# A line segment is bisected by a line with the equation  2 y - 2 x = 2 . If one end of the line segment is at ( 3 , 8 ), where is the other end?

Jul 22, 2017

The other end of the segment is $= \left(7 , 4\right)$

#### Explanation:

The equation of the line is

$y - x = 1$

$y = x + 1$.......................$\left(1\right)$

The slope of the line is $m = 1$

The slope of the segment is $m '$

$m m ' = - 1$

$m ' = - 1$

The equation of the segment is

$y - 8 = - \left(x - 3\right)$

$y = - x + 3 + 8$

$y = - x + 11$.......................$\left(2\right)$

Solving for $x$ and $y$ in the equations $\left(1\right)$ and $\left(2\right)$ gives the point of intersection of the line and the segment

$x + 1 = - x + 11$

$2 x = 10$

$x = 5$

$y = 5 + 1 = 6$

The point of intersection is $= \left(5 , 6\right)$

Let the other end of the segment be $= \left(a , b\right)$

Therefore,

$\left(5 , 6\right) = \left(\frac{a + 3}{2} , \frac{b + 8}{2}\right)$

$\frac{a + 3}{2} = 5$

$a = 10 - 3 = 7$

$\frac{b + 8}{2} = 6$

$b = 12 - 8 = 4$