A line segment is bisected by a line with the equation - y + 3 x = 1 . If one end of the line segment is at (6 ,3 ), where is the other end?

1 Answer
Somebody N.
May 24, 2018

color(blue)((-12/5,29/5)

Explanation:

We know that the line -y+3x=1 and the line containing the point (6.3) are perpendicular. If two lines are perpendicular then the product of their gradients is -1

-y+3x=1=>y=3x-1 \ \ \ [1]

This has a gradient of 3. The line containing (6,3) therefore has a gradient:

m*3=-1=>m=-1/3

Finding the equation of this line using point slope method:

y-3=-1/3(x-6)=>y=-1/3x+5 \ \ \ [2]

Solving [1] and [2] simultaneously:

-1/3x+5-3x+1=0=>x=9/5

Substitute in [1]

y=3(9/5)-1=22/5

(9/5,22/5) are the coordinates of the midpoint.

Coordinates of the midpoint are found using:

((x_1+x_2)/2,(y_1+y_2)/2)

Let the unknown endpoint be (x_2,y_2)

Then:

((6+x_2)/2,(3+y_2)/2)->(9/5,22/5)

(6+x_2)/2=9/5=>x_2=-12/5

(3+y_2)/2=22/5=>y_2=29/5

Coordinates of the other endpoint are:

(-12/5,29/5)

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