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?

Question

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?

Aurora Ayala · Accepted Answer

#(14/3, 7/3)#

The angle that is perpendicular to

#-3x + 3y = 1 #

through #(2,5)# is

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

Let's use subtraction to find the intersection of these.

#6x = 20#

#x = 10/3#

Combining formulas,

#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 halfway point 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 #

Perpendicularity is indicated by a zero dot product.

Parker Allen · Answer

undefined To find the other end of the line segment bisected by the line with the equation $3y - 3x = 1$, we first need to find the equation of the line segment. Since the line bisects the segment, it passes through the midpoint of the segment.

We can find the midpoint using the given endpoint $(2, 5)$ and the formula for the midpoint of a line segment. Then, we find the equation of the line passing through this midpoint and parallel to the given line. Finally, we find the intersection point of this line with the given line to get the other end of the line segment.

Here are the steps:

1. Find the midpoint using the given endpoint and the formula:
   $$ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

2. Find the slope of the given line $3y - 3x = 1$.

3. Since the line bisects the segment, the line passing through the midpoint should have the same slope as the given line.

4. Use the point-slope form of a line to find the equation of the line passing through the midpoint.

5. Find the intersection point of this line with the given line to determine the other end of the line segment.

Performing these calculations will provide the coordinates of the other end of the line segment.