To calculate the area of a polygon defined by the vertices \((0,0)\), \((2,2)\), \((9,-4)\), and \((7,-6)\), we can use the shoelace formula (or surveyor's formula). Here’s how it works:
The shoelace formula states that for a polygon with vertices \((x_1, y_1)\), \((x_2, y_2)\), ..., \((x_n, y_n)\), the area \(A\) can be calculated 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 \((x_1, y_1)\) to close the polygon.
Let's arrange the vertices in order:
- \((0, 0)\)
- \((2, 2)\)
- \((9, -4)\)
- \((7, -6)\)
- Return to the first point \((0, 0)\)
Now, let's set up our sums:
\[ \begin{array}{c|c} x & y \ \hline 0 & 0 \ 2 & 2 \ 9 & -4 \ 7 & -6 \ 0 & 0 \ \end{array} \]
Now, compute \(x_i y_{i+1}\) and \(y_i x_{i+1}\):
- For \(i = 1\): \(0 \cdot 2 = 0\) and \(0 \cdot 2 = 0\)
- For \(i = 2\): \(2 \cdot -4 = -8\) and \(2 \cdot 9 = 18\)
- For \(i = 3\): \(9 \cdot -6 = -54\) and \(-4 \cdot 7 = -28\)
- For \(i = 4\): \(7 \cdot 0 = 0\) and \(-6 \cdot 0 = 0\)
Sum these results:
\[ \text{Sum of } (x_i y_{i+1}) = 0 + (-8) + (-54) + 0 = -62 \]
\[ \text{Sum of } (y_i x_{i+1}) = 0 + 18 + (-28) + 0 = -10 \]
Now plugging into the area formula:
\[ A = \frac{1}{2} \left| -62 - (-10) \right| = \frac{1}{2} \left| -62 + 10 \right| = \frac{1}{2} \left| -52 \right| = \frac{52}{2} = 26 \]
Thus, the area of the polygon is approximately 26 square units.
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