Question

A line segment is bisected by a line with the equation `4 y - 6 x = 8`. If one end of the line segment is at `( 8 , 3 )`, where is the other end?

Answer 1

Exact value: `(-28/13,127/13)`

Decimal value (-2.16 9.77)

Explanation:

Bisectors are perpendicular - they bisect the line at a right angle. This means that the gradient of the line segment is the reciprocal of the other line:
`4y=6x+8`
`y=3/2x+2` has a gradient of `(3/2)`
`:.` unknown segment has a gradient of `color(blue)(-2/3)`

So our line segment has a point at `color(red)(8),color(green)(3)` and gradient `color(blue)(-2/3)`
`y-y_1=m(x-x_1`)
`:. y-color(green)(3)=color(blue)(-2/3)(x-color(red)8)`
`y-color(green)(3)=color(blue)(-2/3)x+color(red)16/3`
`y=color(blue)(-2/3)x+25/3` is the equation of our line segment.

Now to find the other endpoint:

The lines intersect where one equation = the other
`:.` where `3/2x+2=-2/3x+25/3`
`13/6x=19/3`
`x=38/13` or `~~ 2.92`

Distance between point given `(x=8)` and midpoint found `(x=38/13)` is `66/13` or 5.08 units. So other endpoint of line is at `8-2*(66)/(13)=(-28)/(13)`, `=> y=(127/13)`