Question

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?

Answer 1

The other end is at `(8/3, -2/3)`

Explanation:

Write the equation of the perpendicular bisector in slope-intercept form, `y = mx + b`:

`y = x + 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 = nx + b`, the slope, `n = -1`, the given point, `(2,4)`, 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 + 4/3 = -x + 6`

`2x = 14/3`

`x = 7/3`

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

To find the corresponding y coordinate, substitute `8/3` for x in equation [2]

`y = -8/3 + 6`

`y = -2/3`