Question 319054
Can you find the coordinates of B given A(3,8) and M(5,4)? 
M is the midpoint of. 
<pre><b>
There are two ways.  We'll do both.

Graphically:

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),


green(line(3,8,5,4)),  locate(3,9,"A(3,8)"), locate(5.2,5,"M(5,4)") )}}}  

Draw the right triangle with one leg vertical and one leg 
horizontal that has AM as its hypotenuse:

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
line(3,4,3,8), line(3,4,5,4),

green(line(3,8,5,4)),  locate(3,9,"A(3,8)"), locate(5.2,5,"M(5,4)") )}}}

Notice that the vertical leg is 4 units and the horizontal leg is 2 units
long.  So duplicate that triangle below, and you have this, so you see that
M(5,4) is the midpoint between A(3,8) and B(7,0).

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
line(3,4,3,8), line(3,4,5,4),locate(7,1,"B(7,0)"),
line(5,4,5,0), line(5,0,7,0), green(line(5,4,7,0)),
green(line(3,8,5,4)),  locate(3,9,"A(3,8)"), locate(5.2,5,"M(5,4)") )}}}

Second way:

Let B have the coordinates B({{{x[2]}}}, {{{y[2]}}})

Then use the midpoint formula:

Midpoint between ({{{x[1]}}},{{{y[1]}}}) and  ({{{x[2]}}}, {{{y[2]}}})

is  {{{M(matrix(1,3,

           (x[1]+x[2])/2, ",", (y[1]+y[2])/2)) }}} = (5,4)

Let A(3,8) = ({{{x[1]}}},{{{y[1]}}})

so that {{{x[1]=3}}}, and {{{y[1]=8}}}

Let B have the coordinates B({{{x[2]}}}, {{{y[2]}}})

Then  

{{{(x[1]+x[2])/2 =  (3+x[2])/2 = 5}}}

and

{{{(y[1]+y[2])/2 =  (8+y[2])/2 = 4}}}

So you have the system of equations

{{{system(  (3+x[2])/2 = 5, (8+y[2])/2 = 4)}}}

Multiply both equations through by 2

{{{system(  3+x[2] = 10, 8+y[2] = 8)}}}

Solve each and get:

{{{system(x[2]=7,y[2]=0)}}}

So M(5,4) is the midpoint between A(3,8) and B(7,0).

Do it whichever way your teacher wants.

Edwin</pre>