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?

1 Answer
Jacob T.
Jun 18, 2017

(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)
![Generated using Mathematica 11. The blue line indicates the http://bisector.](https://useruploads.socratic.org/US54GFLcQAO0O1fgz7I8_Untitled-1.svg)
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