Question 1154678


first graph it

{{{A}}}({{{0}}},{{{0}}})
{{{B}}}({{{0}}},{{{4}}})
{{{C}}}({{{8}}},{{{8}}})
{{{D}}}({{{14}}},{{{0}}})
{{{E}}}({{{9}}},{{{-2}}})


{{{drawing( 600, 600, -10, 15, -10, 15,
circle(0,0,.14), locate(0.3,0.5,A),
circle(0,4,.13), locate(0.3,4.5,B),
circle(8,8,.13), locate(8.3,8.5,C),
circle(14,0,.13), locate(14.3,0.5,D),
circle(9,-2,.13), locate(9.3,-2.5,E),
green(line(0,4,8,8)),green(line(0,0,9,-2)),green(line(9,-2,14,0)),green(line(14,0,8,8)),
 graph( 600, 600, -10, 15, -10, 15, 0)) }}}

the length of the side {{{AB=4}}}

the length of the side {{{AE}}}:

*[invoke formula_distance 0, 0, 9, -2] 

=>{{{AE=9.2}}}


the length of the side {{{BC}}}:

*[invoke formula_distance 0, 4, 8, 8] 

=>{{{BC=8.9}}}

the length of the side {{{CD}}}:

*[invoke formula_distance 8, 8, 14, 0] 

=>{{{CD=10}}}

the length of the side {{{ED}}}:

*[invoke formula_distance 9, -2, 14, 0] 

=>{{{ED=5,4}}}

The area of any irregular quadrilateral can be calculated by dividing it into triangles.
Heron's Formula for the area of a triangle(Hero's Formula)
A method for calculating the area of a triangle when you know the lengths of all three sides.

Let {{{a}}},{{{b}}},{{{c}}} be the lengths of the sides of a triangle. 

The area is given by:
{{{A=sqrt(p (p-a) (p-b) (p-c) )  }}} where{{{ p}}} is half the perimeter, or   {{{(a+b+c )/2}}}

divide in triangles:

you have triangles ABC, ACD, and AED

the area of  triangles {{{ABC}}}:
sides:
 {{{AB=a=4}}}
{{{BC=b=8.9}}}

find side {{{AC=c}}}

*[invoke formula_distance 0, 0, 8, 8] 

{{{AC=c=11.3}}}

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

{{{p=(4+8.9+11.3 )/2}}}
{{{p=12.1}}}

{{{A=sqrt(12.1 (12.1-4) (12.1-8.9) (12.1-11.3) )  }}}

{{{A=sqrt(250.9056)}}}

{{{A=15.84}}}=>the area of  triangle {{{ABC}}}


find the area of  triangle {{{ACD}}}

{{{AC=c=11.3}}}
{{{AD=a=14}}}
{{{CD=10}}}

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

{{{p=(11.3+14+10 )/2}}}
{{{p=17.65}}}

{{{A=sqrt(17.65 (17.65-11.3) (17.65-14) (17.65-10) )  }}}

{{{A=sqrt(3129.48399375)}}}

{{{A=55.94}}}=>the area of  triangle {{{ACD}}}


find the area of  triangle {{{  AED}}}:

{{{AD=a=14}}}
{{{AE=b=9.2}}}
{{{ ED=c=5.4}}}

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

{{{p=(14+9.2+5.4 )/2}}}
{{{p=14.3}}}

{{{A=sqrt(14.3 (14.3-14) (14.3-9.2) (14.3-5.4) )  }}}

{{{A=sqrt(194.7231)}}}

{{{A=13.95}}}=>the area of  triangle {{{  AED}}}

the area of  triangles {{{ABC+ACD+AED=15.84+55.94+13.95=85.73}}}