9514 1404 393
Answer:
8.5 square units
Step-by-step explanation:
1. Pick's theorem tells you the area of a figure with vertices on a grid is ...
A = i +b/2 -1
where i is the number of interior grid points, and b is the number of grid points on the boundary.
Here, the interior grid points are easily counted. There are 5 of them.
The boundary grid points are only at the vertices of this figure, and there are 9 of them. Then the area is ...
A = 5 +9/2 -1 = 8.5 . . . . square units
__
2. Decomposition of the figure into a rectangle with triangles added and subtracted can be accomplished as follows.
base rectangle = the left 3 columns, 4 units high by 3 units wide = 12 u²
left-side triangle = base of 4 units by height of 1 unit = 1/2(4)(1) = 2 u² (subtracted)
top-side triangle = base of 3 units by height of 1 unit = 1/2(3)(1) = 1.5 u² (subtracted)
right-side triangle = base of 4 units by height of 1 unit = 1/2(4)(1) = 2 u² (added)
bottom-side triangle = base of 2 units by height of 2 units = 1/2(2)(2) = 2 u² (subtracted)
Then the total area is ...
12 u² -2 u² -1.5 u² +2 u² -2u² = (12 -2 -1.5) u² = 8.5 square units
