Question 1132017
 Find the Area and Perimeter for the coordinates below:

{{{K}}}({{{-2}}},{{{3}}}) 
{{{L}}}({{{-4}}},{{{1}}}) 
{{{M}}}({{{-3}}},{{{-3}}}) 
{{{N}}}({{{3}}},{{{-3}}}) 
{{{J}}}({{{4}}},{{{2}}}) 

first sketch it:


{{{drawing ( 600, 600, -10, 10, -10, 10,
line(-2,3,-4,1),line(-2,3,4,2),line(3,-3,4,2),line(-3,-3,-4,1),line(-3,-3,3,-3),
circle(-2,3,.12), locate(-2,3.5,K),
circle(-4,1,.12), locate(-4,1,L),
circle(-3,-3,.12), locate(-3,-3.5,M),
circle(3,-3,.12), locate(3,-3.5,N),
circle(4,2,.12), locate(4,2.5,J),

graph( 600, 600, -10, 10, -10, 10, 0)) }}} 



to find perimeter, first find distance between points

*[invoke formula_distance -3, -3, 3, -3] 
{{{MN=6}}}


*[invoke formula_distance -3, -3, -4, 1] 
{{{LM= 4.12}}}



*[invoke formula_distance -2, 3, -4, 1]

{{{KL=2.83}}}


*[invoke formula_distance -2, 3, 4, 2]

{{{KJ=6.0827}}}


*[invoke formula_distance 3, -3, 4, 2]

{{{JN=5.099}}}

so, perimeter is:
if
{{{MN=6}}}
{{{LM= 4.12}}}
{{{KL=2.83}}}
{{{KJ=6.0827}}}
{{{JN=5.099}}}

{{{P=6+4.12+2.83+6.0827+5.099}}}

{{{P=24.1317}}}


draw two diagonals and cut polygon into three triangles


{{{drawing ( 600, 600, -10, 10, -10, 10,
line(-2,3,-4,1),line(-2,3,4,2),line(3,-3,4,2),line(-3,-3,-4,1),line(-3,-3,3,-3),
circle(-2,3,.12), locate(-2,3.5,K),line(-2,3,-3,-3),line(-2,3,3,-3),
circle(-4,1,.12), locate(-4,1,L),
circle(-3,-3,.12), locate(-3,-3.5,M),
circle(3,-3,.12), locate(3,-3.5,N),
circle(4,2,.12), locate(4,2.5,J),

graph( 600, 600, -10, 10, -10, 10, 0)) }}} 


*[invoke formula_distance -3, -3, -2, 3] 

{{{MK=d[1]=6.083}}}


*[invoke formula_distance 3, -3, -2, 3] 

{{{KN=d[2]=7.81}}}


use Heron's Formula for the area of a triangle:

{{{Area=sqrt( p(p-a) (p-b) (p-c)) }}}where where{{{ p}}} is half the perimeter, or   
{{{(a+b+c )/2}}}

the area of triangle {{{MLK}}} is:
if {{{LM= 4.12}}}, {{{KL=2.83}}}, {{{MK=d[1]=6.083}}}

{{{p=(a+b+c )/2}}}

{{{p=(4.12+2.83+6.083 )/2}}}

{{{p=6.5165}}}


{{{A[1]=sqrt( 6.5165(6.5165-4.12) (6.5165-2.83) (6.5165-6.083)) }}}

{{{A[1]=sqrt( 6.5165(2.3965) (3.6865) (0.4335)) }}}

{{{A[1]=sqrt( 6.5165(3.8298)) }}}

{{{A[1]=sqrt( 24.9568917) }}}

{{{A[1]=4.99568731}}}


the area of triangle {{{MKN}}} is:

{{{MK=d[1]=6.083}}}
{{{MN=d[1]=6}}}
{{{KN=d[2]=7.81}}}

{{{p=(6.083+6+7.8 )/2}}}

{{{p=9.9415}}}

{{{A[2]=sqrt( 9.9415(9.9415-6.083) (9.9415-6) (9.9415-7.8)) }}}

{{{A[2]=sqrt( 9.9415(3.8585) (3.9415) (2.1415)) }}}

{{{A[2]=sqrt( 323.78) }}}

{{{A[2]=17.9939 }}}


the area of triangle {{{KNJ}}} is:
{{{KN=d[2]=7.81}}}
{{{KJ=6.0827}}}
{{{JN=5.099}}}

{{{p=(7.81+6.0827+5.099 )/2}}}

{{{p=9.49585}}}

{{{A[3]=sqrt( 9.49585(9.49585-7.81) (9.49585-6.0827) (9.49585-5.099)) }}}
{{{A[3]=sqrt( 9.49585(25.299733995378375)) }}}

{{{A[3]=sqrt( 240.24247906001374224375) }}}

{{{A[3]=15.499757387133959}}}

so the total area is:

{{{A=A[1]+A[2]+A[3]}}}

{{{A=4.99568731+17.9939+15.499757387133959}}}

{{{A}}}≈{{{38.49}}}