Question

ASAP PLZZZ Find the area of the shaded polygons:

Answer 1

Step-by-step explanation:

You can use the Pick's theorem:

$A=i+\dfrac{b}{2}-1$

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

$1.\\i= 5;\ b=12\\\\A=5+\dfrac{12}{2}-1=5+6-1=10\\\\2.\\i=3;\ b=4\\\\A=3+\dfrac{4}{2}-1=3+2-1=4\\\\3.\\i=5;\ b=10\\\\A=5+\dfrac{10}{2}-1=5+5-1=9$

Answer 2

Answer:

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:

$A=\frac{1}{2} \sum_{i=1}^{n} (x_{i} \cdot y_{i+1}-y_{i} \cdot x_{i+1})$

Taking the purple one:

We have 6 points. I will name them:

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

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

$D=28-8=20$

$A=\frac{20}{2} =10$