# A line segment is bisected by a line with the equation  8 y + 5 x = 4 . If one end of the line segment is at ( 1 , 3 ), where is the other end?

Oct 13, 2017

Other end coordinates $\left(- 1.809 , - 1.4944\right)$

#### Explanation:

Assumption: It’s a perpendicular bisector that intersects the line segment.
Slope of perpendicular bisector m1;
$y = - \left(\frac{5}{8}\right) x + \left(\frac{1}{2}\right)$
$m 1 = - \left(\frac{5}{8}\right)$

Slope of line segment $m 2 = - \left(\frac{1}{m} 1\right) = - \left(\frac{1}{-} \left(\frac{5}{8}\right)\right) = \frac{8}{5}$
Equation of line segment :
$y - 3 = \left(\frac{8}{5}\right) \left(x - 1\right)$
$5 y - 15 = 8 x - 8$
$5 y - 8 x = 7 \textcolor{w h i t e}{a a a a a} E q n \left(1\right)$
$8 y + 5 x = 4 \textcolor{w h i t e}{a a a a a} E q n \left(2\right)$

Solving Eqns (1) & (2) to get the midpoint coordinates,
$25 y - 40 x = 35$
$64 y + 40 x = 32$ color(white)(aaa) Adding both,
$89 y = 67 \mathmr{and} y = \frac{67}{89} = 0.7528$
$x = \frac{4 - \left(8 \cdot \left(\frac{67}{89}\right)\right)}{5} = - 0.4045$
Midpoint coordinates (-0.4045,0.7528)

—0.4045=(x1+1)/2, $0.7528 = \frac{y 1 + 3}{2}$
$x 1 = - 1.809 , \textcolor{w h i t e}{a a a a a} y 1 = - 1.4944$