To find the perimeter of the polygon defined by the points A, B, C, D, and E, we first need to calculate the lengths of the sides connecting each pair of consecutive points.
The formula for the distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate each side length
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Distance AB: \[ AB = \sqrt{(-2 - (-1))^2 + (0 - (-2))^2} = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2 \]
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Distance BC: \[ BC = \sqrt{(0 - (-2))^2 + (2 - 0)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8 \]
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Distance CD: \[ CD = \sqrt{(2 - 0)^2 + (0 - 2)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8 \]
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Distance DE: \[ DE = \sqrt{(1 - 2)^2 + (-2 - 0)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2 \]
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Distance EA: \[ EA = \sqrt{(-1 - 1)^2 + (-2 - (-2))^2} = \sqrt{(-2)^2 + (0)^2} = \sqrt{4} = 2.0 \]
Step 2: Sum the lengths to get the perimeter
Now, we sum the calculated lengths:
\[ \text{Perimeter} = AB + BC + CD + DE + EA \]
Calculating it step-by-step:
\[ \text{Perimeter} \approx 2.2 + 2.8 + 2.8 + 2.2 + 2.0 \]
Calculating each step:
- \(2.2 + 2.8 = 5.0\)
- \(5.0 + 2.8 = 7.8\)
- \(7.8 + 2.2 = 10.0\)
- \(10.0 + 2.0 = 12.0\)
Final Answer
Thus, the perimeter of the polygon is approximately 12.0 units.