A line segment is bisected by line with the equation 3 y - 3 x = 1 . If one end of the line segment is at (2 ,5 ), where is the other end?

1 Answer
Dean R.
May 12, 2018

(14/3, 7/3)

Explanation:

The perpendicular to

-3x + 3y = 1

through (2,5) is

3x + 3y = 3(2)+3(5)=21

Let's find where these meet. Subtracting equations,

6x = 20

x = 10/3

Adding equations,

6y = 22

y = 11/3

So one endpoint is A(2,5) and the bisection point is B(10/3, 11/3). The other endpoint C satisfies:

C - B = B - A

C = 2B -A = (2(10/3) - 2, 2(11/3)-5) = (14/3, 7/3)

Check:

The midpoint is

B = 1/2 (A+C) = ( (2+14/3)/2, (5+7/3)/2) = (10/3, 11/3) quad sqrt

Check the midpoint B is on 3y-3x=1

3( 11/3) -3 (10/3) = 1 quad sqrt

Check perpendicularity. The y intercept is D(0,1/3).

Let's check (D-B) cdot(C -A)

(0 - 10/3, 1/3 - 11/3) cdot ( 14/3-2, 7/3-5)

= (0 - 10/3) ( 14/3-2) + ( 7/3-5)( 1/3 - 11/3) = 0 quad sqrt

Zero dot product means perpendicularity.

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