It looks like there is a typo since the points mentioned include two points labeled as "B." Let's clarify the vertices of the polygon. I assume the vertices are A(-5, 4), B(1, 4), C(6, -4), and D(-5, -4).
We can calculate the area of the polygon using the shoelace formula, which is given as:
\[ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n}(x_iy_{i+1} - y_ix_{i+1}) \right| \]
where \((x_{n+1}, y_{n+1})\) is the same as \((x_1, y_1)\) to close the polygon.
The vertices in order are:
- A(-5, 4)
- B(1, 4)
- C(6, -4)
- D(-5, -4)
We will list them in the shoelace formula:
- A: \((-5, 4)\)
- B: \((1, 4)\)
- C: \((6, -4)\)
- D: \((-5, -4)\)
- Closing back to A: \((-5, 4)\)
Setting up the shoelace formula:
\[ \begin{array}{c|c} x & y \ \hline -5 & 4 \ 1 & 4 \ 6 & -4 \ -5 & -4 \ -5 & 4 \ \end{array} \]
Now we calculate the individual products:
- \(x_1 y_2 = -5 \cdot 4 = -20\)
- \(x_2 y_3 = 1 \cdot -4 = -4\)
- \(x_3 y_4 = 6 \cdot -4 = -24\)
- \(x_4 y_5 = -5 \cdot 4 = -20\)
Now we sum these products:
\[ \text{Sum of } (x_iy_{i+1}) = -20 - 4 - 24 - 20 = -68 \]
Now for the other direction:
- \(y_1 x_2 = 4 \cdot 1 = 4\)
- \(y_2 x_3 = 4 \cdot 6 = 24\)
- \(y_3 x_4 = -4 \cdot -5 = 20\)
- \(y_4 x_5 = -4 \cdot -5 = 20\)
Now we sum these products:
\[ \text{Sum of } (y_ix_{i+1}) = 4 + 24 + 20 + 20 = 68 \]
Now substitute these values into the shoelace formula:
\[ \text{Area} = \frac{1}{2} \left| -68 - 68 \right| = \frac{1}{2} \left| -136 \right| = \frac{1}{2} \cdot 136 = 68 \]
Thus, the area of polygon ABCD is 68 square units.
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