9514 1404 393
Answer:
12 square units
Step-by-step explanation:
There are several ways you can work this. Perhaps easiest to understand is dividing the figure into shapes we have formulas for. This figure can be divided into a triangle and a trapezoid.
The triangle has a base of 4 and a height of 2. Its area is ...
A = 1/2bh
A = 1/2(4)(2) = 4 . . . square units
The trapezoid has bases of 3 and 5, and a height of 2. Its area is ...
A = 1/2(b1 +b2)h
A = 1/2(3 +5)(2) = 8 . . . square units
Then the area of the shape is the sum of these:
Shape Area = 4 + 8 = 12 . . . square units
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<em>Alternate method</em>
Pick's theorem says the area of a polygon with integer vertex coordinates can be found using the formula ...
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.
The slopes of the boundary lines are ±1 or 0 or undefined, so they cross numerous grid points. There are 14 grid points on the boundary. The number of interior grid points is 1 in the center of the triangle, 3 in the middle of the trapezoid, and 2 on line segment FD in the attached figure, for a total of 6.
Using these numbers in the formula, we find the area to be ...
A = 6 + 14/2 -1 = 6 + 7 - 1 = 12
The area is 12 square units by Pick's theorem.
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<em>Alternate solution #2</em>
There are also formulas for finding the area directly from the grid point coordinates. For the coordinates at either end of successive boundary line segments, the value (x1)(y2) -(x2)(y1) is computed. Half the absolute value of the sum of these differences is the area. Taking the points in the given order and repeating the first coordinates at the end, these differences are -16, -8, 2, -2, 0, so the area is ...
(1/2)|-16-8+2-2+0| = 24/2 = 12 . . . square units
This sort of computation is easily implemented in a spreadsheet.