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

Oct 31, 2016

The other end is at $\left(\frac{8}{3} , - \frac{2}{3}\right)$

#### Explanation:

Write the equation of the perpendicular bisector in slope-intercept form, $y = m x + b$:

$y = x + \frac{4}{3}$ [1]

The slope of the bisector is m = 1.

To find the slope of the bisected line, use the fact that the slopes of perpendicular are negative reciprocals of each other:

n = -1/m

Therefore, the slope of the bisected line is n = -1.

Use the slope-intercept form of the line, $y = n x + b$, the slope, $n = - 1$, the given point, $\left(2 , 4\right)$, to write an equation that allows use to find the value of b:

$4 = - 2 + b$

b = 6

The equation of the bisected line is:

$y = - x + 6$ [2]

Because y = y, we can set the right side of equation [1] equal to the right side of equation [2]:

$x + \frac{4}{3} = - x + 6$

$2 x = \frac{14}{3}$

$x = \frac{7}{3}$

This means that the line segment starts at 2, ( which is $\frac{6}{3}$), and goes to $\frac{7}{3}$ to intersect with its bisector, therefore the other end of the segment must be twice that far , $\frac{8}{3}$

To find the corresponding y coordinate, substitute $\frac{8}{3}$ for x in equation [2]

$y = - \frac{8}{3} + 6$

$y = - \frac{2}{3}$