Question 345491
A:({{{0}}},{{{6}}})
B:({{{4}}},{{{0}}})
C:({{{x}}},{{{y}}})
Assume that the final vertex is located at({{{x}}},{{{y}}}).
Then the median connecting ({{{x}}},{{{y}}}) to the midpoint of {{{AB}}}, located at ({{{2}}},{{{3}}}), has a slope,since it also goes through ({{{0}}},{{{2}}}), of,
{{{m=(3-2)/(2-0)=1/2}}}
and a y-intercept of {{{2}}}.
1.{{{y=(1/2)x+2}}}
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The median connecting ({{{4}}},{{{0}}}),({{{0}}},{{{2}}}), and the midpoint of {{{AC}}} has a slope of,
{{{m=(2-0)/(0-4)=-1/2}}}
and a y-intercept of {{{2}}}.
{{{y=-(1/2)x+2}}}
The midpoint of {{{AC}}} is ({{{(x+0)/2}}},{{{(y+6)/2}}}) or ({{{x/2}}},{{{y/2+3}}}).
It also satisfies the line,
{{{y=-(1/2)x+2}}}
{{{y/2+3=-(1/2)(x/2)+2}}}
{{{2y+12=-x+8}}}
2.{{{x+2y=-4}}}
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From eq. 1,
{{{2y=x+4}}}
Substituting into eq. 2,
{{{x+x+4=-4}}}
{{{2x=-8}}}
{{{highlight(x=-4)}}}
Then 
{{{2y=-4+4}}}
{{{2y=0}}}
{{{highlight(y=0)}}}
The final vertex is located at ({{{-4}}},{{{0}}})