# A line segment is bisected by a line with the equation  - 2 y - 5 x = 2 . If one end of the line segment is at ( 8 , 7 ), where is the other end?

Mar 26, 2017

$\left(- \frac{328}{29} , - \frac{21}{29}\right) \approx \left(- 11.31 , - .724\right)$

#### Explanation:

1. Put the line in $y$-intercept form $y = m x + b$:
$- 2 y = 5 x + 2$
$y = - \frac{5}{2} x - 1$
$m = - \frac{5}{2}$
The perpendicular bisector slope $= - \frac{1}{m} = \frac{2}{5}$

2. Find the equation for perpendicular bisector the line with $\left(8 , 7\right)$:
$y = \frac{2}{5} x + b$
$7 = \frac{2}{5} \cdot \left(\frac{8}{1}\right) + b$
$7 = \frac{16}{5} + b$
$\frac{35}{5} - \frac{16}{5} = \frac{19}{5}$
$y = \frac{2}{5} x + \frac{19}{5}$

3. Find the midpoint (intersection point) of the two lines
$- \frac{5}{2} x - 1 = \frac{2}{5} x + \frac{19}{5}$
$- \frac{5}{2} x - \frac{2}{5} x = \frac{19}{5} + 1$
$- \frac{25}{10} x - \frac{4}{10} x = \frac{19}{5} + \frac{5}{5}$
$- \frac{29}{10} x = \frac{24}{5}$
$x = \frac{24}{5} \cdot - \frac{10}{29} = \frac{24}{1} \cdot - \frac{2}{29} = - \frac{48}{29}$
y = -5/2 * -48/29

$y = \left(- \frac{5}{1} \cdot - \frac{24}{29}\right) - \frac{29}{29}$
$y = \frac{120}{29} - \frac{29}{29} = \frac{91}{29}$
midpoint point $\left(- \frac{48}{29} , \frac{91}{29}\right) \approx \left(- 1.655 , 3.138\right)$ 1. Length from midpoint to $\left(8 , 7\right)$, 1/2 length of line segment:
$\sqrt{{\left(7 - \frac{91}{29}\right)}^{2} + {\left(8 - - \frac{48}{29}\right)}^{2}} = \sqrt{{\left(\frac{203}{29} - \frac{91}{29}\right)}^{2} + {\left(\frac{232}{29} + \frac{48}{29}\right)}^{2}} = \sqrt{{\left(\frac{112}{29}\right)}^{2} + {\left(\frac{280}{29}\right)}^{2}} = \sqrt{\frac{{112}^{2} + {280}^{2}}{{29}^{2}}} = \sqrt{\frac{3136}{29}} = \frac{56}{\sqrt{29}} = \frac{56 \sqrt{29}}{29} \approx 10.4$

2. Length of the line segment:
$2 \cdot \frac{56}{\sqrt{29}} = \frac{112}{\sqrt{29}} = \frac{112 \sqrt{29}}{29} \approx 20.8$

3. Use proportions to find the endpoint:
x/(280/29) = (112/sqrt(29))/(56/sqrt(29)); " "x = 560/29

y/(112/29) = (112/sqrt(29))/(56/sqrt(29)); " " y = 224/29 Endpoint $\left(8 - x , 7 - y\right) :$
$\left(8 - \frac{560}{29} , 7 - \frac{224}{29}\right) = \left(- \frac{328}{29} , - \frac{21}{29}\right) \approx \left(- 11.31 , - .724\right)$

CHECK using the line equation & finding length:

$- \frac{21}{29} = \frac{2}{5} \cdot - \frac{328}{29} + \frac{19}{5}$

$- \frac{21}{29} =$-656/145 + 551/145

$- \frac{21}{29} = - \frac{105}{145} = - \frac{21}{29}$

Length of 1/2 line from $\left(- \frac{328}{29} , - \frac{21}{29}\right)$ to midpoint $\left(- \frac{48}{29} , \frac{91}{29}\right)$:

$\sqrt{{\left(\frac{91}{29} - - \frac{21}{29}\right)}^{2} + {\left(- \frac{48}{29} - - \frac{328}{29}\right)}^{2}} = \sqrt{{\left(\frac{112}{29}\right)}^{2} + {\left(\frac{280}{29}\right)}^{2}} = \sqrt{\frac{90944}{29} ^ 2} = \sqrt{\frac{3136}{29}} = \frac{56}{\sqrt{29}}$