Question

A line segment is bisected by a line with the equation `- 3 y + 2 x = 2`. If one end of the line segment is at `( 7 , 9 )`, where is the other end?

Answer 1

(151/13,27/13)

Explanation:

Rewrite the equation of the bisector in the form `y=mx+b`
`y=2/3x - 2/3`
Therefore its gradient equals to `2/3`.
Since the gradient product of two perpendicular lines is `-1`, the slope of the unknown line segment should be `(-1)/(2/3)=-3/2`

Solve for the equation of the line segment (which includes the point `(7,9)`. `y=-3/2(x-7)+9`, simplify (optional) and you get `y=-3/2x+39/2`

Solve for the `x`-coordinate for the intersection of the two perpendicular lines: Let `-3/2(x-7)+9=2/3x - 2/3 ", " x=121/13`

Now you can find the `x`-coordinate of the other end of the segment. Using the formula `x_1-x=-(x_2-x)`, where `x` is the `x`-coordinate of the point in the middle. `x_1=7` and `x_2=2*x-x_1`, `x_2=151/13`

Solve for the `y`-coordinate using the equation for the line segment. `y_2=-3/2(x-7)+9=27/13`
Therefore coordinate for the other end point is `(151/13,27/13)`

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