Question

ASAP PLZZZ Find the area of the shaded polygons:

Answer 1

Of course, the Pick's theorem is the way to solve this question, but consider:

Another approach is using topography:

Gauss's Area Calculation Formula:

`$A=(1)/(2) \sum_(i=1)^(n) (x_(i) \cdot y_(i+1)-y_(i) \cdot x_(i+1))$`

Taking the purple one:

We have 6 points. I will name them:

`A(0, 4);B(0, 0);C(1, 1);D(4, 0);E(4, 4);F(1, 2);`

`$D=\begin{vmatrix}0& 0& 1 & 4& 4 & 1 & 0\\ 4& 0 & 1 & 0& 4 & 2 & 4 \end{vmatrix}$`

`D=28-8=20`

`$A=(20)/(2) =10$`

Answer 2

Explanation:

You can use the Pick's theorem:

`A=i+(b)/(2)-1`

where

i - number of lattice points in the interior located in the polygon

b - number of lattice points on the boundary placed on the polygon's perimeter

`1.\\i= 5;\ b=12\\\\A=5+(12)/(2)-1=5+6-1=10\\\\2.\\i=3;\ b=4\\\\A=3+(4)/(2)-1=3+2-1=4\\\\3.\\i=5;\ b=10\\\\A=5+(10)/(2)-1=5+5-1=9`