Midpoint of a line segment

Question

The point A lies on the line y=2X and the point B lies on the line y=x+3. The coordinates of the mid-point of AB are (2,6). What are the coordinates of A and of B ?

 

Ans: A(5 , 10) B(-1 , 2)

Answer 1

Since we know the midpoint is equidistant from the two endpoints, we know that the x value of one of the endpoints will be a certain distance to the right of (2,6) and the other endpoint will be the same distance to the left of it.  The same applies for the y direction, one will be a certain distance above the midpoint, the other will be the same distance below it.  After graphing the equations we can see that the equation y = 2x will be the equation to the right and above.  Therefore if we say that the distance to the right of the midpoint equals x, then the distance to the left equals -x, if the distance above we call y, then the distance below is -y.  Now we can rewrite the endpoints as follows:

 

(2+x, 6+y) and (2-x, 6-y)

 

We then substitute these values into our two equations:

 

(6 + y) = 2(2 + x)          and           (6 - y) = (2 - x) + 3

  6 + y = 2x + 4             and              6 - y = 5 - x

        y = 2x - 2              and                   y = x + 1

 

Setting the 2 equations equal to each other, we get:

 

2x - 2 = x + 1

       x = 3

 

Since y = x - 1

         y = 4

 

Therefore one point is 3 right and 4 up from the midpoint or (2+3, 6+4) = (5, 10)

and the other point is 3 left and 4 down from the midpoint or (2-3, 6-4) = (-1, 2)

 

 

Answer 2

 

The problem which I have with this question is that the two lines intersect at (3,6).  The mid-point cannot be (2,6).  I know this is where the two lines intersect because...

2x=x+3

x=3

y=2x

y=(2)(3)

y=6

y=x+3

6=3+3

6=6

If I had what I believed to be the correct mid-point, I would proceed by using this point-of-intersection, as well as the mid-point, and I would establish the equation connecting the two.  Then, I would establish the equation perpendicular to this.  Then, I would establish the point at which each line connects with this perpendicular line.

 

 

 

 

 

 

 

Answer 3

 Y = 2x

 Y = 2X +3

 

    A( X_A , 2 X_{A )}   B  _{= (} X _{B ,}2X_B + 3 ) 

 

 

     ( X_A + X_B ) / 2 = 2

       

      ( 2X_{A +} X _B + 3 ) = 6

              2 

      -2 (X _A + X _{B = 4} )

        2X_{A +} X_{B  = 9} 

        - X_{B = 1}               X _{B = -1}  Y_{B = - 1 + 3 = 2}

                                X _{A = 5}   Y_{A = 2(5) =10}

 

   A (  5 ,10 )     B (- 1 , 2 )