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?

1 Answer
TreeReader
Aug 29, 2017

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)

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