Question 334746
Two ways to do this.  Graphically and algebraically.
<pre><b>
Graphical method:

Call the midpoint {{{M(0,1/2)}}}

{{{drawing(400,1400/3,-3,3,-3,4, grid(1),
locate(0.15,1/2+.3,M(0,1/2)), locate(2.1,-1.7,A(2,-2)),
line(0,1/2,2,-2) )}}}

Draw a green line from A(2,-2) left horizontally until you 
are directly underneath the midpoint {{{M(0,1/2)}}}.  That 
requires you to draw the green line 2 units left.  

{{{drawing(400,1400/3,-3,3,-3,4, grid(1),
locate(0.15,1/2+.3,M(0,1/2)), locate(2.1,-1.7,A(2,-2)),
line(0,1/2,2,-2), green(line(2,-2,0,-2))

 )}}}

Now from the end of the green line, draw a vertical 
green line upward to point {{{M(0,1/2)}}}. That 
requires you to draw the vertical green line {{{2&1/2}}} units 
directly upward.  


{{{drawing(400,1400/3,-3,3,-3,4, grid(1),
locate(0.15,1/2+.3,M(0,1/2)), locate(2.1,-1.7,A(2,-2)),
line(0,1/2,2,-2), green(line(2,-2,0,-2), line(0,-2,0,1/2))

 )}}}

Draw a green line from {{{M(0,1/2)}}} left 2 units 
horizontally, the same number of units left that we
drew the other horizontal line. That puts you directly
above -2 on the x-axis.

{{{drawing(400,1400/3,-3,3,-3,4, grid(1),
locate(0.15,1/2+.3,M(0,1/2)), locate(2.1,-1.7,A(2,-2)),
line(0,1/2,2,-2), green(line(2,-2,0,-2), line(0,-2,0,1/2),
line(0,1/2,-2,1/2))

 )}}} 

Now from the end of that green line, draw a green line 
directly upward {{{2&1/2}}} units vertically, the same 
number of units left that we drew the other vertical 
line. That puts you directly above -2 on the x-axis.

{{{drawing(400,1400/3,-3,3,-3,4, grid(1),
locate(0.15,1/2+.3,M(0,1/2)), locate(2.1,-1.7,A(2,-2)),
line(0,1/2,2,-2), green(line(2,-2,0,-2), line(0,-2,0,1/2),
line(0,1/2,-2,1/2), line(-2,1/2,-2,3))

 )}}}

Now we can extend AM to B, and we now have AB.  And we see 
that B is the point B(-2,3)

{{{drawing(400,1400/3,-3,3,-3,4, grid(1),
locate(0.15,1/2+.3,M(0,1/2)), locate(2.1,-1.7,A(2,-2)),
line(-2,3,2,-2), green(line(2,-2,0,-2), line(0,-2,0,1/2),
line(0,1/2,-2,1/2), line(-2,1/2,-2,3)),
locate(-1.9,3.3,B(-2,3))

 )}}}

By algebra:

Use the midpoint formula:

The midpoint between {{{A(x[1],y[1])}}} and {{{B(x[2],y[2])}}}

is {{{M((x[1]+x[2])/2,(y[1]+y[2])/2)}}}

So

The midpoint between {{{A(2,-2)}}} and {{{B(x[2],y[2])}}}

is {{{M((2+x[2])/2,(-2+y[2])/2)}}}

So

{{{M((2+x[2])/2,(-2+y[2])/2)}}}{{{""=""}}}{{{M(0,1/2)}}}

So we set the x-coordinates equal:

{{{(2+x[2])/2}}}{{{""=""}}}{{{0}}}

Multiply both sides by 2

{{{2+x[2]}}}{{{""=""}}}{{{0}}}

{{{x[2]}}}{{{""=""}}}{{{-2}}}

And we set the y-coordinates equal:

{{{(-2+y[2])/2}}}{{{""=""}}}{{{1/2}}}

Multiply both sides by 2

{{{-2+y[2]}}}{{{""=""}}}{{{1}}}

{{{y[2]}}}{{{""=""}}}{{{3}}}

So B has the coordinates {{{B(-2,3)}}}

Edwin</pre>