Answer: Area of triangle is 10 square units and perimeter of polygon is 16.87
Explanation:
We have formula for area of triangle when three vertices are given
![\frac{1}{2}[ (x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5B%20%28x_1%28y_2-y_3%29%2Bx_2%28y_3-y_1%29%2Bx_3%28y_1-y_2%29%5D)
we have values ![x_1=3 ,x_2=3 ,x_3=-2\\y_1=3 ,y_2=-1 ,y_3=-5](https://tex.z-dn.net/?f=x_1%3D3%20%2Cx_2%3D3%20%2Cx_3%3D-2%5C%5Cy_1%3D3%20%2Cy_2%3D-1%20%2Cy_3%3D-5)
substituting the values in the formula we will get
![\frac{1}{2} [3(-1+5)+3(-5-3)+(-2)(3+1)]=\frac{1}{2}[3(4)+3(-8)-2(4)] =\frac{1}{2} [12-24-8]=-10](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%5B3%28-1%2B5%29%2B3%28-5-3%29%2B%28-2%29%283%2B1%29%5D%3D%5Cfrac%7B1%7D%7B2%7D%5B3%284%29%2B3%28-8%29-2%284%29%5D%20%3D%5Cfrac%7B1%7D%7B2%7D%20%5B12-24-8%5D%3D-10)
Hence, area of triangle ABC is 10 square units we will neglect the negative sign since area can not be negative.
To find the perimeter we will add up all the sides
A(-2,-2) , B(-2,-4), C(1,-6),D(4,-6) ,E(3,-3)
First we need to find the sides for that we will be using distance formula below
![\sqrt{(y_2-y_1)^{2}+ (x_2-x_1)^{2}](https://tex.z-dn.net/?f=%5Csqrt%7B%28y_2-y_1%29%5E%7B2%7D%2B%20%28x_2-x_1%29%5E%7B2%7D)
Now to find AB ![x_1 =-2, x_2=-2\\y_1=-2,y_2=-4](https://tex.z-dn.net/?f=x_1%20%3D-2%2C%20x_2%3D-2%5C%5Cy_1%3D-2%2Cy_2%3D-4)
we will get ![\sqrt{(-4+2)^2+(-2+2)^2}= \sqrt{4}=2](https://tex.z-dn.net/?f=%5Csqrt%7B%28-4%2B2%29%5E2%2B%28-2%2B2%29%5E2%7D%3D%20%5Csqrt%7B4%7D%3D2)
Similarly, we will find BC ![x_1=-2,x_2=1\\y_1=-4,y_2=-6](https://tex.z-dn.net/?f=x_1%3D-2%2Cx_2%3D1%5C%5Cy_1%3D-4%2Cy_2%3D-6)
![\sqrt{(-6+4)^2+(1+2)^2}= \sqrt{13}](https://tex.z-dn.net/?f=%5Csqrt%7B%28-6%2B4%29%5E2%2B%281%2B2%29%5E2%7D%3D%20%5Csqrt%7B13%7D)
In same manner CD ![x_1=1,x_2=4\\y_1=-6,y_2=-6](https://tex.z-dn.net/?f=x_1%3D1%2Cx_2%3D4%5C%5Cy_1%3D-6%2Cy_2%3D-6)
![\sqrt{(-6+6)^2+(4-1)^2}= \sqrt{9}=3](https://tex.z-dn.net/?f=%5Csqrt%7B%28-6%2B6%29%5E2%2B%284-1%29%5E2%7D%3D%20%5Csqrt%7B9%7D%3D3)
For DE![x_1=4,x_2=3\\y_1=-6,y_2=-3](https://tex.z-dn.net/?f=x_1%3D4%2Cx_2%3D3%5C%5Cy_1%3D-6%2Cy_2%3D-3)
![\sqrt{(-3+6)^2+(3-4)^2}= \sqrt{10}](https://tex.z-dn.net/?f=%5Csqrt%7B%28-3%2B6%29%5E2%2B%283-4%29%5E2%7D%3D%20%5Csqrt%7B10%7D)
For AE![x_=3,x_2=-2\\y_1=-3,y_2=-2](https://tex.z-dn.net/?f=x_%3D3%2Cx_2%3D-2%5C%5Cy_1%3D-3%2Cy_2%3D-2)
![\sqrt{(-2+3)^2+(-2-3)^2}=\sqrt{26}](https://tex.z-dn.net/?f=%5Csqrt%7B%28-2%2B3%29%5E2%2B%28-2-3%29%5E2%7D%3D%5Csqrt%7B26%7D)
we will add all sides will give![2+3+\sqrt{13}+\sqrt{10}+ \sqrt{26} =2+3+3.6055+3.1622+5.0990=16.8667](https://tex.z-dn.net/?f=2%2B3%2B%5Csqrt%7B13%7D%2B%5Csqrt%7B10%7D%2B%20%5Csqrt%7B26%7D%20%3D2%2B3%2B3.6055%2B3.1622%2B5.0990%3D16.8667)
![\approx16.87](https://tex.z-dn.net/?f=%5Capprox16.87)