Answer:
30 square units
Step-by-step explanation:
There are several ways to determine the area of the figure from its coordinates. A straightforward method is to plot the coordinates on a graph and decompose the resulting figure into simple shapes whose area formulas are known
Area from simpler shapes
A plot reveals the figure to be a parallelogram with a base length 6 and a height of 4, together with a triangle of base 6 and height 2.
The area of the parallelogram is given by the formula ...
A = bh
The area of the triangle is given by the formula ...
A = 1/2bh
In each case, 'b' represents the base of the figure, and 'h' represents its height.
Parallelogram area
A = bh = (6)(4) = 24
Triangle area
A = 1/2bh = 1/2(6)(2) = 6
Total area
The composite area will be the sum of the parallelogram and triangle areas:
A = 24 +6 = 30 . . . . square units
The area of the composite figure is 30 square units.
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Area from coordinates
The area can also be figured directly from the coordinates. The formula for that can be written ...
[tex]\displaystyle A=\dfrac{1}{2}\left|\sum_{k=1}^n (x_k(y_{k+1}-y_{k-1}))\right|\qquad\text{where $y_0=y_n$ and $y_{n+1}=y_1$}[/tex]
Using this formula, we find the area to be ...
A = 1/2|(-4(3-(-1)) +2(-1-3) +4(-3-3) +0(-1-(-1)) -2(3-(-3))|
= 1/2|-16 -8 -24 +0 -12| = 1/2|-60| = 30
The area of the figure is 30 square units.
