To find the area of the shape defined by the vertices \((-4, 0)\), \((0, 4)\), \((2, 2)\), \((-1, 2)\), and \((-1, 0)\), we can use the Shoelace theorem (or formula). This theorem provides a way to calculate the area of a polygon when the vertices are defined in the Cartesian plane.
Step 1: List the vertices
We'll first list the vertices in a systematic order around the shape. To determine the order, we can visualize or sketch the points:
- \((-4, 0)\)
- \((-1, 0)\)
- \((-1, 2)\)
- \((2, 2)\)
- \((0, 4)\)
Step 2: Repeat the first vertex at the end
To apply the Shoelace theorem, we repeat the first vertex at the end of the list:
\[ \begin{array}{c} (-4, 0) \ (-1, 0) \ (-1, 2) \ (2, 2) \ (0, 4) \ (-4, 0) \ \end{array} \]
Step 3: Apply the Shoelace theorem
The Shoelace theorem states that if you have vertices \((x_1, y_1)\), \((x_2, y_2)\), ..., \((x_n, y_n)\), the area \(A\) of the polygon can be computed using the formula:
\[ A = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right| \]
Where \( (x_{n+1}, y_{n+1}) \) is the same as \( (x_1, y_1) \).
Step 4: Set up the table for the calculations
We will now create two columns for our calculations:
\[ \begin{array}{c|c} x & y \ \hline -4 & 0 \ -1 & 0 \ -1 & 2 \ 2 & 2 \ 0 & 4 \ \end{array} \]
And the corresponding \(y\) values shifted down 1 row:
\[ \begin{array}{c|c} x & y \ \hline 0 & -4 \ 0 & -1 \ 2 & -1 \ 2 & 2 \ 4 & 0 \ \end{array} \]
Step 5: Compute the products
We compute the sums:
-
\(x_i y_{i+1}\):
- \((-4)(0) = 0\)
- \((-1)(0) = 0\)
- \((-1)(2) = -2\)
- \(2(4) = 8\)
- \(0(0) = 0\)
The total for \(x_i y_{i+1} = 0 + 0 - 2 + 8 + 0 = 6\)
-
\(y_i x_{i+1}\):
- \(0(-1) = 0\)
- \(0(-1) = 0\)
- \(2(2) = 4\)
- \(2(0) = 0\)
- \(4(-4) = -16\)
The total for \(y_i x_{i+1} = 0 + 0 + 4 + 0 - 16 = -12\)
Step 6: Substitute into the area formula
Now substituting both sums into the area formula:
\[ A = \frac{1}{2} \left| 6 - (-12) \right| = \frac{1}{2} \left| 6 + 12 \right| = \frac{1}{2} \times 18 = 9 \]
Thus, the area of the shape is \(\boxed{9}\).
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