Question 1054654
*[illustration de5.JPG].
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Use the distance formula to find the distance between the points.
{{{a=sqrt((-3-1)^2+(0-7)^2)=sqrt(16+49)=sqrt(65)}}}
{{{b=sqrt((-3-5)^2+(0-4)^2)=sqrt(64+16)=sqrt(80)}}}
{{{c=sqrt((1-5)^2+(7-4)^2)=sqrt(16+9)=sqrt(25)=5}}}
Use Heron's formula,
{{{s=(a+b+c)/2=(sqrt(65)+sqrt(80)+5)/2}}}
{{{A=sqrt(s(s-a)(s-b)(s-c))}}}
{{{s-a=sqrt(65)/2+sqrt(80)/2+5/2-sqrt(65)=sqrt(80)/2+5/2-sqrt(65)/2}}}
{{{s-b=sqrt(65)/2+sqrt(80)/2+5/2-sqrt(80)=sqrt(65)/2+5/2-sqrt(80)/2}}}
{{{s-c=sqrt(65)/2+sqrt(80)/2+5/2-5=sqrt(65)/2+sqrt(80)/2-5/2}}}
Let 
{{{X=sqrt(65)/2}}}
{{{Y=sqrt(80)/2}}}
{{{Z=5/2}}}
{{{s=X+Y+Z}}}
{{{s-a=Y+Z-X}}}
{{{s-b=X+Z-Y}}}
{{{s-c=X+Y-Z}}}
So then,
{{{A=sqrt((X+Y+Z)(Y+Z-X)(X+Z-Y)(X+Y-Z))}}}
{{{A=sqrt(X^2(-X^2+2Y^2+2Z^2)-Y^4+Z^2(2Y^2-Z^2))}}}
Substituting,
{{{A=sqrt((65/4)(-65/4+2(80/4)+2(25/4))-6400/16+(25/4)(2(80/4)-25/4))
{{{A=sqrt((65/4)(-65/4+160/4+50/4)-6400/16+(25/4)(160/4-25/4))
{{{A=sqrt( (65/4)(145/4)-6400/16+(25/4)(135/4))
{{{A=sqrt(9425/16-6400/16+3375/16)}}}
{{{A=sqrt(6400/16)}}}
{{{A=sqrt(400)}}}
{{{A=20}}}