Answer:
Area of Figure = 54 ![unit^2](https://tex.z-dn.net/?f=unit%5E2)
Step-by-step explanation:
Given Graph of a polygon with 7 side.
To find: Area of figure
First we write coordinates of all vertices from left bottom going in clock wise direction.
Say, Vertex A ( -3, -3 ) , Vertex B ( 4, -3 ) , Vertex C ( 4, 3 ) , Vertex D ( 3, 4 ) , Vertex E ( -2, 3 ) , Vertex F ( -4, 5 ) , Vertex G ( -5, 3 )
Now we divide the given figure in a trapezium ABCG and 2 triangle ΔCDE , ΔEFG
Area of Trapezium = ![\frac{1}{2}\times height\times (sum of parallel sides)](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ctimes%20height%5Ctimes%20%28sum%20of%20parallel%20sides%29)
From Figure, Height , BC = 6 unit and Parallel side, AB = 7 unit , CG = 9 units
⇒ Area of Trapezium =
= ![\frac{1}{2}\times 6\times(7+9)](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ctimes%206%5Ctimes%287%2B9%29)
= ![3\times(16)](https://tex.z-dn.net/?f=3%5Ctimes%2816%29)
= 48 ![unit^2](https://tex.z-dn.net/?f=unit%5E2)
Now for area of triangle we use Area of triangle using coordinates. Given by,
![Area\: of\: Triangle\: =\:\frac{1}{2}\times\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|](https://tex.z-dn.net/?f=Area%5C%3A%20of%5C%3A%20Triangle%5C%3A%20%3D%5C%3A%5Cfrac%7B1%7D%7B2%7D%5Ctimes%5Cleft%7Cx_1%28y_2-y_3%29%2Bx_2%28y_3-y_1%29%2Bx_3%28y_1-y_2%29%5Cright%7C)
⇒ Area of ΔCDE = ![\frac{1}{2}\times\left|4(4-3)+3(3-3)+(-2)(3-4)\right|](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ctimes%5Cleft%7C4%284-3%29%2B3%283-3%29%2B%28-2%29%283-4%29%5Cright%7C)
= ![\frac{1}{2}\times\left|4\times1+3\times0+(-2)\times(-1)\right|](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ctimes%5Cleft%7C4%5Ctimes1%2B3%5Ctimes0%2B%28-2%29%5Ctimes%28-1%29%5Cright%7C)
= ![\frac{1}{2}\times\left|4+0+2\right|](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ctimes%5Cleft%7C4%2B0%2B2%5Cright%7C)
= ![\frac{1}{2}\times6](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ctimes6)
= 3 ![unit^2](https://tex.z-dn.net/?f=unit%5E2)
Similarly,
⇒ Area of ΔEFG = ![\frac{1}{2}\times\left|(-5)(3-5)+(-2)(5-3)+(-4)(3-3)\right|](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ctimes%5Cleft%7C%28-5%29%283-5%29%2B%28-2%29%285-3%29%2B%28-4%29%283-3%29%5Cright%7C)
= ![\frac{1}{2}\times\left|(-5)\times(-2)+(-2)\times2+(-4)\times0\right|](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ctimes%5Cleft%7C%28-5%29%5Ctimes%28-2%29%2B%28-2%29%5Ctimes2%2B%28-4%29%5Ctimes0%5Cright%7C)
= ![\frac{1}{2}\times\left|10-4+0\right|](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ctimes%5Cleft%7C10-4%2B0%5Cright%7C)
= ![\frac{1}{2}\times6](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5Ctimes6)
= 3 ![unit^2](https://tex.z-dn.net/?f=unit%5E2)
∴ Area of Figure = area of trapezium ABCG + area of ΔCDE + area of ΔEFG
= 48 + 3 + 3
= 54 ![unit^2](https://tex.z-dn.net/?f=unit%5E2)
Therefore, Area of Figure = 54 ![unit^2](https://tex.z-dn.net/?f=unit%5E2)