end?

Aug 29, 2017

Exact value: $\left(- \frac{28}{13} , \frac{127}{13}\right)$

Decimal value (-2.16 9.77)

#### Explanation:

Bisectors are perpendicular - they bisect the line at a right angle. This means that the gradient of the line segment is the reciprocal of the other line:
$4 y = 6 x + 8$
$y = \frac{3}{2} x + 2$ has a gradient of $\left(\frac{3}{2}\right)$
$\therefore$ unknown segment has a gradient of $\textcolor{b l u e}{- \frac{2}{3}}$

So our line segment has a point at $\textcolor{red}{8} , \textcolor{g r e e n}{3}$ and gradient $\textcolor{b l u e}{- \frac{2}{3}}$
y-y_1=m(x-x_1)
$\therefore y - \textcolor{g r e e n}{3} = \textcolor{b l u e}{- \frac{2}{3}} \left(x - \textcolor{red}{8}\right)$
$y - \textcolor{g r e e n}{3} = \textcolor{b l u e}{- \frac{2}{3}} x + \frac{\textcolor{red}{16}}{3}$
$y = \textcolor{b l u e}{- \frac{2}{3}} x + \frac{25}{3}$ is the equation of our line segment.

Now to find the other endpoint:

The lines intersect where one equation = the other
$\therefore$ where $\frac{3}{2} x + 2 = - \frac{2}{3} x + \frac{25}{3}$
$\frac{13}{6} x = \frac{19}{3}$
$x = \frac{38}{13}$ or $\approx 2.92$

Distance between point given $\left(x = 8\right)$ and midpoint found $\left(x = \frac{38}{13}\right)$ is $\frac{66}{13}$ or 5.08 units. So other endpoint of line is at $8 - 2 \cdot \frac{66}{13} = \frac{- 28}{13}$, $\implies y = \left(\frac{127}{13}\right)$