Question 1189504
The vertices of a triangle are at 
A({{{-3}}}, {{{-2}}})
B({{{2}}}, {{{1}}}) 
C({{{6}}}, {{{5}}}) 


the length of side AB or {{{c}}} is equal to the distance between A and B

{{{c=sqrt((2-(-3))^2+(1-(-2))^2))}}}
{{{c=sqrt(34)}}}
{{{c=5.831}}}


height to {{{c}}} lie on a line perpendicular to AB and passes through C

use coordinates of A and B   first to find a slope of the line passing through A and B 


{{{m=(1+2)/(2+3)=3/5}}}


then line is

{{{y-y[1]=(3/5)(x-x[1])}}}........use slope and ({{{2}}}, {{{1}}}) 
{{{y-1=(3/5)(x-2)}}}
{{{y=(3/5)x-2(3/5)+1}}}
{{{y=(3/5)x - 1/5}}}


then, a line {{{perpendicular}}} to AB will be {{{m= -1/(3/5)=-5/3}}}


{{{y-y[1]=(-5/3)(x-x[1])}}}.......use slope and ({{{6}}}, {{{5}}}) 
{{{y-5=(-5/3)(x-6)}}}
{{{y=(-5/3)(x-6)+5}}}
{{{y= - (5 /3)x+15}}}


find intersection point

{{{(3/5)x - 1/5=- (5 /3)x+15}}}...solving it, you get

{{{x = 114/17}}}

{{{y= - (5 /3)(114/17)+15}}}

 {{{y = 65/17}}}


convert to decimal:

{{{x = 6.706}}}, {{{y = 3.824}}}


height is distance between C and intersection point


{{{h=sqrt((6-6.706)^2+(5-3.824)^2)}}}

{{{h= 4sqrt(2/17)=1.372}}}


Area:

if  {{{h = 1.372}}} and {{{c=5.831}}}, the area will be

{{{A= (1/2)*5.831*1.372}}}
{{{A=4.000066}}}
{{{A = 4}}} square units


{{{ drawing( 600, 600, -10, 10, -10, 10,
circle(-3,-2,.12),circle(2,1,.12),circle(6,5,.12),
locate(-3,-2,A),locate(2,1,B),locate(6,5.5,C),locate(6.5,5,h),
green(line(-3,-2,2,1)),green(line(-3,-2,6,5)),green(line(6,5,2,1)),
circle(6.706,3.824,.12), blue(line(6,5,6.706,3.824)),
graph( 600, 600, -10, 10, -10, 10, (3/5)x - 1/5)) }}}