A line segment is bisected by a line with the equation 5 y -4 x = 1 . If one end of the line segment is at (3 ,4 ), where is the other end?

1 Answer
Narad T.
May 11, 2017

The other end is (179/41,94/41)

Explanation:

Let's rewrite the equation of the line

5y-4x=1

5y=4x+1

y=4/5x+1/5

The slope is

m=4/5

The slope of a line perpendicular is m'=-5/4

as m*m'=-1

The equation of the segment is

y-4=-5/4(x-3)

y=-5/4x+15/4+4=-5/4x+31/4

The point of intersection of the lines

y=4/5x+1/5=-5/4x+31/4

(4/5+5/4)x=-1/5+31/4

41/20x=161/20

x=151/41

y=4/5*151/41+1/5

y=(604+41)/205

y=645/205=129/41

The point of intersection is (151/41,129/41)

Let the other end of the segment be x_1,y_1

Then,

x_1-151/41=151/41-3

x_1=151/41+151/41-3=179/41

and

y_1-129/41=129/41-4

y_1=129/41+129/41-4=94/41

The other end is (179/41,94/41)
graph{(y-4/5x-1/5)(y+5/4x-31/4)((x-3)^2+(y-4)^2-0.01)((x-179/41)^2+(y-94/41)^2-0.01)=0 [-1.985, 9.116, -0.99, 4.56]}

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